answer string | choices dict | id string | kind string | lang string | question string | steps list | subject string | text string | topic string |
|---|---|---|---|---|---|---|---|---|---|
Determinant: 238. | null | mat_0651 | worked_example | en | Compute det [[15,1],[2,16]] step by step. | [
"ad − bc.",
"15×16 − 1×2.",
"= 238."
] | math | Answer: Determinant: 238.
Steps:
1. ad − bc.
2. 15×16 − 1×2.
3. = 238. | linear_algebra |
It averages one variable given another's value, the basis of regression prediction. | null | mat_0652 | concept | en | What is conditional expectation? | [
"It averages one variable given another's value, the basis of regression prediction.",
"So the answer stands: It averages one variable given another's value, the basis of regression prediction."
] | math | Answer: It averages one variable given another's value, the basis of regression prediction.
Steps:
1. It averages one variable given another's value, the basis of regression prediction.
2. So the answer stands: It averages one variable given another's value, the basis of regression prediction. | probability |
lcm = 120. | null | mat_0653 | worked_example | en | Compute lcm(15, 8) step by step. | [
"gcd(15,8) = 1.",
"15×8/1.",
"= 120."
] | math | Answer: lcm = 120.
Steps:
1. gcd(15,8) = 1.
2. 15×8/1.
3. = 120. | number_theory |
They arrange balanced experiments, with each pair meeting exactly lambda times. | null | mat_0654 | concept | en | Why block designs? | [
"They arrange balanced experiments, with each pair meeting exactly lambda times.",
"So the answer stands: They arrange balanced experiments, with each pair meeting exactly lambda times."
] | math | Answer: They arrange balanced experiments, with each pair meeting exactly lambda times.
Steps:
1. They arrange balanced experiments, with each pair meeting exactly lambda times.
2. So the answer stands: They arrange balanced experiments, with each pair meeting exactly lambda times. | combinatorics |
The sum is 66. | {
"A": "55",
"B": "77",
"C": "88",
"D": "66"
} | mat_0655 | multiple_choice | en | What is the sum 1 + 2 + ... + 11? | [
"11×12/2.",
"Sum = 66.",
"Select option D."
] | math | Answer: 66
Steps:
1. 11×12/2.
2. Sum = 66.
3. Select option D.
Answer: (D) | proofs |
Convergence decides whether infinite processes settle, separating meaningful sums from nonsense. | null | mat_0656 | concept | en | Why study convergence? | [
"Convergence decides whether infinite processes settle, separating meaningful sums from nonsense.",
"So the answer stands: Convergence decides whether infinite processes settle, separating meaningful sums from nonsense."
] | math | Answer: Convergence decides whether infinite processes settle, separating meaningful sums from nonsense.
Steps:
1. Convergence decides whether infinite processes settle, separating meaningful sums from nonsense.
2. So the answer stands: Convergence decides whether infinite processes settle, separating meaningful sums f... | calculus |
Trace: 33. | null | mat_0657 | worked_example | en | Compute the trace of [[16,0],[0,17]]. | [
"Sum diagonal.",
"16 + 17.",
"= 33."
] | math | Answer: Trace: 33.
Steps:
1. Sum diagonal.
2. 16 + 17.
3. = 33. | linear_algebra |
E = 8.5. | null | mat_0658 | worked_example | en | Compute E of a uniform draw from {1..16}. | [
"Mean of 1..16.",
"(16+1)/2.",
"E = 8.5."
] | math | Answer: E = 8.5.
Steps:
1. Mean of 1..16.
2. (16+1)/2.
3. E = 8.5. | probability |
It claims every even number above 2 is a prime sum, verified far but unproven. | null | mat_0659 | concept | en | What is the Goldbach conjecture? | [
"It claims every even number above 2 is a prime sum, verified far but unproven.",
"So the answer stands: It claims every even number above 2 is a prime sum, verified far but unproven."
] | math | Answer: It claims every even number above 2 is a prime sum, verified far but unproven.
Steps:
1. It claims every even number above 2 is a prime sum, verified far but unproven.
2. So the answer stands: It claims every even number above 2 is a prime sum, verified far but unproven. | number_theory |
P(6,3) = 120. | null | mat_0660 | worked_example | cs | Ukaž výpočet P(6,3). | [
"n(n−1)(n−2).",
"6×5×4.",
"= 120."
] | math | Answer: P(6,3) = 120.
Steps:
1. n(n−1)(n−2).
2. 6×5×4.
3. = 120. | combinatorics |
Parity splits numbers into even and odd, and tracking it often forces contradictions quickly. | null | mat_0661 | concept | en | Why use parity arguments? | [
"Parity splits numbers into even and odd, and tracking it often forces contradictions quickly.",
"So the answer stands: Parity splits numbers into even and odd, and tracking it often forces contradictions quickly."
] | math | Answer: Parity splits numbers into even and odd, and tracking it often forces contradictions quickly.
Steps:
1. Parity splits numbers into even and odd, and tracking it often forces contradictions quickly.
2. So the answer stands: Parity splits numbers into even and odd, and tracking it often forces contradictions quic... | proofs |
The derivative is 19. | {
"A": "17",
"B": "19",
"C": "21",
"D": "23"
} | mat_0662 | multiple_choice | en | For f(x) = 3x² + 7x, find f'(2). | [
"f'(x) = 6x + 7.",
"At 2: 19.",
"Select option B."
] | math | Answer: 19
Steps:
1. f'(x) = 6x + 7.
2. At 2: 19.
3. Select option B.
Answer: (B) | calculus |
Stopa: 11. | null | mat_0663 | worked_example | cs | Spočítej stopu matice [[5,0],[0,6]]. | [
"Sečti diagonálu.",
"5 + 6.",
"= 11."
] | math | Answer: Stopa: 11.
Steps:
1. Sečti diagonálu.
2. 5 + 6.
3. = 11. | linear_algebra |
P(red) = 14/18. | null | mat_0664 | worked_example | en | A bag holds 14 red and 4 blue marbles. Show P(red). | [
"Red: 14.",
"Total: 18.",
"P = 14/18."
] | math | Answer: P(red) = 14/18.
Steps:
1. Red: 14.
2. Total: 18.
3. P = 14/18. | probability |
gcd = 26. | null | mat_0665 | worked_example | en | Compute gcd(52, 78) step by step. | [
"Factor: 52=4×13, 78=6×13.",
"Common: 13×gcd(4,6).",
"= 13×2 = 26."
] | math | Answer: gcd = 26.
Steps:
1. Factor: 52=4×13, 78=6×13.
2. Common: 13×gcd(4,6).
3. = 13×2 = 26. | number_theory |
C(8,3) = 56. | null | mat_0666 | worked_example | cs | Ukaž výpočet C(8,3). | [
"n(n−1)(n−2)/6.",
"8×7×6/6.",
"= 56."
] | math | Answer: C(8,3) = 56.
Steps:
1. n(n−1)(n−2)/6.
2. 8×7×6/6.
3. = 56. | combinatorics |
Sum: 171. | null | mat_0667 | worked_example | en | Prove by formula: find the sum 1 + 2 + ... + 18. | [
"Pair 1+18, 2+17, ...",
"Use 18×19/2.",
"= 171."
] | math | Answer: Sum: 171.
Steps:
1. Pair 1+18, 2+17, ...
2. Use 18×19/2.
3. = 171. | proofs |
Řeší neurčitosti 0/0 derivováním čitatele i jmenovatele. | null | mat_0668 | concept | cs | Co je l'Hôpitalovo pravidlo? | [
"Řeší neurčitosti 0/0 derivováním čitatele i jmenovatele.",
"Odpověď tedy platí: Řeší neurčitosti 0/0 derivováním čitatele i jmenovatele."
] | math | Answer: Řeší neurčitosti 0/0 derivováním čitatele i jmenovatele.
Steps:
1. Řeší neurčitosti 0/0 derivováním čitatele i jmenovatele.
2. Odpověď tedy platí: Řeší neurčitosti 0/0 derivováním čitatele i jmenovatele. | calculus |
Dot product: 115. | null | mat_0669 | worked_example | en | Compute (15,14) · (3,5) step by step. | [
"Multiply components.",
"15×3 + 14×5.",
"= 115."
] | math | Answer: Dot product: 115.
Steps:
1. Multiply components.
2. 15×3 + 14×5.
3. = 115. | linear_algebra |
P(red) is 6/19. | {
"A": "13/19",
"B": "6/20",
"C": "6/19",
"D": "7/19"
} | mat_0670 | multiple_choice | en | A bag holds 6 red and 13 blue marbles. What is P(red)? | [
"6 of 19.",
"P = 6/19.",
"Select option C."
] | math | Answer: 6/19
Steps:
1. 6 of 19.
2. P = 6/19.
3. Select option C.
Answer: (C) | probability |
The remainder is 0. | {
"A": "1",
"B": "2",
"C": "0",
"D": "3"
} | mat_0671 | multiple_choice | en | Find (135) mod 9. | [
"135 ÷ 9.",
"Remainder 0.",
"Select option C."
] | math | Answer: 0
Steps:
1. 135 ÷ 9.
2. Remainder 0.
3. Select option C.
Answer: (C) | number_theory |
Odpověď je bijekce. | {
"A": "divoká mapa",
"B": "bijekce",
"C": "slepé hádání",
"D": "mlhavá naděje"
} | mat_0672 | multiple_choice | cs | Vyber: co páruje množiny stejně? | [
"Inverze dokazuje velikost.",
"Tedy: bijekce.",
"Vyber možnost B."
] | math | Answer: bijekce
Steps:
1. Inverze dokazuje velikost.
2. Tedy: bijekce.
3. Vyber možnost B.
Answer: (B) | combinatorics |
The value is 90. | {
"A": "90",
"B": "88",
"C": "92",
"D": "94"
} | mat_0673 | multiple_choice | en | Compute N² + N for N = 9. | [
"9² + 9.",
"Value = 90.",
"Select option A."
] | math | Answer: 90
Steps:
1. 9² + 9.
2. Value = 90.
3. Select option A.
Answer: (A) | proofs |
It integrates over infinite bounds or singularities via limits, extending area to the unbounded. | null | mat_0674 | concept | en | What is an improper integral? | [
"It integrates over infinite bounds or singularities via limits, extending area to the unbounded.",
"So the answer stands: It integrates over infinite bounds or singularities via limits, extending area to the unbounded."
] | math | Answer: It integrates over infinite bounds or singularities via limits, extending area to the unbounded.
Steps:
1. It integrates over infinite bounds or singularities via limits, extending area to the unbounded.
2. So the answer stands: It integrates over infinite bounds or singularities via limits, extending area to t... | calculus |
Vlastní čísla odhalují směry natažení zobrazení. | null | mat_0675 | concept | cs | Proč vlastní čísla? | [
"Vlastní čísla odhalují směry natažení zobrazení.",
"Odpověď tedy platí: Vlastní čísla odhalují směry natažení zobrazení."
] | math | Answer: Vlastní čísla odhalují směry natažení zobrazení.
Steps:
1. Vlastní čísla odhalují směry natažení zobrazení.
2. Odpověď tedy platí: Vlastní čísla odhalují směry natažení zobrazení. | linear_algebra |
The answer is Expected value. | {
"A": "First outcome",
"B": "Last digit",
"C": "Wild guess",
"D": "Expected value"
} | mat_0676 | multiple_choice | en | Choose: what averages long-run? | [
"Weighted by chance.",
"So: Expected value.",
"Select option D."
] | math | Answer: Expected value
Steps:
1. Weighted by chance.
2. So: Expected value.
3. Select option D.
Answer: (D) | probability |
Pairs differing by 2 may be infinite, and bounded gaps progress nears the dream. | null | mat_0677 | concept | en | Why twin primes? | [
"Pairs differing by 2 may be infinite, and bounded gaps progress nears the dream.",
"So the answer stands: Pairs differing by 2 may be infinite, and bounded gaps progress nears the dream."
] | math | Answer: Pairs differing by 2 may be infinite, and bounded gaps progress nears the dream.
Steps:
1. Pairs differing by 2 may be infinite, and bounded gaps progress nears the dream.
2. So the answer stands: Pairs differing by 2 may be infinite, and bounded gaps progress nears the dream. | number_theory |
The answer is Linearity of expectation. | {
"A": "Linearity of expectation",
"B": "Wild hope",
"C": "Blind sum",
"D": "Vague wish"
} | mat_0678 | multiple_choice | en | Choose: what adds regardless? | [
"Dependence ignored.",
"So: Linearity of expectation.",
"Select option A."
] | math | Answer: Linearity of expectation
Steps:
1. Dependence ignored.
2. So: Linearity of expectation.
3. Select option A.
Answer: (A) | combinatorics |
Half: 105, so 210 is even. | null | mat_0679 | worked_example | en | Show N² + N is even for N = 14 by computing its half. | [
"14² + 14 = 210.",
"210 / 2 = 105.",
"Integer, so even."
] | math | Answer: Half: 105, so 210 is even.
Steps:
1. 14² + 14 = 210.
2. 210 / 2 = 105.
3. Integer, so even. | proofs |
Limita: 10. | null | mat_0680 | worked_example | cs | Vyčísli limitu (x² − 5²)/(x − 5) pro x → 5. | [
"Rozlož: (x−5)(x+5).",
"Pokrať (x−5).",
"Dosaď 5: 5+5 = 10."
] | math | Answer: Limita: 10.
Steps:
1. Rozlož: (x−5)(x+5).
2. Pokrať (x−5).
3. Dosaď 5: 5+5 = 10. | calculus |
Determinant: 270. | null | mat_0681 | worked_example | en | Compute det [[16,1],[2,17]] step by step. | [
"ad − bc.",
"16×17 − 1×2.",
"= 270."
] | math | Answer: Determinant: 270.
Steps:
1. ad − bc.
2. 16×17 − 1×2.
3. = 270. | linear_algebra |
P = 1/18. | null | mat_0682 | worked_example | en | A fair 18-sided die is rolled. Show P(one specific number). | [
"Favourable: 1.",
"Possible: 18.",
"P = 1/18."
] | math | Answer: P = 1/18.
Steps:
1. Favourable: 1.
2. Possible: 18.
3. P = 1/18. | probability |
Primes thin like n/log n, describing their large-scale density precisely. | null | mat_0683 | concept | en | What is the prime number theorem? | [
"Primes thin like n/log n, describing their large-scale density precisely.",
"So the answer stands: Primes thin like n/log n, describing their large-scale density precisely."
] | math | Answer: Primes thin like n/log n, describing their large-scale density precisely.
Steps:
1. Primes thin like n/log n, describing their large-scale density precisely.
2. So the answer stands: Primes thin like n/log n, describing their large-scale density precisely. | number_theory |
C(9,2) is 36. | {
"A": "27",
"B": "36",
"C": "45",
"D": "54"
} | mat_0684 | multiple_choice | en | Compute C(9,2). | [
"9×8/2.",
"= 36.",
"Select option B."
] | math | Answer: 36
Steps:
1. 9×8/2.
2. = 36.
3. Select option B.
Answer: (B) | combinatorics |
Polovina: 15, číslo 30 je sudé. | null | mat_0685 | worked_example | cs | Ukaž, že N² + N je sudé pro N = 5, výpočtem poloviny. | [
"5² + 5 = 30.",
"30 / 2 = 15.",
"Celé číslo, tedy sudé."
] | math | Answer: Polovina: 15, číslo 30 je sudé.
Steps:
1. 5² + 5 = 30.
2. 30 / 2 = 15.
3. Celé číslo, tedy sudé. | proofs |
Integral: 3375. | null | mat_0686 | worked_example | en | Evaluate the integral of 3x² from 0 to 15. | [
"Antiderivative: x³.",
"Evaluate 0 to 15.",
"15³ − 0 = 3375."
] | math | Answer: Integral: 3375.
Steps:
1. Antiderivative: x³.
2. Evaluate 0 to 15.
3. 15³ − 0 = 3375. | calculus |
Odpověď je skalární součin. | {
"A": "skalární součin",
"B": "divoký odhad",
"C": "slepý součet",
"D": "slepá naděje"
} | mat_0687 | multiple_choice | cs | Vyber: co měří úhly? | [
"Nula znamená kolmost.",
"Tedy: skalární součin.",
"Vyber možnost A."
] | math | Answer: skalární součin
Steps:
1. Nula znamená kolmost.
2. Tedy: skalární součin.
3. Vyber možnost A.
Answer: (A) | linear_algebra |
Regression fits lines minimizing squared error, predicting responses from predictors. | null | mat_0688 | concept | en | Why regression? | [
"Regression fits lines minimizing squared error, predicting responses from predictors.",
"So the answer stands: Regression fits lines minimizing squared error, predicting responses from predictors."
] | math | Answer: Regression fits lines minimizing squared error, predicting responses from predictors.
Steps:
1. Regression fits lines minimizing squared error, predicting responses from predictors.
2. So the answer stands: Regression fits lines minimizing squared error, predicting responses from predictors. | probability |
Remainder: 3. | null | mat_0689 | worked_example | en | Compute (171) mod 7. | [
"Divide 171 by 7.",
"7×24 = 168.",
"171 − 168 = 3."
] | math | Answer: Remainder: 3.
Steps:
1. Divide 171 by 7.
2. 7×24 = 168.
3. 171 − 168 = 3. | number_theory |
P(15,3) = 2730. | null | mat_0690 | worked_example | en | Show the computation of P(15,3). | [
"n(n−1)(n−2).",
"15×14×13.",
"= 2730."
] | math | Answer: P(15,3) = 2730.
Steps:
1. n(n−1)(n−2).
2. 15×14×13.
3. = 2730. | combinatorics |
De Morganovy zákony převracejí a/nebo při negaci. | null | mat_0691 | concept | cs | Co je De Morganův zákon? | [
"De Morganovy zákony převracejí a/nebo při negaci.",
"Odpověď tedy platí: De Morganovy zákony převracejí a/nebo při negaci."
] | math | Answer: De Morganovy zákony převracejí a/nebo při negaci.
Steps:
1. De Morganovy zákony převracejí a/nebo při negaci.
2. Odpověď tedy platí: De Morganovy zákony převracejí a/nebo při negaci. | proofs |
The integral is 343. | {
"A": "336",
"B": "350",
"C": "343",
"D": "357"
} | mat_0692 | multiple_choice | en | Compute the integral of 3x² from 0 to 7. | [
"Antiderivative x³.",
"7³ = 343.",
"Select option C."
] | math | Answer: 343
Steps:
1. Antiderivative x³.
2. 7³ = 343.
3. Select option C.
Answer: (C) | calculus |
Skalární součin: 27. | null | mat_0693 | worked_example | cs | Spočítej (4,3) · (3,5) krok za krokem. | [
"Násob složky.",
"4×3 + 3×5.",
"= 27."
] | math | Answer: Skalární součin: 27.
Steps:
1. Násob složky.
2. 4×3 + 3×5.
3. = 27. | linear_algebra |
The answer is Independent trials. | {
"A": "Independent trials",
"B": "Old coins",
"C": "Tired dice",
"D": "Worn cards"
} | mat_0694 | multiple_choice | en | Choose: what has no memory? | [
"Streaks mean nothing.",
"So: Independent trials.",
"Select option A."
] | math | Answer: Independent trials
Steps:
1. Streaks mean nothing.
2. So: Independent trials.
3. Select option A.
Answer: (A) | probability |
lcm = 16. | null | mat_0695 | worked_example | en | Compute lcm(16, 8) step by step. | [
"gcd(16,8) = 8.",
"16×8/8.",
"= 16."
] | math | Answer: lcm = 16.
Steps:
1. gcd(16,8) = 8.
2. 16×8/8.
3. = 16. | number_theory |
It finds maxima under forbidden patterns, like most edges without triangles. | null | mat_0696 | concept | en | What is extremal combinatorics? | [
"It finds maxima under forbidden patterns, like most edges without triangles.",
"So the answer stands: It finds maxima under forbidden patterns, like most edges without triangles."
] | math | Answer: It finds maxima under forbidden patterns, like most edges without triangles.
Steps:
1. It finds maxima under forbidden patterns, like most edges without triangles.
2. So the answer stands: It finds maxima under forbidden patterns, like most edges without triangles. | combinatorics |
An invariant is a quantity unchanged by allowed moves, so differing invariants prove targets unreachable. | null | mat_0697 | concept | en | What is an invariant? | [
"An invariant is a quantity unchanged by allowed moves, so differing invariants prove targets unreachable.",
"So the answer stands: An invariant is a quantity unchanged by allowed moves, so differing invariants prove targets unreachable."
] | math | Answer: An invariant is a quantity unchanged by allowed moves, so differing invariants prove targets unreachable.
Steps:
1. An invariant is a quantity unchanged by allowed moves, so differing invariants prove targets unreachable.
2. So the answer stands: An invariant is a quantity unchanged by allowed moves, so differi... | proofs |
The limit is 14. | {
"A": "12",
"B": "16",
"C": "18",
"D": "14"
} | mat_0698 | multiple_choice | en | Find the limit of (x² − 7²)/(x − 7) as x → 7. | [
"Factor (x−7)(x+7).",
"Cancel, get 2×7 = 14.",
"Select option D."
] | math | Answer: 14
Steps:
1. Factor (x−7)(x+7).
2. Cancel, get 2×7 = 14.
3. Select option D.
Answer: (D) | calculus |
The answer is SVD. | {
"A": "SVD",
"B": "Lucky cut",
"C": "Blind split",
"D": "Wild guess"
} | mat_0699 | multiple_choice | en | Choose: what splits any matrix? | [
"Rotations plus scalings.",
"So: SVD.",
"Select option A."
] | math | Answer: SVD
Steps:
1. Rotations plus scalings.
2. So: SVD.
3. Select option A.
Answer: (A) | linear_algebra |
E = 9. | null | mat_0700 | worked_example | en | Compute E of a uniform draw from {1..17}. | [
"Mean of 1..17.",
"(17+1)/2.",
"E = 9."
] | math | Answer: E = 9.
Steps:
1. Mean of 1..17.
2. (17+1)/2.
3. E = 9. | probability |
The lcm is 18. | {
"A": "16",
"B": "20",
"C": "22",
"D": "18"
} | mat_0701 | multiple_choice | en | Find lcm(9, 6). | [
"9×6/gcd.",
"lcm = 18.",
"Select option D."
] | math | Answer: 18
Steps:
1. 9×6/gcd.
2. lcm = 18.
3. Select option D.
Answer: (D) | number_theory |
C(8,2) = 28. | null | mat_0702 | worked_example | cs | Ukaž výpočet C(8,2). | [
"Vzorec n(n−1)/2.",
"8×7/2.",
"= 28."
] | math | Answer: C(8,2) = 28.
Steps:
1. Vzorec n(n−1)/2.
2. 8×7/2.
3. = 28. | combinatorics |
Formal induction states base and step explicitly, catching hidden gaps in recursive reasoning. | null | mat_0703 | concept | en | Why formalize induction? | [
"Formal induction states base and step explicitly, catching hidden gaps in recursive reasoning.",
"So the answer stands: Formal induction states base and step explicitly, catching hidden gaps in recursive reasoning."
] | math | Answer: Formal induction states base and step explicitly, catching hidden gaps in recursive reasoning.
Steps:
1. Formal induction states base and step explicitly, catching hidden gaps in recursive reasoning.
2. So the answer stands: Formal induction states base and step explicitly, catching hidden gaps in recursive rea... | proofs |
Parametrization traces curves by a parameter, simplifying motion and arc-length questions. | null | mat_0704 | concept | en | Why parametrize curves? | [
"Parametrization traces curves by a parameter, simplifying motion and arc-length questions.",
"So the answer stands: Parametrization traces curves by a parameter, simplifying motion and arc-length questions."
] | math | Answer: Parametrization traces curves by a parameter, simplifying motion and arc-length questions.
Steps:
1. Parametrization traces curves by a parameter, simplifying motion and arc-length questions.
2. So the answer stands: Parametrization traces curves by a parameter, simplifying motion and arc-length questions. | calculus |
Trace: 35. | null | mat_0705 | worked_example | en | Compute the trace of [[17,0],[0,18]]. | [
"Sum diagonal.",
"17 + 18.",
"= 35."
] | math | Answer: Trace: 35.
Steps:
1. Sum diagonal.
2. 17 + 18.
3. = 35. | linear_algebra |
The probability is 1/11. | {
"A": "1/10",
"B": "1/11",
"C": "1/12",
"D": "2/11"
} | mat_0706 | multiple_choice | en | A fair 11-sided die is rolled. What is the probability of one specific number? | [
"1 face of 11.",
"P = 1/11.",
"Select option B."
] | math | Answer: 1/11
Steps:
1. 1 face of 11.
2. P = 1/11.
3. Select option B.
Answer: (B) | probability |
gcd = 28. | null | mat_0707 | worked_example | en | Compute gcd(56, 84) step by step. | [
"Factor: 56=4×14, 84=6×14.",
"Common: 14×gcd(4,6).",
"= 14×2 = 28."
] | math | Answer: gcd = 28.
Steps:
1. Factor: 56=4×14, 84=6×14.
2. Common: 14×gcd(4,6).
3. = 14×2 = 28. | number_theory |
Počítá rozdělení stejných předmětů do krabic. | null | mat_0708 | concept | cs | Proč metoda hvězdiček? | [
"Počítá rozdělení stejných předmětů do krabic.",
"Odpověď tedy platí: Počítá rozdělení stejných předmětů do krabic."
] | math | Answer: Počítá rozdělení stejných předmětů do krabic.
Steps:
1. Počítá rozdělení stejných předmětů do krabic.
2. Odpověď tedy platí: Počítá rozdělení stejných předmětů do krabic. | combinatorics |
Rows: 32768. | null | mat_0709 | worked_example | en | A proof checks all cases of 15 boolean variables. How many rows? | [
"2 choices each.",
"2^15 = 32768."
] | math | Answer: Rows: 32768.
Steps:
1. 2 choices each.
2. 2^15 = 32768. | proofs |
Arc length integrates speed along a curve, giving the exact distance travelled. | null | mat_0710 | concept | en | What is arc length? | [
"Arc length integrates speed along a curve, giving the exact distance travelled.",
"So the answer stands: Arc length integrates speed along a curve, giving the exact distance travelled."
] | math | Answer: Arc length integrates speed along a curve, giving the exact distance travelled.
Steps:
1. Arc length integrates speed along a curve, giving the exact distance travelled.
2. So the answer stands: Arc length integrates speed along a curve, giving the exact distance travelled. | calculus |
Determinant: 28. | null | mat_0711 | worked_example | cs | Spočítej det [[5,1],[2,6]] krok za krokem. | [
"ad − bc.",
"5×6 − 1×2.",
"= 28."
] | math | Answer: Determinant: 28.
Steps:
1. ad − bc.
2. 5×6 − 1×2.
3. = 28. | linear_algebra |
P(red) = 15/19. | null | mat_0712 | worked_example | en | A bag holds 15 red and 4 blue marbles. Show P(red). | [
"Red: 15.",
"Total: 19.",
"P = 15/19."
] | math | Answer: P(red) = 15/19.
Steps:
1. Red: 15.
2. Total: 19.
3. P = 15/19. | probability |
It crosses multiples to list primes fast, the classic prime-finding engine. | null | mat_0713 | concept | en | Why sieve of Eratosthenes? | [
"It crosses multiples to list primes fast, the classic prime-finding engine.",
"So the answer stands: It crosses multiples to list primes fast, the classic prime-finding engine."
] | math | Answer: It crosses multiples to list primes fast, the classic prime-finding engine.
Steps:
1. It crosses multiples to list primes fast, the classic prime-finding engine.
2. So the answer stands: It crosses multiples to list primes fast, the classic prime-finding engine. | number_theory |
C(17,3) = 680. | null | mat_0714 | worked_example | en | Show the computation of C(17,3). | [
"n(n−1)(n−2)/6.",
"17×16×15/6.",
"= 680."
] | math | Answer: C(17,3) = 680.
Steps:
1. n(n−1)(n−2)/6.
2. 17×16×15/6.
3. = 680. | combinatorics |
Exhaustion checks every case by hand or computer, and it works when the case list is finite. | null | mat_0715 | concept | en | What is proof by exhaustion? | [
"Exhaustion checks every case by hand or computer, and it works when the case list is finite.",
"So the answer stands: Exhaustion checks every case by hand or computer, and it works when the case list is finite."
] | math | Answer: Exhaustion checks every case by hand or computer, and it works when the case list is finite.
Steps:
1. Exhaustion checks every case by hand or computer, and it works when the case list is finite.
2. So the answer stands: Exhaustion checks every case by hand or computer, and it works when the case list is finite... | proofs |
The gradient points to steepest ascent, guiding optimization and physics on surfaces. | null | mat_0716 | concept | en | Why learn gradient? | [
"The gradient points to steepest ascent, guiding optimization and physics on surfaces.",
"So the answer stands: The gradient points to steepest ascent, guiding optimization and physics on surfaces."
] | math | Answer: The gradient points to steepest ascent, guiding optimization and physics on surfaces.
Steps:
1. The gradient points to steepest ascent, guiding optimization and physics on surfaces.
2. So the answer stands: The gradient points to steepest ascent, guiding optimization and physics on surfaces. | calculus |
Dot product: 123. | null | mat_0717 | worked_example | en | Compute (16,15) · (3,5) step by step. | [
"Multiply components.",
"16×3 + 15×5.",
"= 123."
] | math | Answer: Dot product: 123.
Steps:
1. Multiply components.
2. 16×3 + 15×5.
3. = 123. | linear_algebra |
P = 1/19. | null | mat_0718 | worked_example | en | A fair 19-sided die is rolled. Show P(one specific number). | [
"Favourable: 1.",
"Possible: 19.",
"P = 1/19."
] | math | Answer: P = 1/19.
Steps:
1. Favourable: 1.
2. Possible: 19.
3. P = 1/19. | probability |
It tests divisors up to sqrt(n), simple and sure for small primality checks. | null | mat_0719 | concept | en | What is trial division? | [
"It tests divisors up to sqrt(n), simple and sure for small primality checks.",
"So the answer stands: It tests divisors up to sqrt(n), simple and sure for small primality checks."
] | math | Answer: It tests divisors up to sqrt(n), simple and sure for small primality checks.
Steps:
1. It tests divisors up to sqrt(n), simple and sure for small primality checks.
2. So the answer stands: It tests divisors up to sqrt(n), simple and sure for small primality checks. | number_theory |
Random choices prove existence, showing desired objects occur with positive chance. | null | mat_0720 | concept | en | Why probabilistic method? | [
"Random choices prove existence, showing desired objects occur with positive chance.",
"So the answer stands: Random choices prove existence, showing desired objects occur with positive chance."
] | math | Answer: Random choices prove existence, showing desired objects occur with positive chance.
Steps:
1. Random choices prove existence, showing desired objects occur with positive chance.
2. So the answer stands: Random choices prove existence, showing desired objects occur with positive chance. | combinatorics |
Short proofs are easier to check and teach, though finding them often takes more insight. | null | mat_0721 | concept | en | Why prefer short proofs? | [
"Short proofs are easier to check and teach, though finding them often takes more insight.",
"So the answer stands: Short proofs are easier to check and teach, though finding them often takes more insight."
] | math | Answer: Short proofs are easier to check and teach, though finding them often takes more insight.
Steps:
1. Short proofs are easier to check and teach, though finding them often takes more insight.
2. So the answer stands: Short proofs are easier to check and teach, though finding them often takes more insight. | proofs |
The derivative is 14. | {
"A": "12",
"B": "16",
"C": "14",
"D": "18"
} | mat_0722 | multiple_choice | en | For f(x) = 1x² + 8x, find f'(3). | [
"f'(x) = 2x + 8.",
"At 3: 14.",
"Select option C."
] | math | Answer: 14
Steps:
1. f'(x) = 2x + 8.
2. At 3: 14.
3. Select option C.
Answer: (C) | calculus |
The determinant is 48. | {
"A": "45",
"B": "48",
"C": "51",
"D": "54"
} | mat_0723 | multiple_choice | en | Find the determinant of [[6,2],[3,9]]. | [
"6×9 − 2×3.",
"Det = 48.",
"Select option B."
] | math | Answer: 48
Steps:
1. 6×9 − 2×3.
2. Det = 48.
3. Select option B.
Answer: (B) | linear_algebra |
E = 9.5. | null | mat_0724 | worked_example | en | Compute E of a uniform draw from {1..18}. | [
"Mean of 1..18.",
"(18+1)/2.",
"E = 9.5."
] | math | Answer: E = 9.5.
Steps:
1. Mean of 1..18.
2. (18+1)/2.
3. E = 9.5. | probability |
The gcd is 24. | {
"A": "21",
"B": "27",
"C": "24",
"D": "30"
} | mat_0725 | multiple_choice | en | Find gcd(48, 72). | [
"Factor 3 out twice.",
"gcd = 24.",
"Select option C."
] | math | Answer: 24
Steps:
1. Factor 3 out twice.
2. gcd = 24.
3. Select option C.
Answer: (C) | number_theory |
C(17,2) = 136. | null | mat_0726 | worked_example | en | Show the computation of C(17,2). | [
"Formula n(n−1)/2.",
"17×16/2.",
"= 136."
] | math | Answer: C(17,2) = 136.
Steps:
1. Formula n(n−1)/2.
2. 17×16/2.
3. = 136. | combinatorics |
The table has 512 rows. | {
"A": "510",
"B": "514",
"C": "516",
"D": "512"
} | mat_0727 | multiple_choice | en | How many rows does a truth table with 9 variables have? | [
"2^9 rows.",
"Total: 512.",
"Select option D."
] | math | Answer: 512
Steps:
1. 2^9 rows.
2. Total: 512.
3. Select option D.
Answer: (D) | proofs |
It sums over regions in the plane, computing volumes under surfaces slice by slice. | null | mat_0728 | concept | en | What is a double integral? | [
"It sums over regions in the plane, computing volumes under surfaces slice by slice.",
"So the answer stands: It sums over regions in the plane, computing volumes under surfaces slice by slice."
] | math | Answer: It sums over regions in the plane, computing volumes under surfaces slice by slice.
Steps:
1. It sums over regions in the plane, computing volumes under surfaces slice by slice.
2. So the answer stands: It sums over regions in the plane, computing volumes under surfaces slice by slice. | calculus |
The trace is 17. | {
"A": "15",
"B": "19",
"C": "17",
"D": "21"
} | mat_0729 | multiple_choice | en | Find the trace of [[7,1],[2,10]]. | [
"7 + 10.",
"Trace = 17.",
"Select option C."
] | math | Answer: 17
Steps:
1. 7 + 10.
2. Trace = 17.
3. Select option C.
Answer: (C) | linear_algebra |
P(red) = 16/20. | null | mat_0730 | worked_example | en | A bag holds 16 red and 4 blue marbles. Show P(red). | [
"Red: 16.",
"Total: 20.",
"P = 16/20."
] | math | Answer: P(red) = 16/20.
Steps:
1. Red: 16.
2. Total: 20.
3. P = 16/20. | probability |
The remainder is 7. | {
"A": "8",
"B": "0",
"C": "1",
"D": "7"
} | mat_0731 | multiple_choice | en | Find (142) mod 9. | [
"142 ÷ 9.",
"Remainder 7.",
"Select option D."
] | math | Answer: 7
Steps:
1. 142 ÷ 9.
2. Remainder 7.
3. Select option D.
Answer: (D) | number_theory |
P(16,3) = 3360. | null | mat_0732 | worked_example | en | Show the computation of P(16,3). | [
"n(n−1)(n−2).",
"16×15×14.",
"= 3360."
] | math | Answer: P(16,3) = 3360.
Steps:
1. n(n−1)(n−2).
2. 16×15×14.
3. = 3360. | combinatorics |
Odpověď je základní případ. | {
"A": "složitá notace",
"B": "dlouhé řetězy",
"C": "velká čísla",
"D": "základní případ"
} | mat_0733 | multiple_choice | cs | Vyber: co kotví indukci? | [
"Bez základu řetěz nic nedokazuje.",
"Tedy: základní případ.",
"Vyber možnost D."
] | math | Answer: základní případ
Steps:
1. Bez základu řetěz nic nedokazuje.
2. Tedy: základní případ.
3. Vyber možnost D.
Answer: (D) | proofs |
Limit: 30. | null | mat_0734 | worked_example | en | Evaluate the limit of (x² − 15²)/(x − 15) as x → 15. | [
"Factor: (x−15)(x+15).",
"Cancel (x−15).",
"Plug 15: 15+15 = 30."
] | math | Answer: Limit: 30.
Steps:
1. Factor: (x−15)(x+15).
2. Cancel (x−15).
3. Plug 15: 15+15 = 30. | calculus |
New bases simplify matrices, revealing hidden diagonal or triangular structure. | null | mat_0735 | concept | en | Why change basis? | [
"New bases simplify matrices, revealing hidden diagonal or triangular structure.",
"So the answer stands: New bases simplify matrices, revealing hidden diagonal or triangular structure."
] | math | Answer: New bases simplify matrices, revealing hidden diagonal or triangular structure.
Steps:
1. New bases simplify matrices, revealing hidden diagonal or triangular structure.
2. So the answer stands: New bases simplify matrices, revealing hidden diagonal or triangular structure. | linear_algebra |
Overfitting memorizes noise instead of patterns, failing on new data despite perfect training. | null | mat_0736 | concept | en | What is overfitting? | [
"Overfitting memorizes noise instead of patterns, failing on new data despite perfect training.",
"So the answer stands: Overfitting memorizes noise instead of patterns, failing on new data despite perfect training."
] | math | Answer: Overfitting memorizes noise instead of patterns, failing on new data despite perfect training.
Steps:
1. Overfitting memorizes noise instead of patterns, failing on new data despite perfect training.
2. So the answer stands: Overfitting memorizes noise instead of patterns, failing on new data despite perfect tr... | probability |
Remainder: 0. | null | mat_0737 | worked_example | en | Compute (182) mod 7. | [
"Divide 182 by 7.",
"7×26 = 182.",
"182 − 182 = 0."
] | math | Answer: Remainder: 0.
Steps:
1. Divide 182 by 7.
2. 7×26 = 182.
3. 182 − 182 = 0. | number_theory |
P(8,2) is 56. | {
"A": "48",
"B": "64",
"C": "56",
"D": "72"
} | mat_0738 | multiple_choice | en | Compute P(8,2). | [
"8×7.",
"= 56.",
"Select option C."
] | math | Answer: 56
Steps:
1. 8×7.
2. = 56.
3. Select option C.
Answer: (C) | combinatorics |
The sum is 78. | {
"A": "78",
"B": "66",
"C": "90",
"D": "102"
} | mat_0739 | multiple_choice | en | What is the sum 1 + 2 + ... + 12? | [
"12×13/2.",
"Sum = 78.",
"Select option A."
] | math | Answer: 78
Steps:
1. 12×13/2.
2. Sum = 78.
3. Select option A.
Answer: (A) | proofs |
Limita je 6. | {
"A": "4",
"B": "8",
"C": "10",
"D": "6"
} | mat_0740 | multiple_choice | cs | Vypočítej limitu (x² − 3²)/(x − 3) pro x → 3. | [
"Rozlož (x−3)(x+3).",
"Pokrať, dostaneš 2×3 = 6.",
"Vyber možnost D."
] | math | Answer: 6
Steps:
1. Rozlož (x−3)(x+3).
2. Pokrať, dostaneš 2×3 = 6.
3. Vyber možnost D.
Answer: (D) | calculus |
The dot product is 46. | {
"A": "42",
"B": "50",
"C": "54",
"D": "46"
} | mat_0741 | multiple_choice | en | Find the dot product (6,7) · (3,4). | [
"6×3 + 7×4.",
"Product = 46.",
"Select option D."
] | math | Answer: 46
Steps:
1. 6×3 + 7×4.
2. Product = 46.
3. Select option D.
Answer: (D) | linear_algebra |
P(červená) je 3/10. | {
"A": "7/10",
"B": "3/11",
"C": "4/10",
"D": "3/10"
} | mat_0742 | multiple_choice | cs | Sáček má 3 červených a 7 modrých kuliček. Jaká je P(červená)? | [
"3 z 10.",
"P = 3/10.",
"Vyber možnost D."
] | math | Answer: 3/10
Steps:
1. 3 z 10.
2. P = 3/10.
3. Vyber možnost D.
Answer: (D) | probability |
lcm = 136. | null | mat_0743 | worked_example | en | Compute lcm(17, 8) step by step. | [
"gcd(17,8) = 1.",
"17×8/1.",
"= 136."
] | math | Answer: lcm = 136.
Steps:
1. gcd(17,8) = 1.
2. 17×8/1.
3. = 136. | number_theory |
C(10,3) is 120. | {
"A": "110",
"B": "130",
"C": "140",
"D": "120"
} | mat_0744 | multiple_choice | en | Compute C(10,3). | [
"10×9×8/6.",
"= 120.",
"Select option D."
] | math | Answer: 120
Steps:
1. 10×9×8/6.
2. = 120.
3. Select option D.
Answer: (D) | combinatorics |
A non-constructive proof shows existence without exhibiting the object, often via counting or contradiction. | null | mat_0745 | concept | en | What is a non-constructive proof? | [
"A non-constructive proof shows existence without exhibiting the object, often via counting or contradiction.",
"So the answer stands: A non-constructive proof shows existence without exhibiting the object, often via counting or contradiction."
] | math | Answer: A non-constructive proof shows existence without exhibiting the object, often via counting or contradiction.
Steps:
1. A non-constructive proof shows existence without exhibiting the object, often via counting or contradiction.
2. So the answer stands: A non-constructive proof shows existence without exhibiting... | proofs |
f'(3) = 32. | null | mat_0746 | worked_example | en | For f(x) = 3x² + 14x, compute f'(3) step by step. | [
"f'(x) = 6x + 14.",
"Plug x = 3.",
"6×3 + 14 = 32."
] | math | Answer: f'(3) = 32.
Steps:
1. f'(x) = 6x + 14.
2. Plug x = 3.
3. 6×3 + 14 = 32. | calculus |
Determinant je 4. | {
"A": "1",
"B": "4",
"C": "7",
"D": "10"
} | mat_0747 | multiple_choice | cs | Vypočítej determinant matice [[2,2],[3,5]]. | [
"2×5 − 2×3.",
"Det = 4.",
"Vyber možnost B."
] | math | Answer: 4
Steps:
1. 2×5 − 2×3.
2. Det = 4.
3. Vyber možnost B.
Answer: (B) | linear_algebra |
It tests models on held-out data, honestly estimating real-world performance. | null | mat_0748 | concept | en | Why cross-validate? | [
"It tests models on held-out data, honestly estimating real-world performance.",
"So the answer stands: It tests models on held-out data, honestly estimating real-world performance."
] | math | Answer: It tests models on held-out data, honestly estimating real-world performance.
Steps:
1. It tests models on held-out data, honestly estimating real-world performance.
2. So the answer stands: It tests models on held-out data, honestly estimating real-world performance. | probability |
The lcm is 30. | {
"A": "30",
"B": "28",
"C": "32",
"D": "34"
} | mat_0749 | multiple_choice | en | Find lcm(10, 6). | [
"10×6/gcd.",
"lcm = 30.",
"Select option A."
] | math | Answer: 30
Steps:
1. 10×6/gcd.
2. lcm = 30.
3. Select option A.
Answer: (A) | number_theory |
C(10,2) is 45. | {
"A": "35",
"B": "55",
"C": "45",
"D": "65"
} | mat_0750 | multiple_choice | en | Compute C(10,2). | [
"10×9/2.",
"= 45.",
"Select option C."
] | math | Answer: 45
Steps:
1. 10×9/2.
2. = 45.
3. Select option C.
Answer: (C) | combinatorics |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.