answer string | choices dict | id string | kind string | lang string | question string | steps list | subject string | text string | topic string |
|---|---|---|---|---|---|---|---|---|---|
Techniques transfer across fields, so mastering induction and contradiction unlocks all of mathematics. | null | mat_0751 | concept | en | Why study proof techniques? | [
"Techniques transfer across fields, so mastering induction and contradiction unlocks all of mathematics.",
"So the answer stands: Techniques transfer across fields, so mastering induction and contradiction unlocks all of mathematics."
] | math | Answer: Techniques transfer across fields, so mastering induction and contradiction unlocks all of mathematics.
Steps:
1. Techniques transfer across fields, so mastering induction and contradiction unlocks all of mathematics.
2. So the answer stands: Techniques transfer across fields, so mastering induction and contrad... | proofs |
The integral is 512. | {
"A": "504",
"B": "520",
"C": "528",
"D": "512"
} | mat_0752 | multiple_choice | en | Compute the integral of 3x² from 0 to 8. | [
"Antiderivative x³.",
"8³ = 512.",
"Select option D."
] | math | Answer: 512
Steps:
1. Antiderivative x³.
2. 8³ = 512.
3. Select option D.
Answer: (D) | calculus |
Similar matrices represent one map in different bases, sharing eigenvalues and determinant. | null | mat_0753 | concept | en | What is similarity? | [
"Similar matrices represent one map in different bases, sharing eigenvalues and determinant.",
"So the answer stands: Similar matrices represent one map in different bases, sharing eigenvalues and determinant."
] | math | Answer: Similar matrices represent one map in different bases, sharing eigenvalues and determinant.
Steps:
1. Similar matrices represent one map in different bases, sharing eigenvalues and determinant.
2. So the answer stands: Similar matrices represent one map in different bases, sharing eigenvalues and determinant. | linear_algebra |
P = 1/20. | null | mat_0754 | worked_example | en | A fair 20-sided die is rolled. Show P(one specific number). | [
"Favourable: 1.",
"Possible: 20.",
"P = 1/20."
] | math | Answer: P = 1/20.
Steps:
1. Favourable: 1.
2. Possible: 20.
3. P = 1/20. | probability |
The gcd is 27. | {
"A": "24",
"B": "30",
"C": "33",
"D": "27"
} | mat_0755 | multiple_choice | en | Find gcd(54, 81). | [
"Factor 3 out twice.",
"gcd = 27.",
"Select option D."
] | math | Answer: 27
Steps:
1. Factor 3 out twice.
2. gcd = 27.
3. Select option D.
Answer: (D) | number_theory |
Expectations add regardless of dependence, simplifying random counts hugely. | null | mat_0756 | concept | en | What is linearity of expectation? | [
"Expectations add regardless of dependence, simplifying random counts hugely.",
"So the answer stands: Expectations add regardless of dependence, simplifying random counts hugely."
] | math | Answer: Expectations add regardless of dependence, simplifying random counts hugely.
Steps:
1. Expectations add regardless of dependence, simplifying random counts hugely.
2. So the answer stands: Expectations add regardless of dependence, simplifying random counts hugely. | combinatorics |
Řádků: 64. | null | mat_0757 | worked_example | cs | Důkaz prochází všechny případy 6 booleovských proměnných. Kolik řádků? | [
"2 volby každá.",
"2^6 = 64."
] | math | Answer: Řádků: 64.
Steps:
1. 2 volby každá.
2. 2^6 = 64. | proofs |
The limit is 16. | {
"A": "16",
"B": "14",
"C": "18",
"D": "20"
} | mat_0758 | multiple_choice | en | Find the limit of (x² − 8²)/(x − 8) as x → 8. | [
"Factor (x−8)(x+8).",
"Cancel, get 2×8 = 16.",
"Select option A."
] | math | Answer: 16
Steps:
1. Factor (x−8)(x+8).
2. Cancel, get 2×8 = 16.
3. Select option A.
Answer: (A) | calculus |
Determinant: 304. | null | mat_0759 | worked_example | en | Compute det [[17,1],[2,18]] step by step. | [
"ad − bc.",
"17×18 − 1×2.",
"= 304."
] | math | Answer: Determinant: 304.
Steps:
1. ad − bc.
2. 17×18 − 1×2.
3. = 304. | linear_algebra |
The expected value is 5. | {
"A": "6",
"B": "4",
"C": "5",
"D": "7"
} | mat_0760 | multiple_choice | en | Find the expected value of a uniform draw from {1..9}. | [
"(9+1)/2.",
"E = 5.",
"Select option C."
] | math | Answer: 5
Steps:
1. (9+1)/2.
2. E = 5.
3. Select option C.
Answer: (C) | probability |
phi(n) počítá čísla menší než n nesoudělná s ním. | null | mat_0761 | concept | cs | Co je Eulerova funkce? | [
"phi(n) počítá čísla menší než n nesoudělná s ním.",
"Odpověď tedy platí: phi(n) počítá čísla menší než n nesoudělná s ním."
] | math | Answer: phi(n) počítá čísla menší než n nesoudělná s ním.
Steps:
1. phi(n) počítá čísla menší než n nesoudělná s ním.
2. Odpověď tedy platí: phi(n) počítá čísla menší než n nesoudělná s ním. | number_theory |
C(18,3) = 816. | null | mat_0762 | worked_example | en | Show the computation of C(18,3). | [
"n(n−1)(n−2)/6.",
"18×17×16/6.",
"= 816."
] | math | Answer: C(18,3) = 816.
Steps:
1. n(n−1)(n−2)/6.
2. 18×17×16/6.
3. = 816. | combinatorics |
Sum: 190. | null | mat_0763 | worked_example | en | Prove by formula: find the sum 1 + 2 + ... + 19. | [
"Pair 1+19, 2+18, ...",
"Use 19×20/2.",
"= 190."
] | math | Answer: Sum: 190.
Steps:
1. Pair 1+19, 2+18, ...
2. Use 19×20/2.
3. = 190. | proofs |
Integral: 4096. | null | mat_0764 | worked_example | en | Evaluate the integral of 3x² from 0 to 16. | [
"Antiderivative: x³.",
"Evaluate 0 to 16.",
"16³ − 0 = 4096."
] | math | Answer: Integral: 4096.
Steps:
1. Antiderivative: x³.
2. Evaluate 0 to 16.
3. 16³ − 0 = 4096. | calculus |
Stopa: 13. | null | mat_0765 | worked_example | cs | Spočítej stopu matice [[6,0],[0,7]]. | [
"Sečti diagonálu.",
"6 + 7.",
"= 13."
] | math | Answer: Stopa: 13.
Steps:
1. Sečti diagonálu.
2. 6 + 7.
3. = 13. | linear_algebra |
P(červená) = 5/9. | null | mat_0766 | worked_example | cs | Sáček má 5 červených a 4 modré kuličky. Ukaž P(červená). | [
"Červené: 5.",
"Celkem: 9.",
"P = 5/9."
] | math | Answer: P(červená) = 5/9.
Steps:
1. Červené: 5.
2. Celkem: 9.
3. P = 5/9. | probability |
gcd = 30. | null | mat_0767 | worked_example | en | Compute gcd(60, 90) step by step. | [
"Factor: 60=4×15, 90=6×15.",
"Common: 15×gcd(4,6).",
"= 15×2 = 30."
] | math | Answer: gcd = 30.
Steps:
1. Factor: 60=4×15, 90=6×15.
2. Common: 15×gcd(4,6).
3. = 15×2 = 30. | number_theory |
C(18,2) = 153. | null | mat_0768 | worked_example | en | Show the computation of C(18,2). | [
"Formula n(n−1)/2.",
"18×17/2.",
"= 153."
] | math | Answer: C(18,2) = 153.
Steps:
1. Formula n(n−1)/2.
2. 18×17/2.
3. = 153. | combinatorics |
Half: 120, so 240 is even. | null | mat_0769 | worked_example | en | Show N² + N is even for N = 15 by computing its half. | [
"15² + 15 = 240.",
"240 / 2 = 120.",
"Integer, so even."
] | math | Answer: Half: 120, so 240 is even.
Steps:
1. 15² + 15 = 240.
2. 240 / 2 = 120.
3. Integer, so even. | proofs |
The derivative is 25. | {
"A": "23",
"B": "27",
"C": "29",
"D": "25"
} | mat_0770 | multiple_choice | en | For f(x) = 2x² + 9x, find f'(4). | [
"f'(x) = 4x + 9.",
"At 4: 25.",
"Select option D."
] | math | Answer: 25
Steps:
1. f'(x) = 4x + 9.
2. At 4: 25.
3. Select option D.
Answer: (D) | calculus |
Skalární součin: 35. | null | mat_0771 | worked_example | cs | Spočítej (5,4) · (3,5) krok za krokem. | [
"Násob složky.",
"5×3 + 4×5.",
"= 35."
] | math | Answer: Skalární součin: 35.
Steps:
1. Násob složky.
2. 5×3 + 4×5.
3. = 35. | linear_algebra |
P(red) is 7/22. | {
"A": "15/22",
"B": "7/23",
"C": "8/22",
"D": "7/22"
} | mat_0772 | multiple_choice | en | A bag holds 7 red and 15 blue marbles. What is P(red)? | [
"7 of 22.",
"P = 7/22.",
"Select option D."
] | math | Answer: 7/22
Steps:
1. 7 of 22.
2. P = 7/22.
3. Select option D.
Answer: (D) | probability |
Remainder: 4. | null | mat_0773 | worked_example | en | Compute (193) mod 7. | [
"Divide 193 by 7.",
"7×27 = 189.",
"193 − 189 = 4."
] | math | Answer: Remainder: 4.
Steps:
1. Divide 193 by 7.
2. 7×27 = 189.
3. 193 − 189 = 4. | number_theory |
P(17,3) = 4080. | null | mat_0774 | worked_example | en | Show the computation of P(17,3). | [
"n(n−1)(n−2).",
"17×16×15.",
"= 4080."
] | math | Answer: P(17,3) = 4080.
Steps:
1. n(n−1)(n−2).
2. 17×16×15.
3. = 4080. | combinatorics |
Rows: 65536. | null | mat_0775 | worked_example | en | A proof checks all cases of 16 boolean variables. How many rows? | [
"2 choices each.",
"2^16 = 65536."
] | math | Answer: Rows: 65536.
Steps:
1. 2 choices each.
2. 2^16 = 65536. | proofs |
Limit: 32. | null | mat_0776 | worked_example | en | Evaluate the limit of (x² − 16²)/(x − 16) as x → 16. | [
"Factor: (x−16)(x+16).",
"Cancel (x−16).",
"Plug 16: 16+16 = 32."
] | math | Answer: Limit: 32.
Steps:
1. Factor: (x−16)(x+16).
2. Cancel (x−16).
3. Plug 16: 16+16 = 32. | calculus |
Trace: 37. | null | mat_0777 | worked_example | en | Compute the trace of [[18,0],[0,19]]. | [
"Sum diagonal.",
"18 + 19.",
"= 37."
] | math | Answer: Trace: 37.
Steps:
1. Sum diagonal.
2. 18 + 19.
3. = 37. | linear_algebra |
E = 10. | null | mat_0778 | worked_example | en | Compute E of a uniform draw from {1..19}. | [
"Mean of 1..19.",
"(19+1)/2.",
"E = 10."
] | math | Answer: E = 10.
Steps:
1. Mean of 1..19.
2. (19+1)/2.
3. E = 10. | probability |
Zbytek: 6. | null | mat_0779 | worked_example | cs | Spočítej (83) mod 7. | [
"Děl 83 sedmi.",
"7×11 = 77.",
"83 − 77 = 6."
] | math | Answer: Zbytek: 6.
Steps:
1. Děl 83 sedmi.
2. 7×11 = 77.
3. 83 − 77 = 6. | number_theory |
P(9,2) is 72. | {
"A": "63",
"B": "81",
"C": "90",
"D": "72"
} | mat_0780 | multiple_choice | en | Compute P(9,2). | [
"9×8.",
"= 72.",
"Select option D."
] | math | Answer: 72
Steps:
1. 9×8.
2. = 72.
3. Select option D.
Answer: (D) | combinatorics |
Sum: 210. | null | mat_0781 | worked_example | en | Prove by formula: find the sum 1 + 2 + ... + 20. | [
"Pair 1+20, 2+19, ...",
"Use 20×21/2.",
"= 210."
] | math | Answer: Sum: 210.
Steps:
1. Pair 1+20, 2+19, ...
2. Use 20×21/2.
3. = 210. | proofs |
Polar or spherical coordinates match round symmetry, making hard integrals easy. | null | mat_0782 | concept | en | Why change coordinates? | [
"Polar or spherical coordinates match round symmetry, making hard integrals easy.",
"So the answer stands: Polar or spherical coordinates match round symmetry, making hard integrals easy."
] | math | Answer: Polar or spherical coordinates match round symmetry, making hard integrals easy.
Steps:
1. Polar or spherical coordinates match round symmetry, making hard integrals easy.
2. So the answer stands: Polar or spherical coordinates match round symmetry, making hard integrals easy. | calculus |
The determinant is 64. | {
"A": "61",
"B": "67",
"C": "64",
"D": "70"
} | mat_0783 | multiple_choice | en | Find the determinant of [[7,2],[3,10]]. | [
"7×10 − 2×3.",
"Det = 64.",
"Select option C."
] | math | Answer: 64
Steps:
1. 7×10 − 2×3.
2. Det = 64.
3. Select option C.
Answer: (C) | linear_algebra |
P(red) = 17/21. | null | mat_0784 | worked_example | en | A bag holds 17 red and 4 blue marbles. Show P(red). | [
"Red: 17.",
"Total: 21.",
"P = 17/21."
] | math | Answer: P(red) = 17/21.
Steps:
1. Red: 17.
2. Total: 21.
3. P = 17/21. | probability |
It tests primality fast with random bases, practical for cryptographic-size numbers. | null | mat_0785 | concept | en | Why Miller-Rabin? | [
"It tests primality fast with random bases, practical for cryptographic-size numbers.",
"So the answer stands: It tests primality fast with random bases, practical for cryptographic-size numbers."
] | math | Answer: It tests primality fast with random bases, practical for cryptographic-size numbers.
Steps:
1. It tests primality fast with random bases, practical for cryptographic-size numbers.
2. So the answer stands: It tests primality fast with random bases, practical for cryptographic-size numbers. | number_theory |
Derangement share nears 1/e, a lovely limit from inclusion-exclusion sums. | null | mat_0786 | concept | en | Why derangements approximate? | [
"Derangement share nears 1/e, a lovely limit from inclusion-exclusion sums.",
"So the answer stands: Derangement share nears 1/e, a lovely limit from inclusion-exclusion sums."
] | math | Answer: Derangement share nears 1/e, a lovely limit from inclusion-exclusion sums.
Steps:
1. Derangement share nears 1/e, a lovely limit from inclusion-exclusion sums.
2. So the answer stands: Derangement share nears 1/e, a lovely limit from inclusion-exclusion sums. | combinatorics |
The contrapositive is not-Q to not-P, and it is logically equivalent to the original implication. | null | mat_0787 | concept | en | What is the contrapositive of P to Q? | [
"The contrapositive is not-Q to not-P, and it is logically equivalent to the original implication.",
"So the answer stands: The contrapositive is not-Q to not-P, and it is logically equivalent to the original implication."
] | math | Answer: The contrapositive is not-Q to not-P, and it is logically equivalent to the original implication.
Steps:
1. The contrapositive is not-Q to not-P, and it is logically equivalent to the original implication.
2. So the answer stands: The contrapositive is not-Q to not-P, and it is logically equivalent to the origi... | proofs |
The integral is 729. | {
"A": "729",
"B": "720",
"C": "738",
"D": "747"
} | mat_0788 | multiple_choice | en | Compute the integral of 3x² from 0 to 9. | [
"Antiderivative x³.",
"9³ = 729.",
"Select option A."
] | math | Answer: 729
Steps:
1. Antiderivative x³.
2. 9³ = 729.
3. Select option A.
Answer: (A) | calculus |
Báze je nezávislá množina napínající prostor, dává jednoznačné souřadnice. | null | mat_0789 | concept | cs | Co je báze? | [
"Báze je nezávislá množina napínající prostor, dává jednoznačné souřadnice.",
"Odpověď tedy platí: Báze je nezávislá množina napínající prostor, dává jednoznačné souřadnice."
] | math | Answer: Báze je nezávislá množina napínající prostor, dává jednoznačné souřadnice.
Steps:
1. Báze je nezávislá množina napínající prostor, dává jednoznačné souřadnice.
2. Odpověď tedy platí: Báze je nezávislá množina napínající prostor, dává jednoznačné souřadnice. | linear_algebra |
P = 1/21. | null | mat_0790 | worked_example | en | A fair 21-sided die is rolled. Show P(one specific number). | [
"Favourable: 1.",
"Possible: 21.",
"P = 1/21."
] | math | Answer: P = 1/21.
Steps:
1. Favourable: 1.
2. Possible: 21.
3. P = 1/21. | probability |
lcm = 72. | null | mat_0791 | worked_example | en | Compute lcm(18, 8) step by step. | [
"gcd(18,8) = 2.",
"18×8/2.",
"= 72."
] | math | Answer: lcm = 72.
Steps:
1. gcd(18,8) = 2.
2. 18×8/2.
3. = 72. | number_theory |
C(19,3) = 969. | null | mat_0792 | worked_example | en | Show the computation of C(19,3). | [
"n(n−1)(n−2)/6.",
"19×18×17/6.",
"= 969."
] | math | Answer: C(19,3) = 969.
Steps:
1. n(n−1)(n−2)/6.
2. 19×18×17/6.
3. = 969. | combinatorics |
Součet: 55. | null | mat_0793 | worked_example | cs | Dokaž vzorcem: najdi součet 1 + 2 + ... + 10. | [
"Páruj 1+10, 2+9, ...",
"Použij 10×11/2.",
"= 55."
] | math | Answer: Součet: 55.
Steps:
1. Páruj 1+10, 2+9, ...
2. Použij 10×11/2.
3. = 55. | proofs |
The limit is 18. | {
"A": "16",
"B": "18",
"C": "20",
"D": "22"
} | mat_0794 | multiple_choice | en | Find the limit of (x² − 9²)/(x − 9) as x → 9. | [
"Factor (x−9)(x+9).",
"Cancel, get 2×9 = 18.",
"Select option B."
] | math | Answer: 18
Steps:
1. Factor (x−9)(x+9).
2. Cancel, get 2×9 = 18.
3. Select option B.
Answer: (B) | calculus |
Its roots are the eigenvalues, turning eigen-search into polynomial factoring. | null | mat_0795 | concept | en | Why compute characteristic polynomial? | [
"Its roots are the eigenvalues, turning eigen-search into polynomial factoring.",
"So the answer stands: Its roots are the eigenvalues, turning eigen-search into polynomial factoring."
] | math | Answer: Its roots are the eigenvalues, turning eigen-search into polynomial factoring.
Steps:
1. Its roots are the eigenvalues, turning eigen-search into polynomial factoring.
2. So the answer stands: Its roots are the eigenvalues, turning eigen-search into polynomial factoring. | linear_algebra |
The probability is 1/12. | {
"A": "1/11",
"B": "1/13",
"C": "1/12",
"D": "2/12"
} | mat_0796 | multiple_choice | en | A fair 12-sided die is rolled. What is the probability of one specific number? | [
"1 face of 12.",
"P = 1/12.",
"Select option C."
] | math | Answer: 1/12
Steps:
1. 1 face of 12.
2. P = 1/12.
3. Select option C.
Answer: (C) | probability |
gcd = 32. | null | mat_0797 | worked_example | en | Compute gcd(64, 96) step by step. | [
"Factor: 64=4×16, 96=6×16.",
"Common: 16×gcd(4,6).",
"= 16×2 = 32."
] | math | Answer: gcd = 32.
Steps:
1. Factor: 64=4×16, 96=6×16.
2. Common: 16×gcd(4,6).
3. = 16×2 = 32. | number_theory |
C(19,2) = 171. | null | mat_0798 | worked_example | en | Show the computation of C(19,2). | [
"Formula n(n−1)/2.",
"19×18/2.",
"= 171."
] | math | Answer: C(19,2) = 171.
Steps:
1. Formula n(n−1)/2.
2. 19×18/2.
3. = 171. | combinatorics |
Half: 136, so 272 is even. | null | mat_0799 | worked_example | en | Show N² + N is even for N = 16 by computing its half. | [
"16² + 16 = 272.",
"272 / 2 = 136.",
"Integer, so even."
] | math | Answer: Half: 136, so 272 is even.
Steps:
1. 16² + 16 = 272.
2. 272 / 2 = 136.
3. Integer, so even. | proofs |
f'(4) = 47. | null | mat_0800 | worked_example | en | For f(x) = 4x² + 15x, compute f'(4) step by step. | [
"f'(x) = 8x + 15.",
"Plug x = 4.",
"8×4 + 15 = 47."
] | math | Answer: f'(4) = 47.
Steps:
1. f'(x) = 8x + 15.
2. Plug x = 4.
3. 8×4 + 15 = 47. | calculus |
Dot product: 131. | null | mat_0801 | worked_example | en | Compute (17,16) · (3,5) step by step. | [
"Multiply components.",
"17×3 + 16×5.",
"= 131."
] | math | Answer: Dot product: 131.
Steps:
1. Multiply components.
2. 17×3 + 16×5.
3. = 131. | linear_algebra |
A prior encodes beliefs before data, and Bayes updates it into the posterior. | null | mat_0802 | concept | en | What is a prior? | [
"A prior encodes beliefs before data, and Bayes updates it into the posterior.",
"So the answer stands: A prior encodes beliefs before data, and Bayes updates it into the posterior."
] | math | Answer: A prior encodes beliefs before data, and Bayes updates it into the posterior.
Steps:
1. A prior encodes beliefs before data, and Bayes updates it into the posterior.
2. So the answer stands: A prior encodes beliefs before data, and Bayes updates it into the posterior. | probability |
The remainder is 5. | {
"A": "5",
"B": "6",
"C": "7",
"D": "8"
} | mat_0803 | multiple_choice | en | Find (149) mod 9. | [
"149 ÷ 9.",
"Remainder 5.",
"Select option A."
] | math | Answer: 5
Steps:
1. 149 ÷ 9.
2. Remainder 5.
3. Select option A.
Answer: (A) | number_theory |
Opravuje dvojí započtení střídavými součty. | null | mat_0804 | concept | cs | Co je princip inkluze? | [
"Opravuje dvojí započtení střídavými součty.",
"Odpověď tedy platí: Opravuje dvojí započtení střídavými součty."
] | math | Answer: Opravuje dvojí započtení střídavými součty.
Steps:
1. Opravuje dvojí započtení střídavými součty.
2. Odpověď tedy platí: Opravuje dvojí započtení střídavými součty. | combinatorics |
Negating for-all yields exists-not, so sloppy negation flips the claim's meaning entirely. | null | mat_0805 | concept | en | Why negate quantifiers carefully? | [
"Negating for-all yields exists-not, so sloppy negation flips the claim's meaning entirely.",
"So the answer stands: Negating for-all yields exists-not, so sloppy negation flips the claim's meaning entirely."
] | math | Answer: Negating for-all yields exists-not, so sloppy negation flips the claim's meaning entirely.
Steps:
1. Negating for-all yields exists-not, so sloppy negation flips the claim's meaning entirely.
2. So the answer stands: Negating for-all yields exists-not, so sloppy negation flips the claim's meaning entirely. | proofs |
Integral: 4913. | null | mat_0806 | worked_example | en | Evaluate the integral of 3x² from 0 to 17. | [
"Antiderivative: x³.",
"Evaluate 0 to 17.",
"17³ − 0 = 4913."
] | math | Answer: Integral: 4913.
Steps:
1. Antiderivative: x³.
2. Evaluate 0 to 17.
3. 17³ − 0 = 4913. | calculus |
The trace is 19. | {
"A": "17",
"B": "21",
"C": "23",
"D": "19"
} | mat_0807 | multiple_choice | en | Find the trace of [[8,1],[2,11]]. | [
"8 + 11.",
"Trace = 19.",
"Select option D."
] | math | Answer: 19
Steps:
1. 8 + 11.
2. Trace = 19.
3. Select option D.
Answer: (D) | linear_algebra |
P = 1/9. | null | mat_0808 | worked_example | cs | Hází se férovou 9-stěnnou kostkou. Ukaž P(jedno konkrétní číslo). | [
"Příznivé: 1.",
"Možné: 9.",
"P = 1/9."
] | math | Answer: P = 1/9.
Steps:
1. Příznivé: 1.
2. Možné: 9.
3. P = 1/9. | probability |
Remainder: 1. | null | mat_0809 | worked_example | en | Compute (204) mod 7. | [
"Divide 204 by 7.",
"7×29 = 203.",
"204 − 203 = 1."
] | math | Answer: Remainder: 1.
Steps:
1. Divide 204 by 7.
2. 7×29 = 203.
3. 204 − 203 = 1. | number_theory |
Více předmětů než krabic vynutí sdílení. | null | mat_0810 | concept | cs | Proč Dirichletův princip? | [
"Více předmětů než krabic vynutí sdílení.",
"Odpověď tedy platí: Více předmětů než krabic vynutí sdílení."
] | math | Answer: Více předmětů než krabic vynutí sdílení.
Steps:
1. Více předmětů než krabic vynutí sdílení.
2. Odpověď tedy platí: Více předmětů než krabic vynutí sdílení. | combinatorics |
The value is 110. | {
"A": "108",
"B": "110",
"C": "112",
"D": "114"
} | mat_0811 | multiple_choice | en | Compute N² + N for N = 10. | [
"10² + 10.",
"Value = 110.",
"Select option B."
] | math | Answer: 110
Steps:
1. 10² + 10.
2. Value = 110.
3. Select option B.
Answer: (B) | proofs |
Limit: 34. | null | mat_0812 | worked_example | en | Evaluate the limit of (x² − 17²)/(x − 17) as x → 17. | [
"Factor: (x−17)(x+17).",
"Cancel (x−17).",
"Plug 17: 17+17 = 34."
] | math | Answer: Limit: 34.
Steps:
1. Factor: (x−17)(x+17).
2. Cancel (x−17).
3. Plug 17: 17+17 = 34. | calculus |
The trace sums diagonal entries, equal to the eigenvalue sum and invariant under basis change. | null | mat_0813 | concept | en | What is the trace? | [
"The trace sums diagonal entries, equal to the eigenvalue sum and invariant under basis change.",
"So the answer stands: The trace sums diagonal entries, equal to the eigenvalue sum and invariant under basis change."
] | math | Answer: The trace sums diagonal entries, equal to the eigenvalue sum and invariant under basis change.
Steps:
1. The trace sums diagonal entries, equal to the eigenvalue sum and invariant under basis change.
2. So the answer stands: The trace sums diagonal entries, equal to the eigenvalue sum and invariant under basis ... | linear_algebra |
The expected value is 5.5. | {
"A": "6.5",
"B": "4.5",
"C": "7.5",
"D": "5.5"
} | mat_0814 | multiple_choice | en | Find the expected value of a uniform draw from {1..10}. | [
"(10+1)/2.",
"E = 5.5.",
"Select option D."
] | math | Answer: 5.5
Steps:
1. (10+1)/2.
2. E = 5.5.
3. Select option D.
Answer: (D) | probability |
lcm = 152. | null | mat_0815 | worked_example | en | Compute lcm(19, 8) step by step. | [
"gcd(19,8) = 1.",
"19×8/1.",
"= 152."
] | math | Answer: lcm = 152.
Steps:
1. gcd(19,8) = 1.
2. 19×8/1.
3. = 152. | number_theory |
Permutace řadí objekty, n různých jich má právě n!. | null | mat_0816 | concept | cs | Co je permutace? | [
"Permutace řadí objekty, n různých jich má právě n!.",
"Odpověď tedy platí: Permutace řadí objekty, n různých jich má právě n!."
] | math | Answer: Permutace řadí objekty, n různých jich má právě n!.
Steps:
1. Permutace řadí objekty, n různých jich má právě n!.
2. Odpověď tedy platí: Permutace řadí objekty, n různých jich má právě n!. | combinatorics |
The table has 1024 rows. | {
"A": "1024",
"B": "1022",
"C": "1026",
"D": "1028"
} | mat_0817 | multiple_choice | en | How many rows does a truth table with 10 variables have? | [
"2^10 rows.",
"Total: 1024.",
"Select option A."
] | math | Answer: 1024
Steps:
1. 2^10 rows.
2. Total: 1024.
3. Select option A.
Answer: (A) | proofs |
Divergence measures net outflow of a field at a point, flagging sources and sinks. | null | mat_0818 | concept | en | What is divergence? | [
"Divergence measures net outflow of a field at a point, flagging sources and sinks.",
"So the answer stands: Divergence measures net outflow of a field at a point, flagging sources and sinks."
] | math | Answer: Divergence measures net outflow of a field at a point, flagging sources and sinks.
Steps:
1. Divergence measures net outflow of a field at a point, flagging sources and sinks.
2. So the answer stands: Divergence measures net outflow of a field at a point, flagging sources and sinks. | calculus |
The dot product is 53. | {
"A": "53",
"B": "49",
"C": "57",
"D": "61"
} | mat_0819 | multiple_choice | en | Find the dot product (7,8) · (3,4). | [
"7×3 + 8×4.",
"Product = 53.",
"Select option A."
] | math | Answer: 53
Steps:
1. 7×3 + 8×4.
2. Product = 53.
3. Select option A.
Answer: (A) | linear_algebra |
E = 10.5. | null | mat_0820 | worked_example | en | Compute E of a uniform draw from {1..20}. | [
"Mean of 1..20.",
"(20+1)/2.",
"E = 10.5."
] | math | Answer: E = 10.5.
Steps:
1. Mean of 1..20.
2. (20+1)/2.
3. E = 10.5. | probability |
RSA multiplies primes for public keys, secure because factoring stays hard. | null | mat_0821 | concept | en | What is RSA? | [
"RSA multiplies primes for public keys, secure because factoring stays hard.",
"So the answer stands: RSA multiplies primes for public keys, secure because factoring stays hard."
] | math | Answer: RSA multiplies primes for public keys, secure because factoring stays hard.
Steps:
1. RSA multiplies primes for public keys, secure because factoring stays hard.
2. So the answer stands: RSA multiplies primes for public keys, secure because factoring stays hard. | number_theory |
P(7,3) = 210. | null | mat_0822 | worked_example | cs | Ukaž výpočet P(7,3). | [
"n(n−1)(n−2).",
"7×6×5.",
"= 210."
] | math | Answer: P(7,3) = 210.
Steps:
1. n(n−1)(n−2).
2. 7×6×5.
3. = 210. | combinatorics |
Polovina: 21, číslo 42 je sudé. | null | mat_0823 | worked_example | cs | Ukaž, že N² + N je sudé pro N = 6, výpočtem poloviny. | [
"6² + 6 = 42.",
"42 / 2 = 21.",
"Celé číslo, tedy sudé."
] | math | Answer: Polovina: 21, číslo 42 je sudé.
Steps:
1. 6² + 6 = 42.
2. 42 / 2 = 21.
3. Celé číslo, tedy sudé. | proofs |
f'(2) = 12. | null | mat_0824 | worked_example | cs | Pro f(x) = 2x² + 4x spočítej f'(2) krok za krokem. | [
"f'(x) = 4x + 4.",
"Dosaď x = 2.",
"4×2 + 4 = 12."
] | math | Answer: f'(2) = 12.
Steps:
1. f'(x) = 4x + 4.
2. Dosaď x = 2.
3. 4×2 + 4 = 12. | calculus |
The determinant is 82. | {
"A": "79",
"B": "85",
"C": "88",
"D": "82"
} | mat_0825 | multiple_choice | en | Find the determinant of [[8,2],[3,11]]. | [
"8×11 − 2×3.",
"Det = 82.",
"Select option D."
] | math | Answer: 82
Steps:
1. 8×11 − 2×3.
2. Det = 82.
3. Select option D.
Answer: (D) | linear_algebra |
P(red) is 8/25. | {
"A": "8/25",
"B": "17/25",
"C": "8/26",
"D": "9/25"
} | mat_0826 | multiple_choice | en | A bag holds 8 red and 17 blue marbles. What is P(red)? | [
"8 of 25.",
"P = 8/25.",
"Select option A."
] | math | Answer: 8/25
Steps:
1. 8 of 25.
2. P = 8/25.
3. Select option A.
Answer: (A) | probability |
They invert exponentiation mod p, hard enough to secure key exchange protocols. | null | mat_0827 | concept | en | Why discrete logs? | [
"They invert exponentiation mod p, hard enough to secure key exchange protocols.",
"So the answer stands: They invert exponentiation mod p, hard enough to secure key exchange protocols."
] | math | Answer: They invert exponentiation mod p, hard enough to secure key exchange protocols.
Steps:
1. They invert exponentiation mod p, hard enough to secure key exchange protocols.
2. So the answer stands: They invert exponentiation mod p, hard enough to secure key exchange protocols. | number_theory |
P(18,3) = 4896. | null | mat_0828 | worked_example | en | Show the computation of P(18,3). | [
"n(n−1)(n−2).",
"18×17×16.",
"= 4896."
] | math | Answer: P(18,3) = 4896.
Steps:
1. n(n−1)(n−2).
2. 18×17×16.
3. = 4896. | combinatorics |
Odpověď je svědek či počítání. | {
"A": "svědek či počítání",
"B": "dlouhá jména",
"C": "velké fonty",
"D": "samotná naděje"
} | mat_0829 | multiple_choice | cs | Vyber: co dokazuje existenci? | [
"Sestroj nebo spočítej.",
"Tedy: svědek či počítání.",
"Vyber možnost A."
] | math | Answer: svědek či počítání
Steps:
1. Sestroj nebo spočítej.
2. Tedy: svědek či počítání.
3. Vyber možnost A.
Answer: (A) | proofs |
f'(2) = 24. | null | mat_0830 | worked_example | en | For f(x) = 2x² + 16x, compute f'(2) step by step. | [
"f'(x) = 4x + 16.",
"Plug x = 2.",
"4×2 + 16 = 24."
] | math | Answer: f'(2) = 24.
Steps:
1. f'(x) = 4x + 16.
2. Plug x = 2.
3. 4×2 + 16 = 24. | calculus |
The trace is 21. | {
"A": "21",
"B": "19",
"C": "23",
"D": "25"
} | mat_0831 | multiple_choice | en | Find the trace of [[9,1],[2,12]]. | [
"9 + 12.",
"Trace = 21.",
"Select option A."
] | math | Answer: 21
Steps:
1. 9 + 12.
2. Trace = 21.
3. Select option A.
Answer: (A) | linear_algebra |
The probability is 1/13. | {
"A": "1/12",
"B": "1/14",
"C": "2/13",
"D": "1/13"
} | mat_0832 | multiple_choice | en | A fair 13-sided die is rolled. What is the probability of one specific number? | [
"1 face of 13.",
"P = 1/13.",
"Select option D."
] | math | Answer: 1/13
Steps:
1. 1 face of 13.
2. P = 1/13.
3. Select option D.
Answer: (D) | probability |
sigma(n) sums all divisors, multiplicative over prime powers for easy formulas. | null | mat_0833 | concept | en | What is the sigma function? | [
"sigma(n) sums all divisors, multiplicative over prime powers for easy formulas.",
"So the answer stands: sigma(n) sums all divisors, multiplicative over prime powers for easy formulas."
] | math | Answer: sigma(n) sums all divisors, multiplicative over prime powers for easy formulas.
Steps:
1. sigma(n) sums all divisors, multiplicative over prime powers for easy formulas.
2. So the answer stands: sigma(n) sums all divisors, multiplicative over prime powers for easy formulas. | number_theory |
C(20,3) = 1140. | null | mat_0834 | worked_example | en | Show the computation of C(20,3). | [
"n(n−1)(n−2)/6.",
"20×19×18/6.",
"= 1140."
] | math | Answer: C(20,3) = 1140.
Steps:
1. n(n−1)(n−2)/6.
2. 20×19×18/6.
3. = 1140. | combinatorics |
A tautology is true under every assignment, like P-or-not-P, and truth tables detect it. | null | mat_0835 | concept | en | What is a tautology? | [
"A tautology is true under every assignment, like P-or-not-P, and truth tables detect it.",
"So the answer stands: A tautology is true under every assignment, like P-or-not-P, and truth tables detect it."
] | math | Answer: A tautology is true under every assignment, like P-or-not-P, and truth tables detect it.
Steps:
1. A tautology is true under every assignment, like P-or-not-P, and truth tables detect it.
2. So the answer stands: A tautology is true under every assignment, like P-or-not-P, and truth tables detect it. | proofs |
The derivative is 16. | {
"A": "16",
"B": "14",
"C": "18",
"D": "20"
} | mat_0836 | multiple_choice | en | For f(x) = 3x² + 10x, find f'(1). | [
"f'(x) = 6x + 10.",
"At 1: 16.",
"Select option A."
] | math | Answer: 16
Steps:
1. f'(x) = 6x + 10.
2. At 1: 16.
3. Select option A.
Answer: (A) | calculus |
The dot product is 60. | {
"A": "56",
"B": "60",
"C": "64",
"D": "68"
} | mat_0837 | multiple_choice | en | Find the dot product (8,9) · (3,4). | [
"8×3 + 9×4.",
"Product = 60.",
"Select option B."
] | math | Answer: 60
Steps:
1. 8×3 + 9×4.
2. Product = 60.
3. Select option B.
Answer: (B) | linear_algebra |
The expected value is 6. | {
"A": "6",
"B": "7",
"C": "5",
"D": "8"
} | mat_0838 | multiple_choice | en | Find the expected value of a uniform draw from {1..11}. | [
"(11+1)/2.",
"E = 6.",
"Select option A."
] | math | Answer: 6
Steps:
1. (11+1)/2.
2. E = 6.
3. Select option A.
Answer: (A) | probability |
They split over coprime products, reducing all values to prime-power cases. | null | mat_0839 | concept | en | Why multiplicative functions? | [
"They split over coprime products, reducing all values to prime-power cases.",
"So the answer stands: They split over coprime products, reducing all values to prime-power cases."
] | math | Answer: They split over coprime products, reducing all values to prime-power cases.
Steps:
1. They split over coprime products, reducing all values to prime-power cases.
2. So the answer stands: They split over coprime products, reducing all values to prime-power cases. | number_theory |
C(11,3) is 165. | {
"A": "165",
"B": "154",
"C": "176",
"D": "187"
} | mat_0840 | multiple_choice | en | Compute C(11,3). | [
"11×10×9/6.",
"= 165.",
"Select option A."
] | math | Answer: 165
Steps:
1. 11×10×9/6.
2. = 165.
3. Select option A.
Answer: (A) | combinatorics |
The sum is 91. | {
"A": "78",
"B": "91",
"C": "104",
"D": "117"
} | mat_0841 | multiple_choice | en | What is the sum 1 + 2 + ... + 13? | [
"13×14/2.",
"Sum = 91.",
"Select option B."
] | math | Answer: 91
Steps:
1. 13×14/2.
2. Sum = 91.
3. Select option B.
Answer: (B) | proofs |
Integral: 5832. | null | mat_0842 | worked_example | en | Evaluate the integral of 3x² from 0 to 18. | [
"Antiderivative: x³.",
"Evaluate 0 to 18.",
"18³ − 0 = 5832."
] | math | Answer: Integral: 5832.
Steps:
1. Antiderivative: x³.
2. Evaluate 0 to 18.
3. 18³ − 0 = 5832. | calculus |
Symmetric matrices have real eigenvalues and orthogonal eigenvectors, ideal for geometry. | null | mat_0843 | concept | en | Why symmetric matrices? | [
"Symmetric matrices have real eigenvalues and orthogonal eigenvectors, ideal for geometry.",
"So the answer stands: Symmetric matrices have real eigenvalues and orthogonal eigenvectors, ideal for geometry."
] | math | Answer: Symmetric matrices have real eigenvalues and orthogonal eigenvectors, ideal for geometry.
Steps:
1. Symmetric matrices have real eigenvalues and orthogonal eigenvectors, ideal for geometry.
2. So the answer stands: Symmetric matrices have real eigenvalues and orthogonal eigenvectors, ideal for geometry. | linear_algebra |
P(red) is 9/28. | {
"A": "19/28",
"B": "9/28",
"C": "9/29",
"D": "10/28"
} | mat_0844 | multiple_choice | en | A bag holds 9 red and 19 blue marbles. What is P(red)? | [
"9 of 28.",
"P = 9/28.",
"Select option B."
] | math | Answer: 9/28
Steps:
1. 9 of 28.
2. P = 9/28.
3. Select option B.
Answer: (B) | probability |
The lcm is 66. | {
"A": "64",
"B": "66",
"C": "68",
"D": "70"
} | mat_0845 | multiple_choice | en | Find lcm(11, 6). | [
"11×6/gcd.",
"lcm = 66.",
"Select option B."
] | math | Answer: 66
Steps:
1. 11×6/gcd.
2. lcm = 66.
3. Select option B.
Answer: (B) | number_theory |
C(11,2) is 55. | {
"A": "44",
"B": "66",
"C": "77",
"D": "55"
} | mat_0846 | multiple_choice | en | Compute C(11,2). | [
"11×10/2.",
"= 55.",
"Select option D."
] | math | Answer: 55
Steps:
1. 11×10/2.
2. = 55.
3. Select option D.
Answer: (D) | combinatorics |
Rows: 131072. | null | mat_0847 | worked_example | en | A proof checks all cases of 17 boolean variables. How many rows? | [
"2 choices each.",
"2^17 = 131072."
] | math | Answer: Rows: 131072.
Steps:
1. 2 choices each.
2. 2^17 = 131072. | proofs |
Odpověď je l'Hôpitalovo pravidlo. | {
"A": "l'Hôpitalovo pravidlo",
"B": "hádání",
"C": "ignorování",
"D": "mazání"
} | mat_0848 | multiple_choice | cs | Vyber: co řeší 0/0? | [
"Derivuj čitatel i jmenovatel.",
"Tedy: l'Hôpitalovo pravidlo.",
"Vyber možnost A."
] | math | Answer: l'Hôpitalovo pravidlo
Steps:
1. Derivuj čitatel i jmenovatel.
2. Tedy: l'Hôpitalovo pravidlo.
3. Vyber možnost A.
Answer: (A) | calculus |
Positive definite matrices give positive energy for all nonzero vectors, guaranteeing minima. | null | mat_0849 | concept | en | What is positive definite? | [
"Positive definite matrices give positive energy for all nonzero vectors, guaranteeing minima.",
"So the answer stands: Positive definite matrices give positive energy for all nonzero vectors, guaranteeing minima."
] | math | Answer: Positive definite matrices give positive energy for all nonzero vectors, guaranteeing minima.
Steps:
1. Positive definite matrices give positive energy for all nonzero vectors, guaranteeing minima.
2. So the answer stands: Positive definite matrices give positive energy for all nonzero vectors, guaranteeing min... | linear_algebra |
The probability is 1/14. | {
"A": "1/14",
"B": "1/13",
"C": "1/15",
"D": "2/14"
} | mat_0850 | multiple_choice | en | A fair 14-sided die is rolled. What is the probability of one specific number? | [
"1 face of 14.",
"P = 1/14.",
"Select option A."
] | math | Answer: 1/14
Steps:
1. 1 face of 14.
2. P = 1/14.
3. Select option A.
Answer: (A) | probability |
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