row_id int64 0 800k | example_id stringlengths 15 15 | problem_id stringlengths 36 36 | generator stringclasses 483
values | generator_label stringclasses 499
values | operation stringlengths 3 43 | grade_level stringclasses 5
values | difficulty int64 1 5 | problem stringlengths 5 402 | steps listlengths 2 207 | final_answer stringlengths 1 871 | text stringlengths 67 4.7k |
|---|---|---|---|---|---|---|---|---|---|---|---|
0 | train-000000000 | 19c4e64a-eeb3-41f0-a25f-162a1f2800c6 | SpecialRightTriangleGenerator | SpecialRightTriangleGenerator | special_right_triangle_45_from_leg | high | 4 | A 45-45-90 triangle has legs of length 103. Find the hypotenuse. Give an exact answer. | [
"TRI_SETUP|45-45-90 triangle, leg = 103|hypotenuse",
"THEOREM|45-45-90 ratios|leg : leg : hypotenuse = 1 : 1 : √2",
"REWRITE|hypotenuse = 103·√2 = 103√2",
"Z|103√2"
] | 103√2 | Problem:
A 45-45-90 triangle has legs of length 103. Find the hypotenuse. Give an exact answer.
Solution steps:
TRI_SETUP|45-45-90 triangle, leg = 103|hypotenuse
THEOREM|45-45-90 ratios|leg : leg : hypotenuse = 1 : 1 : √2
REWRITE|hypotenuse = 103·√2 = 103√2
Z|103√2
Final answer:
103√2 |
1 | train-000000001 | 155bed3d-e0cb-40ce-86cd-725fea97e9ef | MeanAbsoluteDeviationGenerator | MeanAbsoluteDeviationGenerator | mean_absolute_deviation | middle | 4 | Find the Mean Absolute Deviation (MAD) of the following data set: 37, 30, 50, 23, 46, 26, 33 | [
"STAT_SETUP|37, 30, 50, 23, 46, 26, 33",
"STAT_MEAN|245 / 7|35",
"STAT_DEVIATION|37|35|2",
"STAT_ABS_DEV|2|2",
"STAT_DEVIATION|30|35|-5",
"STAT_ABS_DEV|-5|5",
"STAT_DEVIATION|50|35|15",
"STAT_ABS_DEV|15|15",
"STAT_DEVIATION|23|35|-12",
"STAT_ABS_DEV|-12|12",
"STAT_DEVIATION|46|35|11",
"STAT_AB... | 8 | Problem:
Find the Mean Absolute Deviation (MAD) of the following data set: 37, 30, 50, 23, 46, 26, 33
Solution steps:
STAT_SETUP|37, 30, 50, 23, 46, 26, 33
STAT_MEAN|245 / 7|35
STAT_DEVIATION|37|35|2
STAT_ABS_DEV|2|2
STAT_DEVIATION|30|35|-5
STAT_ABS_DEV|-5|5
STAT_DEVIATION|50|35|15
STAT_ABS_DEV|15|15
STAT_DEVIATION|23... |
2 | train-000000002 | 2157cda9-0736-4b24-a70a-5cf1d930227a | AbacusAdditionGenerator | AbacusAdditionGenerator | abacus_addition | elementary | 2 | 249 + 1962 | [
"AB_SET|249",
"AB_ADD|+1000|249|1249",
"AB_ADD|+900|1249|2149",
"AB_ADD|+60|2149|2209",
"AB_ADD|+2|2209|2211",
"Z|2211"
] | 2211 | Problem:
249 + 1962
Solution steps:
AB_SET|249
AB_ADD|+1000|249|1249
AB_ADD|+900|1249|2149
AB_ADD|+60|2149|2209
AB_ADD|+2|2209|2211
Z|2211
Final answer:
2211 |
3 | train-000000003 | b8ccca44-986a-49f6-8257-c98554b842b0 | ClassifierMetricsGenerator | ClassifierMetricsGenerator | classifier_precision_recall_f1 | college | 2 | Given confusion matrix counts TP=38, FP=26, FN=7, TN=29, compute precision, recall, and F1 for the positive class. | [
"METRICS_SETUP|TP=38, FP=26, FN=7, TN=29",
"METRIC_FORMULA|precision=TP/(TP+FP)",
"A|38|26|64",
"D|38|64|19/32",
"METRIC_FORMULA|recall=TP/(TP+FN)",
"A|38|7|45",
"D|38|45|38/45",
"METRIC_FORMULA|F1=2PR/(P+R)",
"M|19/32|38/45|361/720",
"M|2|361/720|361/360",
"A|19/32|38/45|2071/1440",
"D|361/36... | precision=19/32; recall=38/45; F1=76/109 | Problem:
Given confusion matrix counts TP=38, FP=26, FN=7, TN=29, compute precision, recall, and F1 for the positive class.
Solution steps:
METRICS_SETUP|TP=38, FP=26, FN=7, TN=29
METRIC_FORMULA|precision=TP/(TP+FP)
A|38|26|64
D|38|64|19/32
METRIC_FORMULA|recall=TP/(TP+FN)
A|38|7|45
D|38|45|38/45
METRIC_FORMULA|F1=2PR... |
4 | train-000000004 | 42e4410c-e5c9-452d-89ec-f47171076bee | EquationFromTwoPointsGenerator | EquationFromTwoPointsGenerator | equation_from_two_points | high | 5 | Find the equation of the line passing through (8, -5) and (14, -13) | [
"EQ_2PT_SETUP|(8, -5)|(14, -13)",
"SLOPE_FORMULA|m = (y2 - y1) / (x2 - x1)",
"SLOPE_SUBST|m = (-13 - (-5)) / (14 - 8)",
"SLOPE_RESULT|-4/3",
"POINT_SLOPE_SETUP|y + 5 = -4/3(x - 8)",
"DIST|-4/3|(x - 8)|-4/3x + 32/3",
"COMB_CONST|32/3|-5|17/3",
"Z|y = -4/3x + 17/3"
] | y = -4/3x + 17/3 | Problem:
Find the equation of the line passing through (8, -5) and (14, -13)
Solution steps:
EQ_2PT_SETUP|(8, -5)|(14, -13)
SLOPE_FORMULA|m = (y2 - y1) / (x2 - x1)
SLOPE_SUBST|m = (-13 - (-5)) / (14 - 8)
SLOPE_RESULT|-4/3
POINT_SLOPE_SETUP|y + 5 = -4/3(x - 8)
DIST|-4/3|(x - 8)|-4/3x + 32/3
COMB_CONST|32/3|-5|17/3
Z|y ... |
5 | train-000000005 | de328aa6-38cc-4aab-9935-25a67feb918e | EllipticCurveFiniteFieldGenerator | EllipticCurveFiniteFieldGenerator | elliptic_curve_finite_field_add | graduate | 4 | Work over F_23 on E: y^2 = x^3 + 1x + 4; compute P + Q for P=(10,5) and Q=(0,2). | [
"EC_SETUP|p=23|a=1|b=4",
"EC_POINT_CHECK|P|y^2 mod p = 2|x^3+ax+b mod p = 2",
"EC_POINT_CHECK|Q|y^2 mod p = 4|x^3+ax+b mod p = 4",
"EC_SLOPE_FORMULA|P+Q|(y2-y1)/(x2-x1)",
"MOD_INVERSE|-10 mod 23|16",
"M|-3|16|-48",
"MOD_REDUCE|-48|mod 23|21",
"EC_SLOPE|P+Q|21",
"M|21|21|441",
"S|441|10|431",
"S|... | P+Q = (17,9) | Problem:
Work over F_23 on E: y^2 = x^3 + 1x + 4; compute P + Q for P=(10,5) and Q=(0,2).
Solution steps:
EC_SETUP|p=23|a=1|b=4
EC_POINT_CHECK|P|y^2 mod p = 2|x^3+ax+b mod p = 2
EC_POINT_CHECK|Q|y^2 mod p = 4|x^3+ax+b mod p = 4
EC_SLOPE_FORMULA|P+Q|(y2-y1)/(x2-x1)
MOD_INVERSE|-10 mod 23|16
M|-3|16|-48
MOD_REDUCE|-48|m... |
6 | train-000000006 | 14cda3eb-47d0-4d97-a3ca-85570f02f195 | SegmentPartitionGenerator | SegmentPartitionGenerator | segment_partition | high | 4 | Point P divides the segment from A(0, -1) to B(18, -19) in the ratio 5:1 (measured from A). Find P. | [
"SECTION_SETUP|A(0, -1), B(18, -19); ratio 5:1 from A|point P",
"SECTION_FORMULA|P = (x1 + m/(m+n)·(x2 - x1), y1 + m/(m+n)·(y2 - y1))",
"A|5|1|6",
"S|18|0|18",
"M|5|18|90",
"D|90|6|15",
"A|0|15|15",
"S|-19|-1|-18",
"M|5|-18|-90",
"D|-90|6|-15",
"A|-1|-15|-16",
"Z|(15, -16)"
] | (15, -16) | Problem:
Point P divides the segment from A(0, -1) to B(18, -19) in the ratio 5:1 (measured from A). Find P.
Solution steps:
SECTION_SETUP|A(0, -1), B(18, -19); ratio 5:1 from A|point P
SECTION_FORMULA|P = (x1 + m/(m+n)·(x2 - x1), y1 + m/(m+n)·(y2 - y1))
A|5|1|6
S|18|0|18
M|5|18|90
D|90|6|15
A|0|15|15
S|-19|-1|-18
M|5... |
7 | train-000000007 | 2339975b-6bba-4d6f-adaf-2c7a4ae9d141 | SegmentPartitionGenerator | SegmentPartitionGenerator | segment_partition | high | 4 | Point P divides the segment from A(6, -8) to B(14, -20) in the ratio 1:3 (measured from A). Find P. | [
"SECTION_SETUP|A(6, -8), B(14, -20); ratio 1:3 from A|point P",
"SECTION_FORMULA|P = (x1 + m/(m+n)·(x2 - x1), y1 + m/(m+n)·(y2 - y1))",
"A|1|3|4",
"S|14|6|8",
"M|1|8|8",
"D|8|4|2",
"A|6|2|8",
"S|-20|-8|-12",
"M|1|-12|-12",
"D|-12|4|-3",
"A|-8|-3|-11",
"Z|(8, -11)"
] | (8, -11) | Problem:
Point P divides the segment from A(6, -8) to B(14, -20) in the ratio 1:3 (measured from A). Find P.
Solution steps:
SECTION_SETUP|A(6, -8), B(14, -20); ratio 1:3 from A|point P
SECTION_FORMULA|P = (x1 + m/(m+n)·(x2 - x1), y1 + m/(m+n)·(y2 - y1))
A|1|3|4
S|14|6|8
M|1|8|8
D|8|4|2
A|6|2|8
S|-20|-8|-12
M|1|-12|-1... |
8 | train-000000008 | 89c9913f-c0e7-494a-90ae-9b3868f3789c | SpinHalfGenerator | SpinHalfGenerator | spin_half_apply_pauli | graduate | 4 | For spin state psi=[-5/13,-12/13] in the z basis, apply sigma_z. | [
"SPIN_SETUP|apply_pauli|operator=sigma_z|psi=[-5/13,-12/13]",
"PAULI_MATRIX|sigma_z|[[1,0],[0,-1]]",
"CX_M|1|-5/13|-5/13",
"CX_M|0|-12/13|0",
"CX_A|-5/13|0|-5/13",
"SPIN_COMPONENT|row=1|-5/13",
"CX_M|0|-5/13|0",
"CX_M|-1|-12/13|12/13",
"CX_A|0|12/13|12/13",
"SPIN_COMPONENT|row=2|12/13",
"APPLY_P... | sigma_z psi=[-5/13,12/13] | Problem:
For spin state psi=[-5/13,-12/13] in the z basis, apply sigma_z.
Solution steps:
SPIN_SETUP|apply_pauli|operator=sigma_z|psi=[-5/13,-12/13]
PAULI_MATRIX|sigma_z|[[1,0],[0,-1]]
CX_M|1|-5/13|-5/13
CX_M|0|-12/13|0
CX_A|-5/13|0|-5/13
SPIN_COMPONENT|row=1|-5/13
CX_M|0|-5/13|0
CX_M|-1|-12/13|12/13
CX_A|0|12/13|12/1... |
9 | train-000000009 | 80803027-ba89-4b44-b629-924e50f0e73a | RationalExprMultDivGenerator | RationalExprMultDivGenerator | rational_expr_multiply | high | 5 | Simplify: (x^2 + 4x - 5)/(x^2 - 7x - 8) · (x + 1)/(x - 1) | [
"POLY_SETUP|(x^2 + 4x - 5)/(x^2 - 7x - 8) · (x + 1)/(x - 1)",
"FACTOR_PAIR_GOAL|m·n = -5|m + n = 4",
"TRY|(-1, 5)|(-1)·5=-5, (-1)+5=4",
"ACCEPT|(-1, 5)|product -5 ✓, sum 4 ✓",
"REWRITE|((x - 1)(x + 5))/(x^2 - 7x - 8) · (x + 1)/(x - 1)",
"FACTOR_PAIR_GOAL|m·n = -8|m + n = -7",
"TRY|(1, -8)|1·(-8)=-8, 1+(... | (x + 5)/(x - 8) | Problem:
Simplify: (x^2 + 4x - 5)/(x^2 - 7x - 8) · (x + 1)/(x - 1)
Solution steps:
POLY_SETUP|(x^2 + 4x - 5)/(x^2 - 7x - 8) · (x + 1)/(x - 1)
FACTOR_PAIR_GOAL|m·n = -5|m + n = 4
TRY|(-1, 5)|(-1)·5=-5, (-1)+5=4
ACCEPT|(-1, 5)|product -5 ✓, sum 4 ✓
REWRITE|((x - 1)(x + 5))/(x^2 - 7x - 8) · (x + 1)/(x - 1)
FACTOR_PAIR_GO... |
10 | train-000000010 | cca17a1d-9809-4cde-9675-617c96b16d68 | ElectrostaticsGenerator | ElectrostaticsGenerator | electrostatics_potential_axis | college | 3 | In scaled units with k=1, three point charges are at distances r1=4 m, r2=9 m, r3=9 m from the origin with charges q1=1 C, q2=-1 C, q3=-7 C. Find the electric potential at the origin. | [
"ELEC_SETUP|potential_axis|q1=1, r1=4|q2=-1, r2=9",
"ELEC_SETUP|q3=-7, r3=9|k=1",
"ELEC_FORMULA|V=sum(q_i/r_i)",
"D|1|4|1/4",
"D|-1|9|-1/9",
"A|1/4|-1/9|5/36",
"D|-7|9|-7/9",
"A|5/36|-7/9|-23/36",
"Z|V=-23/36 V"
] | V=-23/36 V | Problem:
In scaled units with k=1, three point charges are at distances r1=4 m, r2=9 m, r3=9 m from the origin with charges q1=1 C, q2=-1 C, q3=-7 C. Find the electric potential at the origin.
Solution steps:
ELEC_SETUP|potential_axis|q1=1, r1=4|q2=-1, r2=9
ELEC_SETUP|q3=-7, r3=9|k=1
ELEC_FORMULA|V=sum(q_i/r_i)
D|1|4|... |
11 | train-000000011 | a888fe61-65c9-46b0-aa2c-c4404bc7da28 | FractionOpGenerator | FractionOpGenerator(-) | fraction_sub | elementary | 3 | 7/8 - 4/9 | [
"L|8|9|72",
"C|7/8|72|63/72",
"C|4/9|72|32/72",
"S|63|32|31",
"Z|31/72"
] | 31/72 | Problem:
7/8 - 4/9
Solution steps:
L|8|9|72
C|7/8|72|63/72
C|4/9|72|32/72
S|63|32|31
Z|31/72
Final answer:
31/72 |
12 | train-000000012 | a14909c4-05b9-4355-ac8b-31fd72231d0c | MSTGenerator | MSTGenerator | mst_prim | college | 4 | Find a minimum spanning tree for the weighted undirected graph with vertices A, B, C, D, E and edges AB=20, AC=10, BC=3, BD=5, BE=16, CE=11, DE=15 using Prim's algorithm starting at C. | [
"MST_SETUP|weighted undirected graph|vertices A, B, C, D, E",
"EDGE_WEIGHT|AB|20",
"EDGE_WEIGHT|AC|10",
"EDGE_WEIGHT|BC|3",
"EDGE_WEIGHT|BD|5",
"EDGE_WEIGHT|BE|16",
"EDGE_WEIGHT|CE|11",
"EDGE_WEIGHT|DE|15",
"PRIM_START|C",
"PRIM_CANDIDATES|visited C|BC=3, AC=10, CE=11",
"EDGE_CHOOSE|BC|weight 3|... | MST weight = 29; edges = AC, BC, BD, CE | Problem:
Find a minimum spanning tree for the weighted undirected graph with vertices A, B, C, D, E and edges AB=20, AC=10, BC=3, BD=5, BE=16, CE=11, DE=15 using Prim's algorithm starting at C.
Solution steps:
MST_SETUP|weighted undirected graph|vertices A, B, C, D, E
EDGE_WEIGHT|AB|20
EDGE_WEIGHT|AC|10
EDGE_WEIGHT|BC... |
13 | train-000000013 | 0e1f15a1-b1b2-4830-8d86-5743445f0787 | RouthHurwitzGenerator | RouthHurwitzGenerator | routh_hurwitz_cubic | graduate | 4 | Build the Routh-Hurwitz array for p(s)=s^3+12s^2+17s+57 and determine stability. | [
"ROUTH_SETUP|p(s)=s^3+12s^2+17s+57",
"ROUTH_ROW|s^3|1, 17",
"ROUTH_ROW|s^2|12, 57",
"M|12|17|204",
"S|204|57|147",
"D|147|12|49/4",
"ROUTH_ROW|s^1|49/4, 0",
"ROUTH_ROW|s^0|57",
"CHECK|first column=[1,12,49/4,57]|stable",
"Z|first column=[1,12,49/4,57]; stable"
] | first column=[1,12,49/4,57]; stable | Problem:
Build the Routh-Hurwitz array for p(s)=s^3+12s^2+17s+57 and determine stability.
Solution steps:
ROUTH_SETUP|p(s)=s^3+12s^2+17s+57
ROUTH_ROW|s^3|1, 17
ROUTH_ROW|s^2|12, 57
M|12|17|204
S|204|57|147
D|147|12|49/4
ROUTH_ROW|s^1|49/4, 0
ROUTH_ROW|s^0|57
CHECK|first column=[1,12,49/4,57]|stable
Z|first column=[1,1... |
14 | train-000000014 | 5950f97f-6228-4374-964a-a15e346abd68 | UndeterminedCoeffGenerator | UndeterminedCoeffGenerator | undetermined_coeff_exponential_forcing | college | 4 | Solve y'' + 3y' + 2y = -18e^x with y(0) = 2 and y'(0) = -11 by undetermined coefficients. | [
"ODE_SETUP|y'' + 3y' + 2y = -18e^x|y(0) = 2, y'(0) = -11",
"CHAR_EQ|assume y=e^(rx)|r^2 + 3r + 2 = 0",
"FACTOR|r^2 + 3r + 2|(r + 2)(r + 1) = 0",
"CHAR_ROOTS|r1 = -2, r2 = -1|complementary",
"HOM_SOL|y_h|y_h = C1e^(-2x) + C2e^(-x)",
"UC_GUESS|exponential forcing|y_p = Ae^x",
"APPLY_OPERATOR|L[Ae^x]|A(1 +... | y = 3e^(-2x) + 2e^(-x) - 3e^x | Problem:
Solve y'' + 3y' + 2y = -18e^x with y(0) = 2 and y'(0) = -11 by undetermined coefficients.
Solution steps:
ODE_SETUP|y'' + 3y' + 2y = -18e^x|y(0) = 2, y'(0) = -11
CHAR_EQ|assume y=e^(rx)|r^2 + 3r + 2 = 0
FACTOR|r^2 + 3r + 2|(r + 2)(r + 1) = 0
CHAR_ROOTS|r1 = -2, r2 = -1|complementary
HOM_SOL|y_h|y_h = C1e^(-2x... |
15 | train-000000015 | 8bc1d468-5d03-477c-9cf3-e6a4295cbfaa | JointDistributionGenerator | JointDistributionGenerator | joint_distribution_binary | college | 4 | For binary variables X,Y with P(X=0,Y=0)=170/1587, P(X=0,Y=1)=520/1587, P(X=1,Y=0)=520/1587, and P(X=1,Y=1)=377/1587, compute the marginals, P(Y=1 given X=1), independence, covariance, and correlation. | [
"JOINT_SETUP|X,Y in {0,1}|p00=170/1587, p01=520/1587|p10=520/1587, p11=377/1587",
"MARGINAL|P(X=0)=p00+p01",
"A|170/1587|520/1587|10/23",
"MARGINAL|P(X=1)=p10+p11",
"A|520/1587|377/1587|13/23",
"MARGINAL|P(Y=0)=p00+p10",
"A|170/1587|520/1587|10/23",
"MARGINAL|P(Y=1)=p01+p11",
"A|520/1587|377/1587|13... | P_X(0)=10/23, P_X(1)=13/23; P_Y(0)=10/23, P_Y(1)=13/23; P(Y=1 given X=1)=29/69; independent=no; covariance=-130/1587; correlation=-1/3 | Problem:
For binary variables X,Y with P(X=0,Y=0)=170/1587, P(X=0,Y=1)=520/1587, P(X=1,Y=0)=520/1587, and P(X=1,Y=1)=377/1587, compute the marginals, P(Y=1 given X=1), independence, covariance, and correlation.
Solution steps:
JOINT_SETUP|X,Y in {0,1}|p00=170/1587, p01=520/1587|p10=520/1587, p11=377/1587
MARGINAL|P(X=... |
16 | train-000000016 | 8d9cba99-4db9-4e8a-944f-4fe542d4cb2d | HamiltonianGenerator | HamiltonianGenerator | hamiltonian_pendulum | graduate | 4 | For a pendulum Hamiltonian with mass m=11, length L=4, and g=10, write H and Hamilton's equations. | [
"HAM_SETUP|pendulum|m=11, L=4|g=10, q=theta",
"E|4|2|16",
"M|11|16|176",
"M|11|10|110",
"M|110|4|440",
"HAMILTONIAN|H=p_theta^2/(2mL^2)+mgL*(1-cos(theta))",
"PARTIAL|dH/dp_theta|p_theta/(mL^2)",
"HAM_EQ|thetadot=dH/dp_theta|thetadot=p_theta/176",
"PARTIAL|dH/dtheta|mgL*sin(theta)",
"HAM_EQ|p_theta... | thetadot=p_theta/176; p_thetadot=-440*sin(theta); thetaddot=-(5/2)*sin(theta) | Problem:
For a pendulum Hamiltonian with mass m=11, length L=4, and g=10, write H and Hamilton's equations.
Solution steps:
HAM_SETUP|pendulum|m=11, L=4|g=10, q=theta
E|4|2|16
M|11|16|176
M|11|10|110
M|110|4|440
HAMILTONIAN|H=p_theta^2/(2mL^2)+mgL*(1-cos(theta))
PARTIAL|dH/dp_theta|p_theta/(mL^2)
HAM_EQ|thetadot=dH/dp... |
17 | train-000000017 | aae0ce56-1d28-4911-b787-1d0cbdc9ad81 | PlanckUnitsGenerator | PlanckUnitsGenerator | planck_units_length | graduate | 4 | Given hbar=144, G=9, and c=36, compute the Planck length sqrt(hbar*G/c^3). | [
"PLANCK_SETUP|length|hbar=144|G=9|c=36",
"M|144|9|1296",
"E|36|3|46656",
"D|1296|46656|1/36",
"ROOT|sqrt(1/36)|1/6",
"Z|l_P = 1/6"
] | l_P = 1/6 | Problem:
Given hbar=144, G=9, and c=36, compute the Planck length sqrt(hbar*G/c^3).
Solution steps:
PLANCK_SETUP|length|hbar=144|G=9|c=36
M|144|9|1296
E|36|3|46656
D|1296|46656|1/36
ROOT|sqrt(1/36)|1/6
Z|l_P = 1/6
Final answer:
l_P = 1/6 |
18 | train-000000018 | 1e95b87a-b41e-4ddb-861a-dcb9be48aa3e | AngleRelationshipsGenerator | AngleRelationshipsGenerator | vertical_angles | middle | 4 | Two vertical angles measure (4x + 28)° and (2x + 50)°. Find the value of x. | [
"ANGLE_SETUP|vertical|Vertical angles are equal",
"ANGLE_RELATION|4x + 28 = 2x + 50",
"ANGLE_SOLVE|2x = 22|x = 11",
"Z|11"
] | 11 | Problem:
Two vertical angles measure (4x + 28)° and (2x + 50)°. Find the value of x.
Solution steps:
ANGLE_SETUP|vertical|Vertical angles are equal
ANGLE_RELATION|4x + 28 = 2x + 50
ANGLE_SOLVE|2x = 22|x = 11
Z|11
Final answer:
11 |
19 | train-000000019 | 6ed790de-754f-470d-9e31-6c83c8b32267 | MutualInformationGenerator | MutualInformationGenerator | mutual_information_joint_entropy | college | 4 | For joint distribution P(X,Y) with rows X=0..2 and columns Y=0..3: rows=[[0,0,1/2,0];[1/8,1/8,0,0];[0,0,0,1/4]]. Find H(X,Y). | [
"MI_SETUP|rows=[[0,0,1/2,0];[1/8,1/8,0,0];[0,0,0,1/4]]|task=H(X,Y)",
"MARGINAL|P(X=0)=row0 sum",
"A|0|0|0",
"A|0|1/2|1/2",
"A|1/2|0|1/2",
"MARGINAL|P(X=1)=row1 sum",
"A|1/8|1/8|1/4",
"A|1/4|0|1/4",
"A|1/4|0|1/4",
"MARGINAL|P(X=2)=row2 sum",
"A|0|0|0",
"A|0|0|0",
"A|0|1/4|1/4",
"MARGINAL|P(... | H(X,Y)=7/4 bits | Problem:
For joint distribution P(X,Y) with rows X=0..2 and columns Y=0..3: rows=[[0,0,1/2,0];[1/8,1/8,0,0];[0,0,0,1/4]]. Find H(X,Y).
Solution steps:
MI_SETUP|rows=[[0,0,1/2,0];[1/8,1/8,0,0];[0,0,0,1/4]]|task=H(X,Y)
MARGINAL|P(X=0)=row0 sum
A|0|0|0
A|0|1/2|1/2
A|1/2|0|1/2
MARGINAL|P(X=1)=row1 sum
A|1/8|1/8|1/4
A|1/4|... |
20 | train-000000020 | b13f3da1-6d14-49dc-af3c-23b46a57755b | MeanGenerator | MeanGenerator | mean | middle | 3 | Find the mean of the following data set: 82, 39, 77, 52, 45 | [
"STAT_SETUP|82, 39, 77, 52, 45",
"STAT_SUM|82 + 39 + 77 + 52 + 45|295",
"STAT_COUNT|5",
"STAT_DIVIDE|295 / 5|59",
"Z|59"
] | 59 | Problem:
Find the mean of the following data set: 82, 39, 77, 52, 45
Solution steps:
STAT_SETUP|82, 39, 77, 52, 45
STAT_SUM|82 + 39 + 77 + 52 + 45|295
STAT_COUNT|5
STAT_DIVIDE|295 / 5|59
Z|59
Final answer:
59 |
21 | train-000000021 | 7b36b0eb-b48f-4d6d-808c-e3fbdc7c020a | TwoStepInequalityGenerator | TwoStepInequalityGenerator | two_step_inequality | middle | 4 | Solve the inequality: 3x + 2 < -22 | [
"INEQ_SETUP|3x + 2 < -22",
"INEQ_OP_BOTH|subtract|2|3x|-24",
"INEQ_SIMPLIFY|3x < -24",
"INEQ_OP_BOTH|divide|3|x|-8",
"INEQ_RESULT|x|<|-8",
"Z|x < -8"
] | x < -8 | Problem:
Solve the inequality: 3x + 2 < -22
Solution steps:
INEQ_SETUP|3x + 2 < -22
INEQ_OP_BOTH|subtract|2|3x|-24
INEQ_SIMPLIFY|3x < -24
INEQ_OP_BOTH|divide|3|x|-8
INEQ_RESULT|x|<|-8
Z|x < -8
Final answer:
x < -8 |
22 | train-000000022 | 7fb8dfd2-1f0b-4391-bf7d-e52fb5827577 | NewtonsLawsGenerator | NewtonsLawsGenerator | newtons_laws_incline_friction | college | 3 | A 30 kg block slides down an incline with supplied sin(theta)=7/25, cos(theta)=24/25, and friction coefficient mu=31/300. Use g=10 m/s^2 to find normal force, friction, and acceleration. | [
"NEWTON_SETUP|incline_friction|m=30, mu=31/300|g=10",
"NEWTON_SETUP|sin=7/25|cos=24/25",
"M|30|10|300",
"FORCE_COMPONENT|parallel=m*g*sin",
"M|300|7/25|84",
"FORCE_COMPONENT|normal=m*g*cos",
"M|300|24/25|288",
"FORCE_COMPONENT|friction=mu*N",
"M|31/300|288|744/25",
"FORCE_EQ|m*a=parallel-friction"... | N=288 N; friction=744/25 N; a=226/125 m/s^2 | Problem:
A 30 kg block slides down an incline with supplied sin(theta)=7/25, cos(theta)=24/25, and friction coefficient mu=31/300. Use g=10 m/s^2 to find normal force, friction, and acceleration.
Solution steps:
NEWTON_SETUP|incline_friction|m=30, mu=31/300|g=10
NEWTON_SETUP|sin=7/25|cos=24/25
M|30|10|300
FORCE_COMPON... |
23 | train-000000023 | 58505546-24e3-4657-a3c0-46adcd3116f7 | RelativisticEnergyGenerator | RelativisticEnergyGenerator | relativistic_energy_rest_energy | college | 4 | Using E=m*c^2, find the rest energy for mass m=19 kg and c=17 m/s. | [
"REL_ENERGY_SETUP|rest_energy|m=19|c=17",
"REL_ENERGY_FORMULA|E=m*c^2",
"E|17|2|289",
"M|19|289|5491",
"Z|E=5491 J"
] | E=5491 J | Problem:
Using E=m*c^2, find the rest energy for mass m=19 kg and c=17 m/s.
Solution steps:
REL_ENERGY_SETUP|rest_energy|m=19|c=17
REL_ENERGY_FORMULA|E=m*c^2
E|17|2|289
M|19|289|5491
Z|E=5491 J
Final answer:
E=5491 J |
24 | train-000000024 | 4886b126-4937-44e6-82cb-d635179c592a | TransferFunctionGenerator | TransferFunctionGenerator | transfer_function_block_feedback | graduate | 4 | Reduce a unity negative-feedback block diagram with G1=7/(s+5) and G2=12/(s+9). | [
"TF_SETUP|block_feedback|G1=7/(s+5), G2=12/(s+9)|H=1",
"SERIES|G=G1*G2",
"M|7|12|84",
"A|5|9|14",
"M|5|9|45",
"TRANSFER|G(s)=84/(s^2+14s+45)",
"FEEDBACK|T=G/(1+G)",
"A|45|84|129",
"TRANSFER|T(s)=84/(s^2+14s+129)",
"Z|T(s)=84/(s^2+14s+129)"
] | T(s)=84/(s^2+14s+129) | Problem:
Reduce a unity negative-feedback block diagram with G1=7/(s+5) and G2=12/(s+9).
Solution steps:
TF_SETUP|block_feedback|G1=7/(s+5), G2=12/(s+9)|H=1
SERIES|G=G1*G2
M|7|12|84
A|5|9|14
M|5|9|45
TRANSFER|G(s)=84/(s^2+14s+45)
FEEDBACK|T=G/(1+G)
A|45|84|129
TRANSFER|T(s)=84/(s^2+14s+129)
Z|T(s)=84/(s^2+14s+129)
Fi... |
25 | train-000000025 | 353acda5-0ff3-4db9-9764-f6719e8c60b8 | VolumeRectPrismGenerator | VolumeRectPrismGenerator | volume_rect_prism | elementary | 3 | Find volume of rectangular prism: L=15, W=5, H=13 | [
"M|15|5|75",
"M|75|13|975",
"VOLUME|975",
"Z|975"
] | 975 | Problem:
Find volume of rectangular prism: L=15, W=5, H=13
Solution steps:
M|15|5|75
M|75|13|975
VOLUME|975
Z|975
Final answer:
975 |
26 | train-000000026 | 07b5313f-8054-432f-b34d-04dae8c1974e | ModularArithmeticGenerator | ModularArithmeticGenerator | modular_arithmetic_isbn10 | middle | 4 | Find the ISBN-10 check digit for prefix 025579224. | [
"MOD_SETUP|ISBN-10 modulus 11|prefix 025579224",
"MOD_TERM|10 * 0|0",
"MOD_TERM|9 * 2|18",
"A|0|18|18",
"MOD_TERM|8 * 5|40",
"A|18|40|58",
"MOD_TERM|7 * 5|35",
"A|58|35|93",
"MOD_TERM|6 * 7|42",
"A|93|42|135",
"MOD_TERM|5 * 9|45",
"A|135|45|180",
"MOD_TERM|4 * 2|8",
"A|180|8|188",
"MOD_T... | 7 | Problem:
Find the ISBN-10 check digit for prefix 025579224.
Solution steps:
MOD_SETUP|ISBN-10 modulus 11|prefix 025579224
MOD_TERM|10 * 0|0
MOD_TERM|9 * 2|18
A|0|18|18
MOD_TERM|8 * 5|40
A|18|40|58
MOD_TERM|7 * 5|35
A|58|35|93
MOD_TERM|6 * 7|42
A|93|42|135
MOD_TERM|5 * 9|45
A|135|45|180
MOD_TERM|4 * 2|8
A|180|8|188
MOD... |
27 | train-000000027 | 8349383f-02b8-448a-8a76-2602b04d101d | MidpointGenerator | MidpointGenerator | midpoint_midpoint | high | 3 | Find the midpoint of the segment from (-9, -13) to (5, 1). | [
"MID_FORMULA|M = ((x1 + x2)/2, (y1 + y2)/2)",
"A|-9|5|-4",
"D|-4|2|-2",
"A|-13|1|-12",
"D|-12|2|-6",
"Z|(-2, -6)"
] | (-2, -6) | Problem:
Find the midpoint of the segment from (-9, -13) to (5, 1).
Solution steps:
MID_FORMULA|M = ((x1 + x2)/2, (y1 + y2)/2)
A|-9|5|-4
D|-4|2|-2
A|-13|1|-12
D|-12|2|-6
Z|(-2, -6)
Final answer:
(-2, -6) |
28 | train-000000028 | b3f4dcab-ed6b-45f9-aaff-e0e10696b996 | SolidRevolutionGenerator | SolidRevolutionGenerator | volume_disk | high | 5 | Find the volume when the region under y = 36x on [0, 43] is rotated about the x-axis. Give an exact answer in terms of π. | [
"VOLUME_SETUP|region under y = 36x on [0, 43], rotated about the x-axis|disk method",
"VOL_FORMULA|V = π ∫ [f(x)]^2 dx",
"REWRITE|[36x]^2 = 1296x^2",
"INTEG_RULE|power rule|∫ x^2 dx = x^3/3",
"ANTIDERIV|1296x^2|(432)x^3",
"EVAL|F(43)|34347024",
"EVAL|F(0)|0",
"S|34347024|0|34347024",
"Z|34347024π"
] | 34347024π | Problem:
Find the volume when the region under y = 36x on [0, 43] is rotated about the x-axis. Give an exact answer in terms of π.
Solution steps:
VOLUME_SETUP|region under y = 36x on [0, 43], rotated about the x-axis|disk method
VOL_FORMULA|V = π ∫ [f(x)]^2 dx
REWRITE|[36x]^2 = 1296x^2
INTEG_RULE|power rule|∫ x^2 dx ... |
29 | train-000000029 | 9e36443e-7485-4892-833a-d2af3c0cbc6e | StandardFormConversionGenerator | StandardFormConversionGenerator | slope_intercept_to_standard | high | 4 | Convert to Standard Form: y = 2/5x - 6/2 | [
"EQ_SETUP|y = 2/5x - 6/2",
"GOAL|Convert to Standard Form (Ax + By = C, integers)",
"EQ_OP_NOTE|multiply|10|to clear fractions",
"REWRITE|10y = 4x - 30",
"MOVE_TERM|4x|to left side|-4x + 10y = -30",
"EQ_OP_NOTE|multiply|-1|to make A positive",
"Z|4x - 10y = 30"
] | 4x - 10y = 30 | Problem:
Convert to Standard Form: y = 2/5x - 6/2
Solution steps:
EQ_SETUP|y = 2/5x - 6/2
GOAL|Convert to Standard Form (Ax + By = C, integers)
EQ_OP_NOTE|multiply|10|to clear fractions
REWRITE|10y = 4x - 30
MOVE_TERM|4x|to left side|-4x + 10y = -30
EQ_OP_NOTE|multiply|-1|to make A positive
Z|4x - 10y = 30
Final answ... |
30 | train-000000030 | d264d849-cd87-4dd6-a5bb-313db0259aab | DopplerGenerator | DopplerGenerator | doppler_relativistic_approach | college | 3 | A light source approaches with beta=35/37 and emits f=365 Hz. Use the relativistic Doppler formula to find f_obs. | [
"DOPPLER_SETUP|relativistic_approach|f=365|beta=35/37",
"DOPPLER_FORMULA|f_obs=f*sqrt((1+beta)/(1-beta))",
"E|6|2|36",
"S|36|1|35",
"A|36|1|37",
"A|1|35/37|72/37",
"S|1|35/37|2/37",
"D|72/37|2/37|36",
"ROOT|sqrt(36)|6",
"M|365|6|2190",
"Z|f_obs=2190 Hz"
] | f_obs=2190 Hz | Problem:
A light source approaches with beta=35/37 and emits f=365 Hz. Use the relativistic Doppler formula to find f_obs.
Solution steps:
DOPPLER_SETUP|relativistic_approach|f=365|beta=35/37
DOPPLER_FORMULA|f_obs=f*sqrt((1+beta)/(1-beta))
E|6|2|36
S|36|1|35
A|36|1|37
A|1|35/37|72/37
S|1|35/37|2/37
D|72/37|2/37|36
ROO... |
31 | train-000000031 | 40200ea2-284e-415d-aebf-36954cd434bb | PhysicsFormulaGenerator | PhysicsFormulaGenerator | physics_formula_power_seconds | middle | 4 | During a test, a machine does 3200 joules of work in 20 seconds. Compute the power. | [
"PHYS_SETUP|W = 3200 joules|t = 20 seconds|power",
"PHYS_FORMULA|P = W/t",
"D|3200|20|160",
"UNIT_ATTACH|160|watts|160 watts",
"Z|160 watts"
] | 160 watts | Problem:
During a test, a machine does 3200 joules of work in 20 seconds. Compute the power.
Solution steps:
PHYS_SETUP|W = 3200 joules|t = 20 seconds|power
PHYS_FORMULA|P = W/t
D|3200|20|160
UNIT_ATTACH|160|watts|160 watts
Z|160 watts
Final answer:
160 watts |
32 | train-000000032 | d56686f8-a25f-4e1f-8262-9335fa113c45 | AbsoluteValueInequalityGenerator | AbsoluteValueInequalityGenerator | absolute_value_ineq | high | 5 | Solve: |2x + 8| ≥ 9 | [
"ABS_INEQ_SETUP|abs(2x + 8) ≥ 9",
"ABS_INEQ_SPLIT|OR case|2x + 8 ≥ 9 or 2x + 8 ≤ -9",
"ABS_INEQ_PART|Part 1|2x + 8 ≥ 9 -> x ≥ 1/2",
"ABS_INEQ_PART|Part 2|2x + 8 ≤ -9 -> x ≤ -17/2",
"Z|x ≥ 1/2 or x ≤ -17/2"
] | x ≥ 1/2 or x ≤ -17/2 | Problem:
Solve: |2x + 8| ≥ 9
Solution steps:
ABS_INEQ_SETUP|abs(2x + 8) ≥ 9
ABS_INEQ_SPLIT|OR case|2x + 8 ≥ 9 or 2x + 8 ≤ -9
ABS_INEQ_PART|Part 1|2x + 8 ≥ 9 -> x ≥ 1/2
ABS_INEQ_PART|Part 2|2x + 8 ≤ -9 -> x ≤ -17/2
Z|x ≥ 1/2 or x ≤ -17/2
Final answer:
x ≥ 1/2 or x ≤ -17/2 |
33 | train-000000033 | cc83db76-7e37-4f86-afbb-77b014687c9c | TwoStepInequalityGenerator | TwoStepInequalityGenerator | two_step_inequality | middle | 4 | Solve the inequality: 4x - 7 < -19 | [
"INEQ_SETUP|4x - 7 < -19",
"INEQ_OP_BOTH|add|7|4x|-12",
"INEQ_SIMPLIFY|4x < -12",
"INEQ_OP_BOTH|divide|4|x|-3",
"INEQ_RESULT|x|<|-3",
"Z|x < -3"
] | x < -3 | Problem:
Solve the inequality: 4x - 7 < -19
Solution steps:
INEQ_SETUP|4x - 7 < -19
INEQ_OP_BOTH|add|7|4x|-12
INEQ_SIMPLIFY|4x < -12
INEQ_OP_BOTH|divide|4|x|-3
INEQ_RESULT|x|<|-3
Z|x < -3
Final answer:
x < -3 |
34 | train-000000034 | c91e4dac-a113-4526-9c39-f63a7bfc6d17 | PercentWordProblemGenerator | PercentWordProblemGenerator | percent_word_problem | elementary | 3 | A quantity of 86 grows by 25%. What is the result? | [
"PERCENT_TO_DEC|25%|0.25",
"M|86|0.25|21.5",
"A|86|21.5|107.5",
"Z|107.5"
] | 107.5 | Problem:
A quantity of 86 grows by 25%. What is the result?
Solution steps:
PERCENT_TO_DEC|25%|0.25
M|86|0.25|21.5
A|86|21.5|107.5
Z|107.5
Final answer:
107.5 |
35 | train-000000035 | 060972a7-886c-4c46-8cc4-50e8a07962a4 | PCAGenerator | PCAGenerator | pca_2d_projection | graduate | 4 | For points [(1,5), (-3,5), (-1,9), (-1,1)], use population covariance (divide by n) to compute 2D PCA and project each centered point onto the principal component. | [
"PCA_SETUP|points=[(1,5), (-3,5), (-1,9), (-1,1)]|population covariance",
"A|0|1|1",
"A|1|-3|-2",
"A|-2|-1|-3",
"A|-3|-1|-4",
"D|-4|4|-1",
"A|0|5|5",
"A|5|5|10",
"A|10|9|19",
"A|19|1|20",
"D|20|4|5",
"S|1|-1|2",
"S|5|5|0",
"CENTER|P1|(2,0)",
"S|-3|-1|-2",
"S|5|5|0",
"CENTER|P2|(-2,0)... | cov=[[2,0], [0,8]]; pc=(0,1); scores=0,0,4,-4 | Problem:
For points [(1,5), (-3,5), (-1,9), (-1,1)], use population covariance (divide by n) to compute 2D PCA and project each centered point onto the principal component.
Solution steps:
PCA_SETUP|points=[(1,5), (-3,5), (-1,9), (-1,1)]|population covariance
A|0|1|1
A|1|-3|-2
A|-2|-1|-3
A|-3|-1|-4
D|-4|4|-1
A|0|5|5
A... |
36 | train-000000036 | 7f65f3cd-32b8-47a2-b1f6-2189e13ce0a5 | ORFormulaGenerator | ORFormulaGenerator | or_formula_eoq | graduate | 3 | For EOQ with annual demand D=782, order cost S=23, and holding cost H=17, compute EOQ and annual relevant costs. | [
"OR_SETUP|EOQ|D=782|S=23, H=17",
"FORMULA|Q=sqrt(2DS/H)",
"M|2|782|1564",
"M|1564|23|35972",
"D|35972|17|2116",
"ROOT|2116|46",
"FORMULA|annual ordering cost=(D/Q)S",
"D|782|46|17",
"M|17|23|391",
"FORMULA|annual holding cost=(Q/2)H",
"D|46|2|23",
"M|23|17|391",
"A|391|391|782",
"CHECK|ord... | EOQ=46; ordering cost=391; holding cost=391; total cost=782 | Problem:
For EOQ with annual demand D=782, order cost S=23, and holding cost H=17, compute EOQ and annual relevant costs.
Solution steps:
OR_SETUP|EOQ|D=782|S=23, H=17
FORMULA|Q=sqrt(2DS/H)
M|2|782|1564
M|1564|23|35972
D|35972|17|2116
ROOT|2116|46
FORMULA|annual ordering cost=(D/Q)S
D|782|46|17
M|17|23|391
FORMULA|ann... |
37 | train-000000037 | 02181a99-9c72-4bdd-8fe8-98535455f8ee | SoftmaxGradientGenerator | SoftmaxGradientGenerator | softmax_gradient_exact | graduate | 4 | Given logits z=(3*ln(4),3*ln(6),3*ln(8)) with temperature T=3 and target class 1, compute the temperature-scaled softmax, log-softmax, cross-entropy, and gradient p-y. | [
"SOFTMAX_SETUP|z=(3*ln(4),3*ln(6),3*ln(8))|T=3|target=1",
"TEMP_SCALE|z1/T|ln(4)",
"SOFTMAX_EXP|1|4",
"TEMP_SCALE|z2/T|ln(6)",
"SOFTMAX_EXP|2|6",
"TEMP_SCALE|z3/T|ln(8)",
"SOFTMAX_EXP|3|8",
"A|0|4|4",
"A|4|6|10",
"A|10|8|18",
"D|4|18|2/9",
"SOFTMAX_PROB|1|2/9",
"LOG_SOFTMAX|1|ln(2/9)",
"D|... | p=(2/9,1/3,4/9); log_softmax=(ln(2/9),ln(1/3),ln(4/9)); CE=ln(9/2); grad=(-7/9,1/3,4/9) | Problem:
Given logits z=(3*ln(4),3*ln(6),3*ln(8)) with temperature T=3 and target class 1, compute the temperature-scaled softmax, log-softmax, cross-entropy, and gradient p-y.
Solution steps:
SOFTMAX_SETUP|z=(3*ln(4),3*ln(6),3*ln(8))|T=3|target=1
TEMP_SCALE|z1/T|ln(4)
SOFTMAX_EXP|1|4
TEMP_SCALE|z2/T|ln(6)
SOFTMAX_EXP... |
38 | train-000000038 | c784a02a-fda2-4b92-904a-7086b48cd54c | EntropyGenerator | EntropyGenerator | entropy_distribution_entropy | college | 3 | Compute Shannon entropy in bits for distribution P=[1/8,1/8,1/8,1/8,1/8,1/4,1/16,1/16]. | [
"ENTROPY_SETUP|P=[1/8,1/8,1/8,1/8,1/8,1/4,1/16,1/16]|H=-sum p log2(p)",
"LOG2|1/8|-3",
"M|1/8|3|3/8",
"A|0|3/8|3/8",
"LOG2|1/8|-3",
"M|1/8|3|3/8",
"A|3/8|3/8|3/4",
"LOG2|1/8|-3",
"M|1/8|3|3/8",
"A|3/4|3/8|9/8",
"LOG2|1/8|-3",
"M|1/8|3|3/8",
"A|9/8|3/8|3/2",
"LOG2|1/8|-3",
"M|1/8|3|3/8",
... | H=23/8 bits | Problem:
Compute Shannon entropy in bits for distribution P=[1/8,1/8,1/8,1/8,1/8,1/4,1/16,1/16].
Solution steps:
ENTROPY_SETUP|P=[1/8,1/8,1/8,1/8,1/8,1/4,1/16,1/16]|H=-sum p log2(p)
LOG2|1/8|-3
M|1/8|3|3/8
A|0|3/8|3/8
LOG2|1/8|-3
M|1/8|3|3/8
A|3/8|3/8|3/4
LOG2|1/8|-3
M|1/8|3|3/8
A|3/4|3/8|9/8
LOG2|1/8|-3
M|1/8|3|3/8
A... |
39 | train-000000039 | e9dbff93-0fdd-4dbb-a227-cf348ea48681 | KernelRidgeGenerator | KernelRidgeGenerator | kernel_ridge_linear_2point | graduate | 4 | For kernel ridge regression with linear kernel K(x,z)=xz, training data [(-5,0), (5,-2)], lambda=2, and x*=-6, solve (K + lambda I) alpha = y and predict f(x*). | [
"KRR_SETUP|kernel=linear|data=[(-5,0), (5,-2)]|lambda=2,x*=-6",
"M|-5|-5|25",
"KERNEL_VALUE|1,1|25",
"M|-5|5|-25",
"KERNEL_VALUE|1,2|-25",
"M|5|-5|-25",
"KERNEL_VALUE|2,1|-25",
"M|5|5|25",
"KERNEL_VALUE|2,2|25",
"RIDGE_ENTRY|K|[[25,-25], [-25,25]]",
"A|25|2|27",
"RIDGE_ENTRY|1,1|27",
"RIDGE_... | alpha=(-25/52,-27/52); prediction=15/13 | Problem:
For kernel ridge regression with linear kernel K(x,z)=xz, training data [(-5,0), (5,-2)], lambda=2, and x*=-6, solve (K + lambda I) alpha = y and predict f(x*).
Solution steps:
KRR_SETUP|kernel=linear|data=[(-5,0), (5,-2)]|lambda=2,x*=-6
M|-5|-5|25
KERNEL_VALUE|1,1|25
M|-5|5|-25
KERNEL_VALUE|1,2|-25
M|5|-5|-2... |
40 | train-000000040 | 2d3589ff-af9b-4fbe-aa1e-90e5dffb1e32 | FiveNumberSummaryGenerator | FiveNumberSummaryGenerator | five_number_summary_outliers | middle | 3 | Using the 1.5×IQR rule, find the outliers in the data set: 21, 14, 17, 37, 32, 7, 13, 32, 15, 37, 38, 12, 10, 31, 19. | [
"SORT|21,14,17,37,32,7,13,32,15,37,38,12,10,31,19|7,10,12,13,14,15,17,19,21,31,32,32,37,37,38",
"MEDIAN_PICK|19|19",
"QUARTILE|Q1|7,10,12,13,14,15,17|13",
"QUARTILE|Q3|21,31,32,32,37,37,38|32",
"S|32|13|19",
"M|1.5|19|28.5",
"S|13|28.5|-15.5",
"A|32|28.5|60.5",
"CHECK|1.5×IQR rule|outside [-15.5, 60... | none | Problem:
Using the 1.5×IQR rule, find the outliers in the data set: 21, 14, 17, 37, 32, 7, 13, 32, 15, 37, 38, 12, 10, 31, 19.
Solution steps:
SORT|21,14,17,37,32,7,13,32,15,37,38,12,10,31,19|7,10,12,13,14,15,17,19,21,31,32,32,37,37,38
MEDIAN_PICK|19|19
QUARTILE|Q1|7,10,12,13,14,15,17|13
QUARTILE|Q3|21,31,32,32,37,37,... |
41 | train-000000041 | 54dfd232-eb80-4f82-8302-6c10e6ef152b | BackpropGenerator | BackpropGenerator | backprop_relu_step | graduate | 5 | For a 2-2-1 ReLU network with x=(3,3), y=1, eta=1/5, W1=[[1,-1], [-2,-1]], b1=(2,-2), v=(1,-2), c=2. Do one SGD backprop step using L=1/2*(y_hat-y)^2. | [
"BACKPROP_SETUP|x=(3,3)|y=1|eta=1/5",
"PARAMS|W1=[[1,-1], [-2,-1]]|b1=(2,-2)|v=(1,-2), c=2",
"M|1|3|3",
"M|-1|3|-3",
"A|3|-3|0",
"A|0|2|2",
"HIDDEN_PRE|h1|z=2",
"RELU|z=2|h=2|deriv=1",
"M|-2|3|-6",
"M|-1|3|-3",
"A|-6|-3|-9",
"A|-9|-2|-11",
"HIDDEN_PRE|h2|z=-11",
"RELU|z=-11|h=0|deriv=0",
... | y_hat=4; loss=9/2; W1_new=[[-4/5,-14/5], [-2,-1]]; b1_new=(7/5,-2); v_new=(-1/5,-2); c_new=7/5 | Problem:
For a 2-2-1 ReLU network with x=(3,3), y=1, eta=1/5, W1=[[1,-1], [-2,-1]], b1=(2,-2), v=(1,-2), c=2. Do one SGD backprop step using L=1/2*(y_hat-y)^2.
Solution steps:
BACKPROP_SETUP|x=(3,3)|y=1|eta=1/5
PARAMS|W1=[[1,-1], [-2,-1]]|b1=(2,-2)|v=(1,-2), c=2
M|1|3|3
M|-1|3|-3
A|3|-3|0
A|0|2|2
HIDDEN_PRE|h1|z=2
REL... |
42 | train-000000042 | eebfce25-c489-4a55-ad76-bbd36a5e9955 | PascalTriangleGenerator | PascalTriangleGenerator | pascal_triangle_ncr | middle | 3 | Use Pascal's triangle to find 10C7 (row 0 is 1). | [
"PASCAL_SETUP|10C7",
"PASCAL_ROW|0|1",
"PASCAL_ROW|1|1, 1",
"A|1|1|2",
"PASCAL_ROW|2|1, 2, 1",
"A|1|2|3",
"A|2|1|3",
"PASCAL_ROW|3|1, 3, 3, 1",
"A|1|3|4",
"A|3|3|6",
"A|3|1|4",
"PASCAL_ROW|4|1, 4, 6, 4, 1",
"A|1|4|5",
"A|4|6|10",
"A|6|4|10",
"A|4|1|5",
"PASCAL_ROW|5|1, 5, 10, 10, 5, ... | 120 | Problem:
Use Pascal's triangle to find 10C7 (row 0 is 1).
Solution steps:
PASCAL_SETUP|10C7
PASCAL_ROW|0|1
PASCAL_ROW|1|1, 1
A|1|1|2
PASCAL_ROW|2|1, 2, 1
A|1|2|3
A|2|1|3
PASCAL_ROW|3|1, 3, 3, 1
A|1|3|4
A|3|3|6
A|3|1|4
PASCAL_ROW|4|1, 4, 6, 4, 1
A|1|4|5
A|4|6|10
A|6|4|10
A|4|1|5
PASCAL_ROW|5|1, 5, 10, 10, 5, 1
A|1|5|6
... |
43 | train-000000043 | 9a07b6a7-9c2c-4bdd-83d0-aa57273b53d2 | ErrorSpottingGenerator | ErrorSpottingGenerator | error_spotting_equation | middle | 4 | The worked solution below contains exactly one arithmetic mistake. Check it line by line, identify the wrong line, and redo the work from that point.
Problem: Solve for x: 7x + 4 = 46
1) EQ_SETUP|7x + 4 = 46
2) EQ_OP_BOTH|subtract|4|7x|42
3) EQ_SIMPLIFY|7x = 42
4) EQ_OP_BOTH|divide|7|x|4
5) EQ_RESULT|x|4
6) Z|4 | [
"VERIFY|1|ok",
"VERIFY|2|ok",
"VERIFY|3|ok",
"FLAG|4|42 ÷ 7 = 6, not 4",
"EQ_OP_BOTH|divide|7|x|6",
"EQ_RESULT|x|6",
"CHECK|substitute|7·6 + 4 = 46|46",
"Z|step 4; 6"
] | step 4; 6 | Problem:
The worked solution below contains exactly one arithmetic mistake. Check it line by line, identify the wrong line, and redo the work from that point.
Problem: Solve for x: 7x + 4 = 46
1) EQ_SETUP|7x + 4 = 46
2) EQ_OP_BOTH|subtract|4|7x|42
3) EQ_SIMPLIFY|7x = 42
4) EQ_OP_BOTH|divide|7|x|4
5) EQ_RESULT|x|4
6) Z|... |
44 | train-000000044 | 0aae6ddc-26b6-4cdd-9eb8-70e92cb26335 | LUDecompositionGenerator | LUDecompositionGenerator | lu_decomposition | college | 3 | Find an LU decomposition A = L*U with unit lower triangular L for A = [[2, -2, 4], [-8, 10, -17], [-2, -4, -2]]. | [
"LU_SETUP|A = [[2, -2, 4], [-8, 10, -17], [-2, -4, -2]]|unit lower L",
"LU_ENTRY|u11|a11 = 2|2",
"LU_ENTRY|u12|a12 = -2|-2",
"LU_ENTRY|u13|a13 = 4|4",
"LU_ENTRY|l21|(-8)/2|-4",
"LU_ENTRY|l31|(-2)/2|-1",
"LU_ENTRY|u22|10 - (-4)*(-2)|2",
"LU_ENTRY|u23|(-17) - (-4)*4|-1",
"LU_ENTRY|l32|((-4) - (-1)*(-2... | L=[[1, 0, 0], [-4, 1, 0], [-1, -3, 1]]; U=[[2, -2, 4], [0, 2, -1], [0, 0, -1]] | Problem:
Find an LU decomposition A = L*U with unit lower triangular L for A = [[2, -2, 4], [-8, 10, -17], [-2, -4, -2]].
Solution steps:
LU_SETUP|A = [[2, -2, 4], [-8, 10, -17], [-2, -4, -2]]|unit lower L
LU_ENTRY|u11|a11 = 2|2
LU_ENTRY|u12|a12 = -2|-2
LU_ENTRY|u13|a13 = 4|4
LU_ENTRY|l21|(-8)/2|-4
LU_ENTRY|l31|(-2)/2... |
45 | train-000000045 | 0b5a01f6-5b91-49ed-b7bd-4c987272d6a1 | LPCornerGenerator | LPCornerGenerator | lp_corner_point | college | 3 | Use the corner-point method to maximize z = 3x + 9y subject to 0 <= x <= 22, 0 <= y <= 10, and x + y <= 28. | [
"LP_CORNER_SETUP|max z=3x+9y|0<=x<=22, 0<=y<=10|x+y<=28",
"VERTEX_SOLVE|x=0|y=0",
"VERTEX|(0,0)",
"VERTEX_SOLVE|x=22|y=0",
"VERTEX|(22,0)",
"VERTEX_SOLVE|x=22|x+y=28",
"S|28|22|6",
"VERTEX|(22,6)",
"VERTEX_SOLVE|y=10|x+y=28",
"S|28|10|18",
"VERTEX|(18,10)",
"VERTEX_SOLVE|x=0|y=10",
"VERTEX|(... | optimal vertex=(18,10), max z=144 | Problem:
Use the corner-point method to maximize z = 3x + 9y subject to 0 <= x <= 22, 0 <= y <= 10, and x + y <= 28.
Solution steps:
LP_CORNER_SETUP|max z=3x+9y|0<=x<=22, 0<=y<=10|x+y<=28
VERTEX_SOLVE|x=0|y=0
VERTEX|(0,0)
VERTEX_SOLVE|x=22|y=0
VERTEX|(22,0)
VERTEX_SOLVE|x=22|x+y=28
S|28|22|6
VERTEX|(22,6)
VERTEX_SOLVE... |
46 | train-000000046 | 8ac3d875-4752-4d6b-a19f-3c4334460efe | OrderOfOperationsGenerator | OrderOfOperationsGenerator(integers) | order_of_operations | elementary | 3 | Compute 7 + 9 / 3 | [
"D|9|3|3",
"REWRITE|7 + 3",
"A|7|3|10",
"Z|10"
] | 10 | Problem:
Compute 7 + 9 / 3
Solution steps:
D|9|3|3
REWRITE|7 + 3
A|7|3|10
Z|10
Final answer:
10 |
47 | train-000000047 | cb330c4c-d772-4ff4-af72-a147f8079ef0 | OptimizationGenerator | OptimizationGenerator | optimization_product | high | 5 | Two positive numbers x and y satisfy x + y = 2121. Maximize x·y². | [
"OPT_SETUP|x + y = 2121, x, y > 0|maximize P = x·y^2",
"SUBST|x|2121 - y|P = (2121 - y)y^2 = 2121y^2 - y^3",
"POWER_RULE|2121y^2 - y^3|4242y - 3y^2",
"REWRITE|P' = y(4242 - 3y)",
"ZERO_PRODUCT|y(4242 - 3y) = 0|y = 0 or y = 1414",
"REJECT|y = 0|gives zero product",
"ACCEPT|y = 1414|the interior critical ... | x = 707, y = 1414; maximum product 1413572972 | Problem:
Two positive numbers x and y satisfy x + y = 2121. Maximize x·y².
Solution steps:
OPT_SETUP|x + y = 2121, x, y > 0|maximize P = x·y^2
SUBST|x|2121 - y|P = (2121 - y)y^2 = 2121y^2 - y^3
POWER_RULE|2121y^2 - y^3|4242y - 3y^2
REWRITE|P' = y(4242 - 3y)
ZERO_PRODUCT|y(4242 - 3y) = 0|y = 0 or y = 1414
REJECT|y = 0|... |
48 | train-000000048 | 12b83609-f510-49f9-a0a8-6bf3f9fbc865 | SpinHalfGenerator | SpinHalfGenerator | spin_half_apply_pauli | graduate | 4 | For spin state psi=[3/5,-4/5] in the z basis, apply sigma_y. | [
"SPIN_SETUP|apply_pauli|operator=sigma_y|psi=[3/5,-4/5]",
"PAULI_MATRIX|sigma_y|[[0,-i],[i,0]]",
"CX_M|0|3/5|0",
"CX_M|-i|-4/5|4i/5",
"CX_A|0|4i/5|4i/5",
"SPIN_COMPONENT|row=1|4i/5",
"CX_M|i|3/5|3i/5",
"CX_M|0|-4/5|0",
"CX_A|3i/5|0|3i/5",
"SPIN_COMPONENT|row=2|3i/5",
"APPLY_PAULI|sigma_y psi|[4i... | sigma_y psi=[4i/5,3i/5] | Problem:
For spin state psi=[3/5,-4/5] in the z basis, apply sigma_y.
Solution steps:
SPIN_SETUP|apply_pauli|operator=sigma_y|psi=[3/5,-4/5]
PAULI_MATRIX|sigma_y|[[0,-i],[i,0]]
CX_M|0|3/5|0
CX_M|-i|-4/5|4i/5
CX_A|0|4i/5|4i/5
SPIN_COMPONENT|row=1|4i/5
CX_M|i|3/5|3i/5
CX_M|0|-4/5|0
CX_A|3i/5|0|3i/5
SPIN_COMPONENT|row=2|... |
49 | train-000000049 | 350e7062-3a4c-4f01-82a8-97a81ef972b4 | EulerCircuitGenerator | EulerCircuitGenerator | euler_path | college | 3 | Use Hierholzer's algorithm to find an Euler path in the connected undirected graph with vertices A, B, C, D, E, F and edges AC, AD, AE, AF, BD, CE, CF, DE, DF, EF. Start at B; when extending the current walk, choose the alphabetically first unused neighbor. | [
"GRAPH_SETUP|connected undirected graph|vertices A, B, C, D, E, F",
"EDGE_LIST|AC, AD, AE, AF, BD, CE, CF, DE, DF, EF",
"CHECK|connected|yes",
"EDGE_COUNT|unused|10",
"ADJ_LIST|A|C, D, E, F",
"DEGREE|A|4",
"ADJ_LIST|B|D",
"DEGREE|B|1",
"ADJ_LIST|C|A, E, F",
"DEGREE|C|3",
"ADJ_LIST|D|A, B, E, F",... | Euler path = B-D-A-C-E-A-F-D-E-F-C | Problem:
Use Hierholzer's algorithm to find an Euler path in the connected undirected graph with vertices A, B, C, D, E, F and edges AC, AD, AE, AF, BD, CE, CF, DE, DF, EF. Start at B; when extending the current walk, choose the alphabetically first unused neighbor.
Solution steps:
GRAPH_SETUP|connected undirected gra... |
50 | train-000000050 | 0f6e0599-4a22-4099-9d50-49ab0e84156e | ComplexNumberOpsGenerator | ComplexNumberOpsGenerator | complex_add | high | 4 | Add: (4 + 3i) + (4 - 8i). | [
"CX_SETUP|(4 + 3i) + (4 - 8i)|add",
"REWRITE|(4 + 4) + (3 + (-8))i",
"A|4|4|8",
"A|3|-8|-5",
"Z|8 - 5i"
] | 8 - 5i | Problem:
Add: (4 + 3i) + (4 - 8i).
Solution steps:
CX_SETUP|(4 + 3i) + (4 - 8i)|add
REWRITE|(4 + 4) + (3 + (-8))i
A|4|4|8
A|3|-8|-5
Z|8 - 5i
Final answer:
8 - 5i |
51 | train-000000051 | a8c37119-0866-4ed3-9916-4021e5cb173a | PiecewiseEvaluationGenerator | PiecewiseEvaluationGenerator | piecewise_evaluation | high | 4 | Given h(x) = { 2x - 2 if x < -1; x^2 if -1 <= x <= 2; -1 if x > 2 }, find h(-1). | [
"FUNC_SETUP|h(x) = { 2x - 2 if x < -1; x^2 if -1 <= x <= 2; -1 if x > 2 }|h(-1)",
"BRANCH_TEST|-1 < -1|no",
"BRANCH_TEST|-1 <= -1 <= 2|yes",
"SUBST|x|-1|(-1)^2",
"E|(-1)|2|1",
"Z|1"
] | 1 | Problem:
Given h(x) = { 2x - 2 if x < -1; x^2 if -1 <= x <= 2; -1 if x > 2 }, find h(-1).
Solution steps:
FUNC_SETUP|h(x) = { 2x - 2 if x < -1; x^2 if -1 <= x <= 2; -1 if x > 2 }|h(-1)
BRANCH_TEST|-1 < -1|no
BRANCH_TEST|-1 <= -1 <= 2|yes
SUBST|x|-1|(-1)^2
E|(-1)|2|1
Z|1
Final answer:
1 |
52 | train-000000052 | 7414110c-5195-4510-9d15-0b06bb1682be | HawkingGenerator | HawkingGenerator | hawking_entropy | graduate | 4 | Given k_B=7, c=6, A=31, hbar=2, and G=6, compute the Bekenstein-Hawking entropy S_BH=k_B*c^3*A/(4*hbar*G). | [
"HAWKING_SETUP|entropy|S_BH=k_B*c^3*A/(4*hbar*G)|k_B=7,c=6,A=31,hbar=2,G=6",
"E|6|3|216",
"M|7|216|1512",
"M|1512|31|46872",
"M|4|2|8",
"M|8|6|48",
"D|46872|48|1953/2",
"Z|S_BH = 1953/2"
] | S_BH = 1953/2 | Problem:
Given k_B=7, c=6, A=31, hbar=2, and G=6, compute the Bekenstein-Hawking entropy S_BH=k_B*c^3*A/(4*hbar*G).
Solution steps:
HAWKING_SETUP|entropy|S_BH=k_B*c^3*A/(4*hbar*G)|k_B=7,c=6,A=31,hbar=2,G=6
E|6|3|216
M|7|216|1512
M|1512|31|46872
M|4|2|8
M|8|6|48
D|46872|48|1953/2
Z|S_BH = 1953/2
Final answer:
S_BH = 1... |
53 | train-000000053 | a55948f3-e40a-4ad0-8bd3-c210e24cfc1f | OneStepEquationGenerator | OneStepEquationGenerator | one_step_equation_mult | middle | 3 | Solve for x: 8x = -8 | [
"EQ_SETUP|8x = -8",
"EQ_OP_BOTH|divide|8|x|-1",
"EQ_RESULT|x|-1",
"Z|-1"
] | -1 | Problem:
Solve for x: 8x = -8
Solution steps:
EQ_SETUP|8x = -8
EQ_OP_BOTH|divide|8|x|-1
EQ_RESULT|x|-1
Z|-1
Final answer:
-1 |
54 | train-000000054 | d49db1c5-3c79-4768-af69-226e334f84dc | BaseConversionGenerator | BaseConversionGenerator | base_conversion_binary_to_decimal | middle | 3 | Convert binary 1100_2 to decimal. | [
"BASE_SETUP|1100_2|decimal",
"PLACE_VALUE|0 * 2^0|0",
"PLACE_VALUE|0 * 2^1|0",
"A|0|0|0",
"PLACE_VALUE|1 * 2^2|4",
"A|0|4|4",
"PLACE_VALUE|1 * 2^3|8",
"A|4|8|12",
"Z|12"
] | 12 | Problem:
Convert binary 1100_2 to decimal.
Solution steps:
BASE_SETUP|1100_2|decimal
PLACE_VALUE|0 * 2^0|0
PLACE_VALUE|0 * 2^1|0
A|0|0|0
PLACE_VALUE|1 * 2^2|4
A|0|4|4
PLACE_VALUE|1 * 2^3|8
A|4|8|12
Z|12
Final answer:
12 |
55 | train-000000055 | 47f42ed0-7c6b-4211-8e2c-855258337a83 | ECDSAGenerator | ECDSAGenerator | ecdsa_sign_verify | graduate | 5 | On E: y^2=x^3+2x+2 over F_17 with G=(5,1) of order n=19, private key d=8, message hash z=6, and nonce k=15. Compute the ECDSA signature and verify it. | [
"ECDSA_SETUP|E/F_17, G=(5,1), n=19|d=8|z=6|k=15",
"ECDSA_PUBLIC|Q=dG=(13,7)",
"ECDSA_NONCE|kG=(3,16)|r=3",
"MOD_INVERSE|15 mod 19|14",
"ECDSA_SIGN|s=k^-1(z+rd) mod n|s=2",
"MOD_INVERSE|2 mod 19|10",
"ECDSA_VERIFY|u1=3|u2=11",
"EC_SCALAR|u1G=(10,6)|u2Q=(0,11)",
"EC_ADD|(3,16)",
"CHECK|x(X) mod n = ... | signature = (r=3, s=2); verification = valid | Problem:
On E: y^2=x^3+2x+2 over F_17 with G=(5,1) of order n=19, private key d=8, message hash z=6, and nonce k=15. Compute the ECDSA signature and verify it.
Solution steps:
ECDSA_SETUP|E/F_17, G=(5,1), n=19|d=8|z=6|k=15
ECDSA_PUBLIC|Q=dG=(13,7)
ECDSA_NONCE|kG=(3,16)|r=3
MOD_INVERSE|15 mod 19|14
ECDSA_SIGN|s=k^-1(z+... |
56 | train-000000056 | 6d07d0d2-ef4f-48e6-9ffc-a9b17125d617 | QuantizationGenerator | QuantizationGenerator | quantization_int8_affine | college | 3 | Quantize tensor x=(3/50,141/100,7/50) with int8 scale=1/20 and zero_point=9 using q=round(x/scale)+zero_point, then dequantize and compute sum absolute round-trip error. | [
"QUANT_SETUP|x=(3/50,141/100,7/50)|scale=1/20|zero_point=9",
"D|3/50|1/20|6/5",
"A|6/5|9|51/5",
"ROUND|51/5|10",
"QUANT_VALUE|1|10",
"S|10|9|1",
"M|1|1/20|1/20",
"DEQUANT_VALUE|1|1/20",
"S|3/50|1/20|1/100",
"ABS_ERROR|1|1/100",
"D|141/100|1/20|141/5",
"A|141/5|9|186/5",
"ROUND|186/5|37",
"... | q=(10,37,12); dequant=(1/20,7/5,3/20); sum_abs_error=3/100 | Problem:
Quantize tensor x=(3/50,141/100,7/50) with int8 scale=1/20 and zero_point=9 using q=round(x/scale)+zero_point, then dequantize and compute sum absolute round-trip error.
Solution steps:
QUANT_SETUP|x=(3/50,141/100,7/50)|scale=1/20|zero_point=9
D|3/50|1/20|6/5
A|6/5|9|51/5
ROUND|51/5|10
QUANT_VALUE|1|10
S|10|9... |
57 | train-000000057 | d004cff3-96a4-4ce8-a22f-71abf28de1b3 | GeometryAreaPerimeterGenerator | GeometryAreaPerimeterGenerator | geometry_parallelogram | elementary | 3 | Parallelogram base 15, side 16, height 11: find perimeter and area | [
"A|15|16|31",
"M|2|31|62",
"PERIM|62",
"M|15|11|165",
"AREA|165",
"Z|Perimeter=62, Area=165"
] | Perimeter=62, Area=165 | Problem:
Parallelogram base 15, side 16, height 11: find perimeter and area
Solution steps:
A|15|16|31
M|2|31|62
PERIM|62
M|15|11|165
AREA|165
Z|Perimeter=62, Area=165
Final answer:
Perimeter=62, Area=165 |
58 | train-000000058 | 843a7f9a-b79f-416c-8192-94294ccd7dc1 | MSTGenerator | MSTGenerator | mst_prim | college | 4 | Find a minimum spanning tree for the weighted undirected graph with vertices A, B, C, D, E and edges AB=11, AD=24, AE=15, BC=13, BE=12 using Prim's algorithm starting at E. | [
"MST_SETUP|weighted undirected graph|vertices A, B, C, D, E",
"EDGE_WEIGHT|AB|11",
"EDGE_WEIGHT|AD|24",
"EDGE_WEIGHT|AE|15",
"EDGE_WEIGHT|BC|13",
"EDGE_WEIGHT|BE|12",
"PRIM_START|E",
"PRIM_CANDIDATES|visited E|BE=12, AE=15",
"EDGE_CHOOSE|BE|weight 12|add B",
"A|0|12|12",
"MST_ADD|BE|total 12",
... | MST weight = 60; edges = AB, AD, BC, BE | Problem:
Find a minimum spanning tree for the weighted undirected graph with vertices A, B, C, D, E and edges AB=11, AD=24, AE=15, BC=13, BE=12 using Prim's algorithm starting at E.
Solution steps:
MST_SETUP|weighted undirected graph|vertices A, B, C, D, E
EDGE_WEIGHT|AB|11
EDGE_WEIGHT|AD|24
EDGE_WEIGHT|AE|15
EDGE_WEI... |
59 | train-000000059 | 17799b7f-57d2-4f7b-a190-cd93743bdef3 | GraphTraversalGenerator | GraphTraversalGenerator | graph_traversal_dfs | college | 3 | Run DFS from D on the undirected graph with vertices A, B, C, D, E, F and edges AB, AD, AE, BC, CE, CF. Visit neighbors in alphabetical order. | [
"GRAPH_SETUP|undirected graph|vertices A, B, C, D, E, F",
"ADJ_LIST|A|B, D, E",
"ADJ_LIST|B|A, C",
"ADJ_LIST|C|B, E, F",
"ADJ_LIST|D|A",
"ADJ_LIST|E|A, C",
"ADJ_LIST|F|C",
"VISIT|D|D",
"DFS_EDGE|D->A|tree",
"VISIT|A|D, A",
"DFS_EDGE|A->B|tree",
"VISIT|B|D, A, B",
"DFS_EDGE|B->A|skip visited"... | DFS order = D, A, B, C, E, F | Problem:
Run DFS from D on the undirected graph with vertices A, B, C, D, E, F and edges AB, AD, AE, BC, CE, CF. Visit neighbors in alphabetical order.
Solution steps:
GRAPH_SETUP|undirected graph|vertices A, B, C, D, E, F
ADJ_LIST|A|B, D, E
ADJ_LIST|B|A, C
ADJ_LIST|C|B, E, F
ADJ_LIST|D|A
ADJ_LIST|E|A, C
ADJ_LIST|F|C
... |
60 | train-000000060 | c06cc6ec-2fb4-41b9-9dac-4613c527196a | ProjectorGenerator | ProjectorGenerator | projector_plus_projector | graduate | 3 | Verify that P=[[400/10201,1980/10201],[1980/10201,9801/10201]] is a projector. | [
"PROJECTOR_SETUP|v=(20/101, 99/101)|P=vv^T=[[400/10201,1980/10201],[1980/10201,9801/10201]]",
"MATRIX_MULT|row1 dot col1|400/10201*400/10201+1980/10201*1980/10201|400/10201",
"MATRIX_MULT|row1 dot col2|400/10201*1980/10201+1980/10201*9801/10201|1980/10201",
"MATRIX_MULT|row2 dot col2|1980/10201*1980/10201+980... | projector yes; P^2 = P | Problem:
Verify that P=[[400/10201,1980/10201],[1980/10201,9801/10201]] is a projector.
Solution steps:
PROJECTOR_SETUP|v=(20/101, 99/101)|P=vv^T=[[400/10201,1980/10201],[1980/10201,9801/10201]]
MATRIX_MULT|row1 dot col1|400/10201*400/10201+1980/10201*1980/10201|400/10201
MATRIX_MULT|row1 dot col2|400/10201*1980/10201... |
61 | train-000000061 | 7ce3cd38-f24d-42b7-a9be-88b777d0b293 | ProportionalRelationshipGenerator | ProportionalRelationshipGenerator | proportional_relationship | middle | 3 | If 1 is to 6, what is 2 proportional to? | [
"PROP_SETUP|1/6 = 2/x",
"M|6|2|12",
"EQ_SETUP|x = 12/1",
"D|12|1|12",
"Z|12"
] | 12 | Problem:
If 1 is to 6, what is 2 proportional to?
Solution steps:
PROP_SETUP|1/6 = 2/x
M|6|2|12
EQ_SETUP|x = 12/1
D|12|1|12
Z|12
Final answer:
12 |
62 | train-000000062 | 80e5f4e9-7730-4cc4-a024-f74a1bf49bc3 | SecondOrderODEGenerator | SecondOrderODEGenerator | second_order_ode_complex_roots | college | 3 | Solve y'' - 2y' + 5y = 0 with y(0) = 2 and y'(0) = -6. | [
"ODE_SETUP|y'' - 2y' + 5y = 0|y(0) = 2, y'(0) = -6",
"CHAR_EQ|assume y=e^(rx)|r^2 - 2r + 5 = 0",
"CHAR_ROOTS|r = 1 ± 2i|complex conjugates",
"SOL_FORM|y = e^x(C1 cos(2x) + C2 sin(2x))",
"SUBST|x=0|C1 = 2",
"DERIV_FORM|y'(0)|C1 + 2C2",
"SUBST|x=0|C1 + 2C2 = -6",
"M|1|2|2",
"S|-6|2|-8",
"D|-8|2|-4",... | y = e^x(2cos(2x) - 4sin(2x)) | Problem:
Solve y'' - 2y' + 5y = 0 with y(0) = 2 and y'(0) = -6.
Solution steps:
ODE_SETUP|y'' - 2y' + 5y = 0|y(0) = 2, y'(0) = -6
CHAR_EQ|assume y=e^(rx)|r^2 - 2r + 5 = 0
CHAR_ROOTS|r = 1 ± 2i|complex conjugates
SOL_FORM|y = e^x(C1 cos(2x) + C2 sin(2x))
SUBST|x=0|C1 = 2
DERIV_FORM|y'(0)|C1 + 2C2
SUBST|x=0|C1 + 2C2 = -... |
63 | train-000000063 | 7d871ca5-7e9b-479c-95a6-479ef5ab9dde | OrderOfOperationsGenerator | OrderOfOperationsGenerator(mixed_numbers) | order_of_operations_mixed_numbers | elementary | 4 | Compute 6 1/2 + 6 1/2 * 4 | [
"MIX_IMPROPER|6 1/2|13/2",
"M|13/2|4|52/2",
"REWRITE|6 1/2 + 52/2",
"MIX_IMPROPER|6 1/2|13/2",
"A|13/2|52/2|65/2",
"IMPROPER_TO_MIX|65/2|32 1/2",
"Z|32 1/2"
] | 32 1/2 | Problem:
Compute 6 1/2 + 6 1/2 * 4
Solution steps:
MIX_IMPROPER|6 1/2|13/2
M|13/2|4|52/2
REWRITE|6 1/2 + 52/2
MIX_IMPROPER|6 1/2|13/2
A|13/2|52/2|65/2
IMPROPER_TO_MIX|65/2|32 1/2
Z|32 1/2
Final answer:
32 1/2 |
64 | train-000000064 | 4eaf1aa9-4370-4792-9f9b-ee70864b93a0 | RemainderFactorTheoremGenerator | RemainderFactorTheoremGenerator | factor_theorem_find_k | high | 4 | Find k so that x + 3 is a factor of P(x) = x^3 + 3x^2 - 4x + k. | [
"THEOREM|factor theorem|x + 3 is a factor iff P(-3) = 0",
"SUBST|x|-3|(-3)^3 + 3(-3)^2 - 4(-3) + k",
"E|(-3)|3|-27",
"E|(-3)|2|9",
"M|3|9|27",
"M|-4|-3|12",
"A|-27|27|0",
"A|0|12|12",
"EQ_SETUP|12 + k = 0|solve for k",
"EQ_OP_BOTH|subtract|12|k|-12",
"Z|k = -12"
] | k = -12 | Problem:
Find k so that x + 3 is a factor of P(x) = x^3 + 3x^2 - 4x + k.
Solution steps:
THEOREM|factor theorem|x + 3 is a factor iff P(-3) = 0
SUBST|x|-3|(-3)^3 + 3(-3)^2 - 4(-3) + k
E|(-3)|3|-27
E|(-3)|2|9
M|3|9|27
M|-4|-3|12
A|-27|27|0
A|0|12|12
EQ_SETUP|12 + k = 0|solve for k
EQ_OP_BOTH|subtract|12|k|-12
Z|k = -12... |
65 | train-000000065 | 9c38d7dd-9350-47b3-898e-1993e38a0cd6 | FiniteFieldGenerator | FiniteFieldGenerator | finite_field_gf2_division | graduate | 4 | Over GF(2), divide x^5 + x^2 + x by x^2 + 1. Use XOR for coefficient arithmetic. | [
"FIELD_SETUP|GF(2)[x]|addition is XOR",
"POLYDIV_SETUP|x^5 + x^2 + x|x^2 + 1",
"DIV_TERM|x^5|x^2|x^3",
"GF2_XOR|quotient x^3|0 xor 1|1",
"GF2_XOR|remainder x^3|0 xor 1|1",
"GF2_XOR|remainder x^5|1 xor 1|0",
"POLY_REMAINDER|x^3 + x^2 + x",
"DIV_TERM|x^3|x^2|x",
"GF2_XOR|quotient x|0 xor 1|1",
"GF2_... | quotient = x^3 + x + 1; remainder = 1 | Problem:
Over GF(2), divide x^5 + x^2 + x by x^2 + 1. Use XOR for coefficient arithmetic.
Solution steps:
FIELD_SETUP|GF(2)[x]|addition is XOR
POLYDIV_SETUP|x^5 + x^2 + x|x^2 + 1
DIV_TERM|x^5|x^2|x^3
GF2_XOR|quotient x^3|0 xor 1|1
GF2_XOR|remainder x^3|0 xor 1|1
GF2_XOR|remainder x^5|1 xor 1|0
POLY_REMAINDER|x^3 + x^2... |
66 | train-000000066 | 2558d213-4f2e-41fd-9707-3e563d98c4e2 | DecimalMultGenerator | DecimalMultGenerator | decimal_mul | elementary | 3 | 35.91 * 19.1 | [
"MUL_SETUP|3591|191",
"MUL_PARTIAL|1|3591|3591",
"MUL_PARTIAL|9|3591|323190",
"MUL_PARTIAL|1|3591|359100",
"ADD_PARTIALS|3591+323190+359100|685881",
"COUNT_DP|2|1|3",
"PLACE_DP|685881|3|685.881",
"Z|685.881"
] | 685.881 | Problem:
35.91 * 19.1
Solution steps:
MUL_SETUP|3591|191
MUL_PARTIAL|1|3591|3591
MUL_PARTIAL|9|3591|323190
MUL_PARTIAL|1|3591|359100
ADD_PARTIALS|3591+323190+359100|685881
COUNT_DP|2|1|3
PLACE_DP|685881|3|685.881
Z|685.881
Final answer:
685.881 |
67 | train-000000067 | 482f23b2-c009-4c6a-8c0e-e3c63af96ca0 | AlgorithmTraceGenerator | AlgorithmTraceGenerator | algorithm_trace_insertion_sort | college | 3 | Trace insertion sort on values 23, 22, 2, 37, 14, 28, 11 for 4 passes. What is the array after those passes? | [
"ALG_SETUP|insertion sort|passes 4|values 23, 22, 2, 37, 14, 28, 11",
"INSERT_KEY|pass 1|22|index 1",
"COMPARE|arr[0]=23|key 22|shift",
"SHIFT|0->1|23, 23, 2, 37, 14, 28, 11",
"INSERT_PLACE|index 0|22, 23, 2, 37, 14, 28, 11",
"ARRAY_STATE|pass 1|22, 23, 2, 37, 14, 28, 11",
"INSERT_KEY|pass 2|2|index 2",... | array = [2, 14, 22, 23, 37, 28, 11] | Problem:
Trace insertion sort on values 23, 22, 2, 37, 14, 28, 11 for 4 passes. What is the array after those passes?
Solution steps:
ALG_SETUP|insertion sort|passes 4|values 23, 22, 2, 37, 14, 28, 11
INSERT_KEY|pass 1|22|index 1
COMPARE|arr[0]=23|key 22|shift
SHIFT|0->1|23, 23, 2, 37, 14, 28, 11
INSERT_PLACE|index 0|... |
68 | train-000000068 | 37b27adb-56b3-4afb-a3b3-2fbb26c11eae | HawkingGenerator | HawkingGenerator | hawking_entropy | graduate | 4 | Given k_B=10, c=1, A=18, hbar=4, and G=11, compute the Bekenstein-Hawking entropy S_BH=k_B*c^3*A/(4*hbar*G). | [
"HAWKING_SETUP|entropy|S_BH=k_B*c^3*A/(4*hbar*G)|k_B=10,c=1,A=18,hbar=4,G=11",
"E|1|3|1",
"M|10|1|10",
"M|10|18|180",
"M|4|4|16",
"M|16|11|176",
"D|180|176|45/44",
"Z|S_BH = 45/44"
] | S_BH = 45/44 | Problem:
Given k_B=10, c=1, A=18, hbar=4, and G=11, compute the Bekenstein-Hawking entropy S_BH=k_B*c^3*A/(4*hbar*G).
Solution steps:
HAWKING_SETUP|entropy|S_BH=k_B*c^3*A/(4*hbar*G)|k_B=10,c=1,A=18,hbar=4,G=11
E|1|3|1
M|10|1|10
M|10|18|180
M|4|4|16
M|16|11|176
D|180|176|45/44
Z|S_BH = 45/44
Final answer:
S_BH = 45/44 |
69 | train-000000069 | 432b5b25-31da-42eb-ace0-25a5545a0c47 | TemperatureConversionGenerator | TemperatureConversionGenerator | convert_temperature | elementary | 3 | Convert 122 F to C | [
"S|122|32|90",
"M|5|90|450",
"D|450|9|50",
"CONV_RESULT|122 F|50 C",
"Z|50 C"
] | 50 C | Problem:
Convert 122 F to C
Solution steps:
S|122|32|90
M|5|90|450
D|450|9|50
CONV_RESULT|122 F|50 C
Z|50 C
Final answer:
50 C |
70 | train-000000070 | b50e2460-c0a3-426f-9bb6-149dd87c080b | DPTableGenerator | DPTableGenerator | dp_table_knapsack | college | 4 | Fill the 0/1 knapsack DP table for capacity 5 with items 1:(w=5,v=6); 2:(w=5,v=10); 3:(w=3,v=11). What maximum value fits? | [
"DP_SETUP|0/1 knapsack|capacity 5",
"DP_ITEMS|1:(w=5,v=6); 2:(w=5,v=10); 3:(w=3,v=11)",
"DP_ROW|i=0|0, 0, 0, 0, 0, 0",
"DP_CELL|i=1,c=0|base|0",
"DP_CELL|i=1,c=1|skip w=5 > c|0",
"DP_CELL|i=1,c=2|skip w=5 > c|0",
"DP_CELL|i=1,c=3|skip w=5 > c|0",
"DP_CELL|i=1,c=4|skip w=5 > c|0",
"A|6|0|6",
"MAX|0... | maximum value = 11 | Problem:
Fill the 0/1 knapsack DP table for capacity 5 with items 1:(w=5,v=6); 2:(w=5,v=10); 3:(w=3,v=11). What maximum value fits?
Solution steps:
DP_SETUP|0/1 knapsack|capacity 5
DP_ITEMS|1:(w=5,v=6); 2:(w=5,v=10); 3:(w=3,v=11)
DP_ROW|i=0|0, 0, 0, 0, 0, 0
DP_CELL|i=1,c=0|base|0
DP_CELL|i=1,c=1|skip w=5 > c|0
DP_CELL... |
71 | train-000000071 | 8fa48a72-1d97-4863-bf1a-e58747591dfa | PerceptronGenerator | PerceptronGenerator | perceptron_three_point_epoch | graduate | 3 | Run one perceptron epoch with eta=2, starting weights w=(-2,2,1) for samples [(-3,-2,1), (2,-2,-1), (-3,-3,-1)]. Use bias feature x0=1, score=w0+w1*x1+w2*x2, and update when y*score <= 0. | [
"PERCEPTRON_SETUP|eta=2|w=(-2,2,1)|samples=[(-3,-2,1), (2,-2,-1), (-3,-3,-1)]",
"PERCEPTRON_RULE|score=w0+w1*x1+w2*x2|if y*score <= 0 update",
"PERCEPTRON_SAMPLE|i=1|x=(-3,-2)|y=1",
"M|2|-3|-6",
"A|-2|-6|-8",
"M|1|-2|-2",
"A|-8|-2|-10",
"PERCEPTRON_SCORE|i=1|score=-10",
"M|1|-10|-10",
"CHECK|i=1|y... | w_final=(-2,2,3); updates=2 | Problem:
Run one perceptron epoch with eta=2, starting weights w=(-2,2,1) for samples [(-3,-2,1), (2,-2,-1), (-3,-3,-1)]. Use bias feature x0=1, score=w0+w1*x1+w2*x2, and update when y*score <= 0.
Solution steps:
PERCEPTRON_SETUP|eta=2|w=(-2,2,1)|samples=[(-3,-2,1), (2,-2,-1), (-3,-3,-1)]
PERCEPTRON_RULE|score=w0+w1*x... |
72 | train-000000072 | 0db1dd9d-637b-4cf4-a368-2a656898d387 | LPCornerGenerator | LPCornerGenerator | lp_corner_point | college | 3 | Use the corner-point method to maximize z = 5x + 11y subject to 0 <= x <= 7, 0 <= y <= 8, and x + y <= 13. | [
"LP_CORNER_SETUP|max z=5x+11y|0<=x<=7, 0<=y<=8|x+y<=13",
"VERTEX_SOLVE|x=0|y=0",
"VERTEX|(0,0)",
"VERTEX_SOLVE|x=7|y=0",
"VERTEX|(7,0)",
"VERTEX_SOLVE|x=7|x+y=13",
"S|13|7|6",
"VERTEX|(7,6)",
"VERTEX_SOLVE|y=8|x+y=13",
"S|13|8|5",
"VERTEX|(5,8)",
"VERTEX_SOLVE|x=0|y=8",
"VERTEX|(0,8)",
"OB... | optimal vertex=(5,8), max z=113 | Problem:
Use the corner-point method to maximize z = 5x + 11y subject to 0 <= x <= 7, 0 <= y <= 8, and x + y <= 13.
Solution steps:
LP_CORNER_SETUP|max z=5x+11y|0<=x<=7, 0<=y<=8|x+y<=13
VERTEX_SOLVE|x=0|y=0
VERTEX|(0,0)
VERTEX_SOLVE|x=7|y=0
VERTEX|(7,0)
VERTEX_SOLVE|x=7|x+y=13
S|13|7|6
VERTEX|(7,6)
VERTEX_SOLVE|y=8|x+... |
73 | train-000000073 | bbd03990-6610-42ba-8fda-7cc0b855e125 | BondPricingGenerator | BondPricingGenerator | bond_pricing_current_yield | college | 4 | A bond has face value $700, annual coupon rate 20%, yield to maturity 20%, and 3 years to maturity with annual coupons. Compute the bond price and current yield. | [
"BOND_SETUP|face=700|coupon=20%,ytm=20%,years=3",
"PERCENT_TO_DEC|20%|0.2",
"PERCENT_TO_DEC|20%|0.2",
"BOND_FORMULA|price=sum coupon/(1+y)^t + face/(1+y)^n",
"M|700|0.2|140",
"COUPON|140",
"A|1|0.2|1.2",
"E|1.2|1|1.2",
"D|140|1.2|350/3",
"CASHFLOW_PV|coupon_t1|350/3",
"A|0|350/3|350/3",
"E|1.2... | price $700.00; current_yield=0.2 | Problem:
A bond has face value $700, annual coupon rate 20%, yield to maturity 20%, and 3 years to maturity with annual coupons. Compute the bond price and current yield.
Solution steps:
BOND_SETUP|face=700|coupon=20%,ytm=20%,years=3
PERCENT_TO_DEC|20%|0.2
PERCENT_TO_DEC|20%|0.2
BOND_FORMULA|price=sum coupon/(1+y)^t +... |
74 | train-000000074 | d52df7bd-466c-4692-9827-cac386c2eaeb | OrderOfOperationsGenerator | OrderOfOperationsGenerator(integers) | order_of_operations | elementary | 3 | Compute 1 - 2 + 10 * 8 | [
"M|10|8|80",
"REWRITE|1 - 2 + 80",
"S|1|2|-1",
"REWRITE|-1 + 80",
"A|-1|80|79",
"Z|79"
] | 79 | Problem:
Compute 1 - 2 + 10 * 8
Solution steps:
M|10|8|80
REWRITE|1 - 2 + 80
S|1|2|-1
REWRITE|-1 + 80
A|-1|80|79
Z|79
Final answer:
79 |
75 | train-000000075 | 30857d23-73eb-4b8f-9493-a78e698b4066 | DistanceFormulaGenerator | DistanceFormulaGenerator | distance_formula | high | 3 | Find the distance between (5, 0) and (3, 1). | [
"DIST_FORMULA|d = √((x2 - x1)^2 + (y2 - y1)^2)",
"S|3|5|-2",
"S|1|0|1",
"E|(-2)|2|4",
"E|1|2|1",
"A|4|1|5",
"Z|d = √5"
] | d = √5 | Problem:
Find the distance between (5, 0) and (3, 1).
Solution steps:
DIST_FORMULA|d = √((x2 - x1)^2 + (y2 - y1)^2)
S|3|5|-2
S|1|0|1
E|(-2)|2|4
E|1|2|1
A|4|1|5
Z|d = √5
Final answer:
d = √5 |
76 | train-000000076 | 230b70b0-43c9-4693-be3b-5701dbc8701e | PhysicsFormulaGenerator | PhysicsFormulaGenerator | physics_formula_work | middle | 4 | During a lab, a force of 70 newtons moves an object 12 meters. How much work is done? | [
"PHYS_SETUP|F = 70 newtons|d = 12 meters|work",
"PHYS_FORMULA|W = F*d",
"M|70|12|840",
"UNIT_ATTACH|840|joules|840 joules",
"Z|840 joules"
] | 840 joules | Problem:
During a lab, a force of 70 newtons moves an object 12 meters. How much work is done?
Solution steps:
PHYS_SETUP|F = 70 newtons|d = 12 meters|work
PHYS_FORMULA|W = F*d
M|70|12|840
UNIT_ATTACH|840|joules|840 joules
Z|840 joules
Final answer:
840 joules |
77 | train-000000077 | 3d2c8d48-cba5-4c1d-bf64-547c6ba026bf | SecondOrderODEGenerator | SecondOrderODEGenerator | second_order_ode_distinct_real | college | 3 | Solve y'' + y' - 6y = 0 with y(0) = 0 and y'(0) = 10. | [
"ODE_SETUP|y'' + y' - 6y = 0|y(0) = 0, y'(0) = 10",
"CHAR_EQ|assume y=e^(rx)|r^2 + r - 6 = 0",
"FACTOR|r^2 + r - 6|(r + 3)(r - 2) = 0",
"CHAR_ROOTS|r1 = -3, r2 = 2|distinct real",
"SOL_FORM|y = C1e^(-3x) + C2e^(2x)",
"SUBST|x=0|C1 + C2 = 0",
"DERIV_FORM|y'|-3C1e^(-3x) + 2C2e^(2x)",
"SUBST|x=0|-3C1 + 2... | y = -2e^(-3x) + 2e^(2x) | Problem:
Solve y'' + y' - 6y = 0 with y(0) = 0 and y'(0) = 10.
Solution steps:
ODE_SETUP|y'' + y' - 6y = 0|y(0) = 0, y'(0) = 10
CHAR_EQ|assume y=e^(rx)|r^2 + r - 6 = 0
FACTOR|r^2 + r - 6|(r + 3)(r - 2) = 0
CHAR_ROOTS|r1 = -3, r2 = 2|distinct real
SOL_FORM|y = C1e^(-3x) + C2e^(2x)
SUBST|x=0|C1 + C2 = 0
DERIV_FORM|y'|-3... |
78 | train-000000078 | 07fb43b0-8274-4128-a315-a49d8d4cc698 | LayerNormGenerator | LayerNormGenerator | layer_norm_exact_2d | college | 3 | Apply LayerNorm to x=(-2,0) with gamma=(4,3) and beta=(0,5). Use population variance and epsilon=0. | [
"LAYERNORM_SETUP|x=(-2,0)|gamma=(4,3)|beta=(0,5)",
"A|-2|0|-2",
"D|-2|2|-1",
"MEAN|-1",
"S|-2|-1|-1",
"E|-1|2|1",
"S|0|-1|1",
"E|1|2|1",
"A|1|1|2",
"D|2|2|1",
"VARIANCE|1",
"ROOT|sqrt(1)|1",
"STD|1",
"D|-1|1|-1",
"NORMALIZE|1|-1",
"M|4|-1|-4",
"A|-4|0|-4",
"SCALE_SHIFT|1|-4",
"D|... | mean=-1; variance=1; normalized=(-1,1); y=(-4,8) | Problem:
Apply LayerNorm to x=(-2,0) with gamma=(4,3) and beta=(0,5). Use population variance and epsilon=0.
Solution steps:
LAYERNORM_SETUP|x=(-2,0)|gamma=(4,3)|beta=(0,5)
A|-2|0|-2
D|-2|2|-1
MEAN|-1
S|-2|-1|-1
E|-1|2|1
S|0|-1|1
E|1|2|1
A|1|1|2
D|2|2|1
VARIANCE|1
ROOT|sqrt(1)|1
STD|1
D|-1|1|-1
NORMALIZE|1|-1
M|4|-1|-... |
79 | train-000000079 | 0081e337-eb50-47e1-9fea-53878531ceab | LPCornerGenerator | LPCornerGenerator | lp_corner_point | college | 3 | Use the corner-point method to maximize z = 5x + 11y subject to 0 <= x <= 5, 0 <= y <= 21, and x + y <= 25. | [
"LP_CORNER_SETUP|max z=5x+11y|0<=x<=5, 0<=y<=21|x+y<=25",
"VERTEX_SOLVE|x=0|y=0",
"VERTEX|(0,0)",
"VERTEX_SOLVE|x=5|y=0",
"VERTEX|(5,0)",
"VERTEX_SOLVE|x=5|x+y=25",
"S|25|5|20",
"VERTEX|(5,20)",
"VERTEX_SOLVE|y=21|x+y=25",
"S|25|21|4",
"VERTEX|(4,21)",
"VERTEX_SOLVE|x=0|y=21",
"VERTEX|(0,21)... | optimal vertex=(4,21), max z=251 | Problem:
Use the corner-point method to maximize z = 5x + 11y subject to 0 <= x <= 5, 0 <= y <= 21, and x + y <= 25.
Solution steps:
LP_CORNER_SETUP|max z=5x+11y|0<=x<=5, 0<=y<=21|x+y<=25
VERTEX_SOLVE|x=0|y=0
VERTEX|(0,0)
VERTEX_SOLVE|x=5|y=0
VERTEX|(5,0)
VERTEX_SOLVE|x=5|x+y=25
S|25|5|20
VERTEX|(5,20)
VERTEX_SOLVE|y=... |
80 | train-000000080 | ea98c783-a398-4464-9c2d-b247c7a0b065 | ProjectileMotionGenerator | ProjectileMotionGenerator | projectile_motion_components | college | 2 | A projectile is launched from ground level with horizontal velocity 14 m/s and vertical velocity 10 m/s. Use g=10 m/s^2 to compute time of flight, range, and maximum height. | [
"PROJECTILE_SETUP|vx=14|vy=10|g=10",
"FORMULA|t_up=vy/g",
"D|10|10|1",
"FORMULA|T=2*t_up",
"M|2|1|2",
"FORMULA|range=vx*T",
"M|14|2|28",
"FORMULA|h_max=vy^2/(2g)",
"E|10|2|100",
"M|2|10|20",
"D|100|20|5",
"Z|time=2 s; range=28 m; max height=5 m"
] | time=2 s; range=28 m; max height=5 m | Problem:
A projectile is launched from ground level with horizontal velocity 14 m/s and vertical velocity 10 m/s. Use g=10 m/s^2 to compute time of flight, range, and maximum height.
Solution steps:
PROJECTILE_SETUP|vx=14|vy=10|g=10
FORMULA|t_up=vy/g
D|10|10|1
FORMULA|T=2*t_up
M|2|1|2
FORMULA|range=vx*T
M|14|2|28
FORM... |
81 | train-000000081 | 42688af9-87c5-4d27-88e9-28db916c6c1b | BisectionGenerator | BisectionGenerator | bisection_interval | college | 2 | Use bisection for f(x)=x^2-22 on [4, 5] for 4 iterations. Give the final bracket. | [
"BISECTION_SETUP|f(x)=x^2-22|interval=[4, 5]|iterations=4",
"M|4|4|16",
"S|16|22|-6",
"SIGN|left|-6|negative",
"M|5|5|25",
"S|25|22|3",
"SIGN|right|3|positive",
"A|4|5|9",
"D|9|2|9/2",
"M|9/2|9/2|81/4",
"S|81/4|22|-7/4",
"SIGN|mid1|-7/4|negative",
"M|-6|-7/4|21/2",
"SIGN|product_1|21/2|pos... | root in [75/16, 19/4] | Problem:
Use bisection for f(x)=x^2-22 on [4, 5] for 4 iterations. Give the final bracket.
Solution steps:
BISECTION_SETUP|f(x)=x^2-22|interval=[4, 5]|iterations=4
M|4|4|16
S|16|22|-6
SIGN|left|-6|negative
M|5|5|25
S|25|22|3
SIGN|right|3|positive
A|4|5|9
D|9|2|9/2
M|9/2|9/2|81/4
S|81/4|22|-7/4
SIGN|mid1|-7/4|negative
... |
82 | train-000000082 | aa38f117-57a0-4197-b8cb-c9addef9a655 | EllipseFeaturesGenerator | EllipseFeaturesGenerator | ellipse_features | high | 5 | Find the center, vertices, and foci of the ellipse (x + 2)^2/256 + y^2/400 = 1. | [
"CONIC_SETUP|(x + 2)^2/256 + y^2/400 = 1|center, vertices, foci",
"FORM_IDENTIFY|(x - h)^2/a^2 + (y - k)^2/b^2 = 1 (ellipse)|major axis vertical (400 > 256)",
"CENTER|(-2, 0)",
"E|20|2|400",
"EVAL|a|20",
"E|16|2|256",
"EVAL|b|16",
"S|0|20|-20",
"A|0|20|20",
"VERTEX|(-2, -20)",
"VERTEX|(-2, 20)",... | center (-2, 0); vertices (-2, -20) and (-2, 20); foci (-2, -12) and (-2, 12) | Problem:
Find the center, vertices, and foci of the ellipse (x + 2)^2/256 + y^2/400 = 1.
Solution steps:
CONIC_SETUP|(x + 2)^2/256 + y^2/400 = 1|center, vertices, foci
FORM_IDENTIFY|(x - h)^2/a^2 + (y - k)^2/b^2 = 1 (ellipse)|major axis vertical (400 > 256)
CENTER|(-2, 0)
E|20|2|400
EVAL|a|20
E|16|2|256
EVAL|b|16
S|0|... |
83 | train-000000083 | 34711ef5-51f9-4329-b0c3-a082d6704295 | ConservationLawGenerator | ConservationLawGenerator | conservation_law_allowed | college | 3 | Audit conservation of Q, B, Le, Lmu for reaction gamma + pi0 + n -> anti_nu_e + p + pi0 + e- + gamma. Quantum numbers: gamma(Q=0,B=0,Le=0,Lmu=0); pi0(Q=0,B=0,Le=0,Lmu=0); n(Q=0,B=1,Le=0,Lmu=0); anti_nu_e(Q=0,B=0,Le=-1,Lmu=0); p(Q=1,B=1,Le=0,Lmu=0); e-(Q=-1,B=0,Le=1,Lmu=0). | [
"CONSERVATION_SETUP|gamma + pi0 + n -> anti_nu_e + p + pi0 + e- + gamma|check=Q,B,Le,Lmu",
"PARTICLE_TABLE|gamma(Q=0,B=0,Le=0,Lmu=0); pi0(Q=0,B=0,Le=0,Lmu=0); n(Q=0,B=1,Le=0,Lmu=0); anti_nu_e(Q=0,B=0,Le=-1,Lmu=0); p(Q=1,B=1,Le=0,Lmu=0); e-(Q=-1,B=0,Le=1,Lmu=0)",
"QN_ADD|Q|left|0 + gamma(0)|0",
"QN_ADD|Q|left|... | allowed - Q, B, Le, Lmu conserved | Problem:
Audit conservation of Q, B, Le, Lmu for reaction gamma + pi0 + n -> anti_nu_e + p + pi0 + e- + gamma. Quantum numbers: gamma(Q=0,B=0,Le=0,Lmu=0); pi0(Q=0,B=0,Le=0,Lmu=0); n(Q=0,B=1,Le=0,Lmu=0); anti_nu_e(Q=0,B=0,Le=-1,Lmu=0); p(Q=1,B=1,Le=0,Lmu=0); e-(Q=-1,B=0,Le=1,Lmu=0).
Solution steps:
CONSERVATION_SETUP|g... |
84 | train-000000084 | 2708065b-6291-446b-ae4f-76b02599c953 | SetOperationsGenerator | SetOperationsGenerator | set_operations_algebra | college | 2 | Given A = {a, c, e} and B = {d}, find A union B. | [
"SET_SETUP|A = {a, c, e}|B = {d}|union",
"ELEMENT_SCAN|a|in A=True, in B=False|keep",
"ELEMENT_SCAN|b|in A=False, in B=False|skip",
"ELEMENT_SCAN|c|in A=True, in B=False|keep",
"ELEMENT_SCAN|d|in A=False, in B=True|keep",
"ELEMENT_SCAN|e|in A=True, in B=False|keep",
"ELEMENT_SCAN|f|in A=False, in B=Fals... | {a, c, d, e} | Problem:
Given A = {a, c, e} and B = {d}, find A union B.
Solution steps:
SET_SETUP|A = {a, c, e}|B = {d}|union
ELEMENT_SCAN|a|in A=True, in B=False|keep
ELEMENT_SCAN|b|in A=False, in B=False|skip
ELEMENT_SCAN|c|in A=True, in B=False|keep
ELEMENT_SCAN|d|in A=False, in B=True|keep
ELEMENT_SCAN|e|in A=True, in B=False|k... |
85 | train-000000085 | 6e0e8a36-9ca3-4c94-8e31-958ab9890fdf | HermitianCheckGenerator | HermitianCheckGenerator | hermitian_check_unitary | college | 3 | Check whether U=[[260/269,-69/269],[69/269,260/269]] is unitary. | [
"MATRIX_SETUP|unitary|U=[[260/269,-69/269],[69/269,260/269]]",
"ADJOINT|U^dagger=[[260/269,69/269],[-69/269,260/269]]",
"E|260/269|2|67600/72361",
"E|69/269|2|4761/72361",
"A|67600/72361|4761/72361|1",
"M|260/269|-69/269|-17940/72361",
"M|69/269|260/269|17940/72361",
"A|-17940/72361|17940/72361|0",
... | unitary yes; U^dagger U = I | Problem:
Check whether U=[[260/269,-69/269],[69/269,260/269]] is unitary.
Solution steps:
MATRIX_SETUP|unitary|U=[[260/269,-69/269],[69/269,260/269]]
ADJOINT|U^dagger=[[260/269,69/269],[-69/269,260/269]]
E|260/269|2|67600/72361
E|69/269|2|4761/72361
A|67600/72361|4761/72361|1
M|260/269|-69/269|-17940/72361
M|69/269|26... |
86 | train-000000086 | d8f00c73-b95b-496e-b356-cc55df5de309 | ActivationGenerator | ActivationGenerator | activation_chain_sigmoid | college | 3 | For the two-layer scalar model y=w2*a(w1*x+b1)+b2 with x=4, w1=-5, b1=20, w2=-5, b2=1, use sigmoid activation with provided exp(-z)=1. Compute activation value, activation derivative, y, and dy/dx. | [
"ACT_SETUP|activation=sigmoid|x=4|w1=-5,b1=20,w2=-5,b2=1",
"M|-5|4|-20",
"A|-20|20|0",
"EXP_VALUE|exp(-z)|1",
"A|1|1|2",
"D|1|2|1/2",
"S|1|1/2|1/2",
"M|1/2|1/2|1/4",
"ACT_VALUE|sigmoid|0|1/2",
"ACT_DERIV|sigmoid|0|1/4",
"M|-5|1/2|-5/2",
"A|-5/2|1|-3/2",
"MODEL_OUTPUT|-3/2",
"M|-5|1/4|-5/4"... | z=0; a=1/2; a_prime=1/4; y=-3/2; dy_dx=25/4 | Problem:
For the two-layer scalar model y=w2*a(w1*x+b1)+b2 with x=4, w1=-5, b1=20, w2=-5, b2=1, use sigmoid activation with provided exp(-z)=1. Compute activation value, activation derivative, y, and dy/dx.
Solution steps:
ACT_SETUP|activation=sigmoid|x=4|w1=-5,b1=20,w2=-5,b2=1
M|-5|4|-20
A|-20|20|0
EXP_VALUE|exp(-z)|... |
87 | train-000000087 | 1bc8a033-d352-4cc3-9e97-d880b0864983 | AngleRelationshipsGenerator | AngleRelationshipsGenerator | supplementary_angles_algebraic | middle | 4 | Two supplementary angles measure (3x + 20)° and (5x - 40)°. Find the value of x. | [
"ANGLE_SETUP|supplementary|(3x + 20)° + (5x - 40)° = 180°",
"ANGLE_RELATION|8x - 20 = 180",
"ANGLE_SOLVE|8x = 200|x = 25",
"Z|25"
] | 25 | Problem:
Two supplementary angles measure (3x + 20)° and (5x - 40)°. Find the value of x.
Solution steps:
ANGLE_SETUP|supplementary|(3x + 20)° + (5x - 40)° = 180°
ANGLE_RELATION|8x - 20 = 180
ANGLE_SOLVE|8x = 200|x = 25
Z|25
Final answer:
25 |
88 | train-000000088 | 38ada27b-cb79-4965-8945-517f0da93459 | CRTGenerator | CRTGenerator | crt | college | 4 | Solve the CRT system x congruent to 0 modulo 4; x congruent to 4 modulo 7; x congruent to 5 modulo 11. Give the least nonnegative solution modulo the product. | [
"CRT_SETUP|3 congruences",
"CRT_TOTAL_MODULUS|4, 7, 11|308",
"CRT_CONGRUENCE|i=1|x=0|mod 4",
"CRT_CONGRUENCE|i=2|x=4|mod 7",
"CRT_CONGRUENCE|i=3|x=5|mod 11",
"D|308|4|77",
"CRT_FACTOR|i=1|M_i=77|mod 4",
"MOD_INVERSE|77 mod 4|1",
"M|0|77|0",
"M|0|1|0",
"CRT_TERM|i=1|0",
"A|0|0|0",
"D|308|7|44... | x = 60 mod 308 | Problem:
Solve the CRT system x congruent to 0 modulo 4; x congruent to 4 modulo 7; x congruent to 5 modulo 11. Give the least nonnegative solution modulo the product.
Solution steps:
CRT_SETUP|3 congruences
CRT_TOTAL_MODULUS|4, 7, 11|308
CRT_CONGRUENCE|i=1|x=0|mod 4
CRT_CONGRUENCE|i=2|x=4|mod 7
CRT_CONGRUENCE|i=3|x=5... |
89 | train-000000089 | 3038ed41-db3f-4f46-ba30-3747c368fa58 | EllipticCurveFiniteFieldGenerator | EllipticCurveFiniteFieldGenerator | elliptic_curve_finite_field_add | graduate | 4 | On the elliptic curve E: y^2 = x^3 + 1x + 4 over F_23, compute P + Q for P=(9,12) and Q=(4,16). | [
"EC_SETUP|p=23|a=1|b=4",
"EC_POINT_CHECK|P|y^2 mod p = 6|x^3+ax+b mod p = 6",
"EC_POINT_CHECK|Q|y^2 mod p = 3|x^3+ax+b mod p = 3",
"EC_SLOPE_FORMULA|P+Q|(y2-y1)/(x2-x1)",
"MOD_INVERSE|-5 mod 23|9",
"M|4|9|36",
"MOD_REDUCE|36|mod 23|13",
"EC_SLOPE|P+Q|13",
"M|13|13|169",
"S|169|9|160",
"S|160|4|1... | P+Q = (18,9) | Problem:
On the elliptic curve E: y^2 = x^3 + 1x + 4 over F_23, compute P + Q for P=(9,12) and Q=(4,16).
Solution steps:
EC_SETUP|p=23|a=1|b=4
EC_POINT_CHECK|P|y^2 mod p = 6|x^3+ax+b mod p = 6
EC_POINT_CHECK|Q|y^2 mod p = 3|x^3+ax+b mod p = 3
EC_SLOPE_FORMULA|P+Q|(y2-y1)/(x2-x1)
MOD_INVERSE|-5 mod 23|9
M|4|9|36
MOD_RE... |
90 | train-000000090 | 083601ac-1d1e-4c6a-a4b5-a3ebafa7fc3b | QuaternionGenerator | QuaternionGenerator | quaternion_arithmetic | graduate | 4 | Let p=(2,-3,-2,2) and q=(-2,2,3,2) represent coefficients of 1,i,j,k. With i^2=j^2=k^2=ijk=-1, compute p*q, q*p, conjugate(p), norm^2(p), and p^-1. | [
"QUAT_SETUP|p=(2,-3,-2,2)|q=(-2,2,3,2)",
"HAMILTON|i*i|-1",
"HAMILTON|j*j|-1",
"HAMILTON|k*k|-1",
"HAMILTON|i*j|k",
"HAMILTON|j*i|-k",
"QUAT_MUL_START|p*q|p|q",
"M|2|-2|-4",
"A|0|-4|-4",
"M|-3|2|-6",
"S|0|-6|6",
"A|-4|6|2",
"M|-2|3|-6",
"S|0|-6|6",
"A|2|6|8",
"M|2|2|4",
"S|0|4|-4",
... | p*q = (4,0,20,-5); q*p = (4,20,0,5); conjugate(p) = (2,3,2,-2); norm^2(p) = 21; p^-1 = (2/21,1/7,2/21,-2/21) | Problem:
Let p=(2,-3,-2,2) and q=(-2,2,3,2) represent coefficients of 1,i,j,k. With i^2=j^2=k^2=ijk=-1, compute p*q, q*p, conjugate(p), norm^2(p), and p^-1.
Solution steps:
QUAT_SETUP|p=(2,-3,-2,2)|q=(-2,2,3,2)
HAMILTON|i*i|-1
HAMILTON|j*j|-1
HAMILTON|k*k|-1
HAMILTON|i*j|k
HAMILTON|j*i|-k
QUAT_MUL_START|p*q|p|q
M|2|-2... |
91 | train-000000091 | 905bd9f2-1c8c-4135-806e-8325cfb80859 | StandardDeviationGenerator | StandardDeviationGenerator | standard_deviation_sample_variance | middle | 4 | Find the sample variance of the data set: 28, 23, 33, 20, 26. Give an exact answer. | [
"A|28|23|51",
"A|51|33|84",
"A|84|20|104",
"A|104|26|130",
"MEAN_DIV|130|5|26",
"DEV_ROW|28|2|4",
"DEV_ROW|23|-3|9",
"DEV_ROW|33|7|49",
"DEV_ROW|20|-6|36",
"DEV_ROW|26|0|0",
"A|4|9|13",
"A|13|49|62",
"A|62|36|98",
"A|98|0|98",
"EVAL|n - 1|4",
"D|98|4|49/2",
"Z|49/2"
] | 49/2 | Problem:
Find the sample variance of the data set: 28, 23, 33, 20, 26. Give an exact answer.
Solution steps:
A|28|23|51
A|51|33|84
A|84|20|104
A|104|26|130
MEAN_DIV|130|5|26
DEV_ROW|28|2|4
DEV_ROW|23|-3|9
DEV_ROW|33|7|49
DEV_ROW|20|-6|36
DEV_ROW|26|0|0
A|4|9|13
A|13|49|62
A|62|36|98
A|98|0|98
EVAL|n - 1|4
D|98|4|49/2
... |
92 | train-000000092 | 03cab47a-3876-4def-a36d-57151d1b3391 | DecimalMultGenerator | DecimalMultGenerator | decimal_mul | elementary | 3 | 59.6 * 38.4 | [
"MUL_SETUP|596|384",
"MUL_PARTIAL|4|596|2384",
"MUL_PARTIAL|8|596|47680",
"MUL_PARTIAL|3|596|178800",
"ADD_PARTIALS|2384+47680+178800|228864",
"COUNT_DP|1|1|2",
"PLACE_DP|228864|2|2288.64",
"Z|2288.64"
] | 2288.64 | Problem:
59.6 * 38.4
Solution steps:
MUL_SETUP|596|384
MUL_PARTIAL|4|596|2384
MUL_PARTIAL|8|596|47680
MUL_PARTIAL|3|596|178800
ADD_PARTIALS|2384+47680+178800|228864
COUNT_DP|1|1|2
PLACE_DP|228864|2|2288.64
Z|2288.64
Final answer:
2288.64 |
93 | train-000000093 | eda6e4d6-d1f4-411b-944d-24a7c31821d7 | SolutionChemGenerator | SolutionChemGenerator | solution_chem_mixing_molarity | high | 3 | Mix Va=214 mL of Ma=7/5 M solution with Vb=53 mL of Mb=3/5 M solution. Find final molarity M_final. | [
"SOLUTION_SETUP|mixing_molarity|Ma=7/5, Va=214|Mb=3/5, Vb=53",
"SOLUTION_FORMULA|M_final=(Ma*Va+Mb*Vb)/(Va+Vb)",
"M|7/5|214|1498/5",
"M|3/5|53|159/5",
"A|1498/5|159/5|1657/5",
"A|214|53|267",
"D|1657/5|267|1657/1335",
"Z|M_final=1657/1335 M"
] | M_final=1657/1335 M | Problem:
Mix Va=214 mL of Ma=7/5 M solution with Vb=53 mL of Mb=3/5 M solution. Find final molarity M_final.
Solution steps:
SOLUTION_SETUP|mixing_molarity|Ma=7/5, Va=214|Mb=3/5, Vb=53
SOLUTION_FORMULA|M_final=(Ma*Va+Mb*Vb)/(Va+Vb)
M|7/5|214|1498/5
M|3/5|53|159/5
A|1498/5|159/5|1657/5
A|214|53|267
D|1657/5|267|1657/13... |
94 | train-000000094 | 4896e4e5-fea4-41d2-a732-e783f056779f | BlackbodyGenerator | BlackbodyGenerator | blackbody_stefan_power | college | 3 | A blackbody has area A=12 m^2 and temperature T=15 K. Using Stefan-Boltzmann constant sigma=10, find radiated power P. | [
"BLACKBODY_SETUP|stefan_power|sigma=10, A=12|T=15",
"BLACKBODY_FORMULA|P=sigma*A*T^4",
"E|15|4|50625",
"M|10|12|120",
"M|120|50625|6075000",
"Z|P=6075000 W"
] | P=6075000 W | Problem:
A blackbody has area A=12 m^2 and temperature T=15 K. Using Stefan-Boltzmann constant sigma=10, find radiated power P.
Solution steps:
BLACKBODY_SETUP|stefan_power|sigma=10, A=12|T=15
BLACKBODY_FORMULA|P=sigma*A*T^4
E|15|4|50625
M|10|12|120
M|120|50625|6075000
Z|P=6075000 W
Final answer:
P=6075000 W |
95 | train-000000095 | 70515a7d-b1e6-4d19-a4bd-eeffb5a6ed88 | DijkstraGenerator | DijkstraGenerator | dijkstra_trace | college | 4 | Use Dijkstra's algorithm on the weighted undirected graph with vertices A, B, C, D and edges AB=6, AC=2, BC=9, BD=8, CD=2. Start at B and find shortest distances to all vertices. | [
"GRAPH_SETUP|weighted undirected graph|vertices A, B, C, D",
"EDGE_WEIGHT|AB|6",
"EDGE_WEIGHT|AC|2",
"EDGE_WEIGHT|BC|9",
"EDGE_WEIGHT|BD|8",
"EDGE_WEIGHT|CD|2",
"DIJKSTRA_INIT|start B|A=inf, B=0, C=inf, D=inf",
"SELECT_MIN|B|0",
"A|0|6|6",
"RELAX|B->A|update inf to 6|via weight 6",
"A|0|9|9",
... | distances = A:6, B:0, C:8, D:8 | Problem:
Use Dijkstra's algorithm on the weighted undirected graph with vertices A, B, C, D and edges AB=6, AC=2, BC=9, BD=8, CD=2. Start at B and find shortest distances to all vertices.
Solution steps:
GRAPH_SETUP|weighted undirected graph|vertices A, B, C, D
EDGE_WEIGHT|AB|6
EDGE_WEIGHT|AC|2
EDGE_WEIGHT|BC|9
EDGE_W... |
96 | train-000000096 | 1d920d23-b873-4f10-b6b4-f853c55f03c5 | RationalExprSimplifyGenerator | RationalExprSimplifyGenerator | rational_expr_simplify | high | 4 | Simplify: (35x^2 - 15x)/(5x) | [
"POLY_SETUP|(35x^2 - 15x)/(5x)",
"GCF_COEFF|35, 15|5",
"GCF_VAR|x^2, x|x",
"GCF_RESULT|5x",
"REWRITE|(5x(7x - 3))/(5x)",
"CANCEL|5x|7x - 3",
"Z|7x - 3"
] | 7x - 3 | Problem:
Simplify: (35x^2 - 15x)/(5x)
Solution steps:
POLY_SETUP|(35x^2 - 15x)/(5x)
GCF_COEFF|35, 15|5
GCF_VAR|x^2, x|x
GCF_RESULT|5x
REWRITE|(5x(7x - 3))/(5x)
CANCEL|5x|7x - 3
Z|7x - 3
Final answer:
7x - 3 |
97 | train-000000097 | 1048a6be-7ab2-4df5-a654-2b69b34fd275 | QRDecompositionGenerator | QRDecompositionGenerator | qr_decomposition_three | college | 4 | Find a QR decomposition A = QR for A = [[4, -3, -2], [0, 3, -1], [0, 0, 2]]. | [
"QR_SETUP|A = [[4, -3, -2], [0, 3, -1], [0, 0, 2]]|Gram-Schmidt columns",
"QR_ENTRY|q1|[1, 0, 0]",
"GS_SUBTRACT|v2 - (q1·v2)q1|[0, 3, 0]",
"QR_ENTRY|q2|[0, 1, 0]",
"GS_SUBTRACT|v3 - projections|[0, 0, 2]",
"QR_ENTRY|q3|[0, 0, 1]",
"QR_ENTRY|Q|[[1, 0, 0], [0, 1, 0], [0, 0, 1]]",
"QR_ENTRY|R|[[4, -3, -2... | Q=[[1, 0, 0], [0, 1, 0], [0, 0, 1]]; R=[[4, -3, -2], [0, 3, -1], [0, 0, 2]] | Problem:
Find a QR decomposition A = QR for A = [[4, -3, -2], [0, 3, -1], [0, 0, 2]].
Solution steps:
QR_SETUP|A = [[4, -3, -2], [0, 3, -1], [0, 0, 2]]|Gram-Schmidt columns
QR_ENTRY|q1|[1, 0, 0]
GS_SUBTRACT|v2 - (q1·v2)q1|[0, 3, 0]
QR_ENTRY|q2|[0, 1, 0]
GS_SUBTRACT|v3 - projections|[0, 0, 2]
QR_ENTRY|q3|[0, 0, 1]
QR_E... |
98 | train-000000098 | d328f7cc-351f-4519-887d-52ea3af3d6a6 | DiffieHellmanGenerator | DiffieHellmanGenerator | diffie_hellman | college | 3 | For Diffie-Hellman with prime p=29, generator g=3, Alice secret a=25, and Bob secret b=15, compute both public keys and the shared secret. | [
"DH_SETUP|p=29|g=3",
"DH_SECRET|Alice|25",
"DH_SECRET|Bob|15",
"MOD_POWER|3^25|mod 29|14",
"DH_PUBLIC|Alice|14",
"MOD_POWER|3^15|mod 29|26",
"DH_PUBLIC|Bob|26",
"MOD_POWER|26^25|mod 29|15",
"DH_SHARED|Alice|15",
"MOD_POWER|14^15|mod 29|15",
"DH_SHARED|Bob|15",
"CHECK|shared secrets match|15",
... | Alice public = 14; Bob public = 26; shared secret = 15 | Problem:
For Diffie-Hellman with prime p=29, generator g=3, Alice secret a=25, and Bob secret b=15, compute both public keys and the shared secret.
Solution steps:
DH_SETUP|p=29|g=3
DH_SECRET|Alice|25
DH_SECRET|Bob|15
MOD_POWER|3^25|mod 29|14
DH_PUBLIC|Alice|14
MOD_POWER|3^15|mod 29|26
DH_PUBLIC|Bob|26
MOD_POWER|26^25... |
99 | train-000000099 | d9a59976-26a4-4bce-89ed-518537778cc5 | RadicalAddSubGenerator | RadicalAddSubGenerator | radical_add_sub | high | 4 | Simplify: √2 - 2√18 - 4√18 | [
"ROOT_SETUP|√2 - 2√18 - 4√18",
"SQUARE_FACTOR|18|9 × 2|9",
"ROOT|9|3",
"REWRITE|√2 - 6√2 - 4√18",
"SQUARE_FACTOR|18|9 × 2|9",
"ROOT|9|3",
"REWRITE|√2 - 6√2 - 12√2",
"S|√2|6√2|-5√2",
"S|-5√2|12√2|-17√2",
"Z|-17√2"
] | -17√2 | Problem:
Simplify: √2 - 2√18 - 4√18
Solution steps:
ROOT_SETUP|√2 - 2√18 - 4√18
SQUARE_FACTOR|18|9 × 2|9
ROOT|9|3
REWRITE|√2 - 6√2 - 4√18
SQUARE_FACTOR|18|9 × 2|9
ROOT|9|3
REWRITE|√2 - 6√2 - 12√2
S|√2|6√2|-5√2
S|-5√2|12√2|-17√2
Z|-17√2
Final answer:
-17√2 |
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