answer string | choices dict | id string | kind string | lang string | question string | steps list | subject string | text string | topic string |
|---|---|---|---|---|---|---|---|---|---|
The lcm is 6. | {
"A": "6",
"B": "4",
"C": "8",
"D": "10"
} | mat_0251 | multiple_choice | en | Find lcm(6, 6). | [
"6×6/gcd.",
"lcm = 6.",
"Select option A."
] | math | Answer: 6
Steps:
1. 6×6/gcd.
2. lcm = 6.
3. Select option A.
Answer: (A) | number_theory |
C(10,3) = 120. | null | mat_0252 | worked_example | en | Show the computation of C(10,3). | [
"n(n−1)(n−2)/6.",
"10×9×8/6.",
"= 120."
] | math | Answer: C(10,3) = 120.
Steps:
1. n(n−1)(n−2)/6.
2. 10×9×8/6.
3. = 120. | combinatorics |
Half: 21, so 42 is even. | null | mat_0253 | worked_example | en | Show N² + N is even for N = 6 by computing its half. | [
"6² + 6 = 42.",
"42 / 2 = 21.",
"Integer, so even."
] | math | Answer: Half: 21, so 42 is even.
Steps:
1. 6² + 6 = 42.
2. 42 / 2 = 21.
3. Integer, so even. | proofs |
Limita: 6. | null | mat_0254 | worked_example | cs | Vyčísli limitu (x² − 3²)/(x − 3) pro x → 3. | [
"Rozlož: (x−3)(x+3).",
"Pokrať (x−3).",
"Dosaď 3: 3+3 = 6."
] | math | Answer: Limita: 6.
Steps:
1. Rozlož: (x−3)(x+3).
2. Pokrať (x−3).
3. Dosaď 3: 3+3 = 6. | calculus |
The determinant is 12. | {
"A": "9",
"B": "15",
"C": "12",
"D": "18"
} | mat_0255 | multiple_choice | en | Find the determinant of [[3,2],[3,6]]. | [
"3×6 − 2×3.",
"Det = 12.",
"Select option C."
] | math | Answer: 12
Steps:
1. 3×6 − 2×3.
2. Det = 12.
3. Select option C.
Answer: (C) | linear_algebra |
Bayes obrací podmíněnosti a usuzuje příčiny z následků. | null | mat_0256 | concept | cs | Proč Bayesova věta? | [
"Bayes obrací podmíněnosti a usuzuje příčiny z následků.",
"Odpověď tedy platí: Bayes obrací podmíněnosti a usuzuje příčiny z následků."
] | math | Answer: Bayes obrací podmíněnosti a usuzuje příčiny z následků.
Steps:
1. Bayes obrací podmíněnosti a usuzuje příčiny z následků.
2. Odpověď tedy platí: Bayes obrací podmíněnosti a usuzuje příčiny z následků. | probability |
NSN je 12. | {
"A": "10",
"B": "14",
"C": "12",
"D": "16"
} | mat_0257 | multiple_choice | cs | Najdi nejmenší společný násobek NSN(4, 6). | [
"4×6/NSD.",
"NSN = 12.",
"Vyber možnost C."
] | math | Answer: 12
Steps:
1. 4×6/NSD.
2. NSN = 12.
3. Vyber možnost C.
Answer: (C) | number_theory |
C(10,2) = 45. | null | mat_0258 | worked_example | en | Show the computation of C(10,2). | [
"Formula n(n−1)/2.",
"10×9/2.",
"= 45."
] | math | Answer: C(10,2) = 45.
Steps:
1. Formula n(n−1)/2.
2. 10×9/2.
3. = 45. | combinatorics |
Součet: 28. | null | mat_0259 | worked_example | cs | Dokaž vzorcem: najdi součet 1 + 2 + ... + 7. | [
"Páruj 1+7, 2+6, ...",
"Použij 7×8/2.",
"= 28."
] | math | Answer: Součet: 28.
Steps:
1. Páruj 1+7, 2+6, ...
2. Použij 7×8/2.
3. = 28. | proofs |
Integration recovers totals from rates, computing areas, volumes, and accumulated change. | null | mat_0260 | concept | en | Why integrate? | [
"Integration recovers totals from rates, computing areas, volumes, and accumulated change.",
"So the answer stands: Integration recovers totals from rates, computing areas, volumes, and accumulated change."
] | math | Answer: Integration recovers totals from rates, computing areas, volumes, and accumulated change.
Steps:
1. Integration recovers totals from rates, computing areas, volumes, and accumulated change.
2. So the answer stands: Integration recovers totals from rates, computing areas, volumes, and accumulated change. | calculus |
Trace: 15. | null | mat_0261 | worked_example | en | Compute the trace of [[7,0],[0,8]]. | [
"Sum diagonal.",
"7 + 8.",
"= 15."
] | math | Answer: Trace: 15.
Steps:
1. Sum diagonal.
2. 7 + 8.
3. = 15. | linear_algebra |
The probability is 1/8. | {
"A": "1/7",
"B": "1/9",
"C": "1/8",
"D": "2/8"
} | mat_0262 | multiple_choice | en | A fair 8-sided die is rolled. What is the probability of one specific number? | [
"1 face of 8.",
"P = 1/8.",
"Select option C."
] | math | Answer: 1/8
Steps:
1. 1 face of 8.
2. P = 1/8.
3. Select option C.
Answer: (C) | probability |
NSN = 24. | null | mat_0263 | worked_example | cs | Spočítej NSN(6, 8) krok za krokem. | [
"NSD(6,8) = 2.",
"6×8/2.",
"= 24."
] | math | Answer: NSN = 24.
Steps:
1. NSD(6,8) = 2.
2. 6×8/2.
3. = 24. | number_theory |
Derangements count fixed-point-free orders, modelling secret Santas and mismatches. | null | mat_0264 | concept | en | Why derangements? | [
"Derangements count fixed-point-free orders, modelling secret Santas and mismatches.",
"So the answer stands: Derangements count fixed-point-free orders, modelling secret Santas and mismatches."
] | math | Answer: Derangements count fixed-point-free orders, modelling secret Santas and mismatches.
Steps:
1. Derangements count fixed-point-free orders, modelling secret Santas and mismatches.
2. So the answer stands: Derangements count fixed-point-free orders, modelling secret Santas and mismatches. | combinatorics |
The sum is 28. | {
"A": "21",
"B": "35",
"C": "42",
"D": "28"
} | mat_0265 | multiple_choice | en | What is the sum 1 + 2 + ... + 7? | [
"7×8/2.",
"Sum = 28.",
"Select option D."
] | math | Answer: 28
Steps:
1. 7×8/2.
2. Sum = 28.
3. Select option D.
Answer: (D) | proofs |
Integral: 512. | null | mat_0266 | worked_example | en | Evaluate the integral of 3x² from 0 to 8. | [
"Antiderivative: x³.",
"Evaluate 0 to 8.",
"8³ − 0 = 512."
] | math | Answer: Integral: 512.
Steps:
1. Antiderivative: x³.
2. Evaluate 0 to 8.
3. 8³ − 0 = 512. | calculus |
The trace is 11. | {
"A": "9",
"B": "13",
"C": "15",
"D": "11"
} | mat_0267 | multiple_choice | en | Find the trace of [[4,1],[2,7]]. | [
"4 + 7.",
"Trace = 11.",
"Select option D."
] | math | Answer: 11
Steps:
1. 4 + 7.
2. Trace = 11.
3. Select option D.
Answer: (D) | linear_algebra |
P = 1/6. | null | mat_0268 | worked_example | cs | Hází se férovou 6-stěnnou kostkou. Ukaž P(jedno konkrétní číslo). | [
"Příznivé: 1.",
"Možné: 6.",
"P = 1/6."
] | math | Answer: P = 1/6.
Steps:
1. Příznivé: 1.
2. Možné: 6.
3. P = 1/6. | probability |
Každé číslo nad 1 se jednoznačně rozkládá na prvočísla. | null | mat_0269 | concept | cs | Co je základní věta aritmetiky? | [
"Každé číslo nad 1 se jednoznačně rozkládá na prvočísla.",
"Odpověď tedy platí: Každé číslo nad 1 se jednoznačně rozkládá na prvočísla."
] | math | Answer: Každé číslo nad 1 se jednoznačně rozkládá na prvočísla.
Steps:
1. Každé číslo nad 1 se jednoznačně rozkládá na prvočísla.
2. Odpověď tedy platí: Každé číslo nad 1 se jednoznačně rozkládá na prvočísla. | number_theory |
C(6,2) is 15. | {
"A": "9",
"B": "21",
"C": "15",
"D": "27"
} | mat_0270 | multiple_choice | en | Compute C(6,2). | [
"6×5/2.",
"= 15.",
"Select option C."
] | math | Answer: 15
Steps:
1. 6×5/2.
2. = 15.
3. Select option C.
Answer: (C) | combinatorics |
The value is 30. | {
"A": "30",
"B": "28",
"C": "32",
"D": "34"
} | mat_0271 | multiple_choice | en | Compute N² + N for N = 5. | [
"5² + 5.",
"Value = 30.",
"Select option A."
] | math | Answer: 30
Steps:
1. 5² + 5.
2. Value = 30.
3. Select option A.
Answer: (A) | proofs |
It links derivatives and integrals, so definite integrals equal antiderivative differences. | null | mat_0272 | concept | en | What is the fundamental theorem of calculus? | [
"It links derivatives and integrals, so definite integrals equal antiderivative differences.",
"So the answer stands: It links derivatives and integrals, so definite integrals equal antiderivative differences."
] | math | Answer: It links derivatives and integrals, so definite integrals equal antiderivative differences.
Steps:
1. It links derivatives and integrals, so definite integrals equal antiderivative differences.
2. So the answer stands: It links derivatives and integrals, so definite integrals equal antiderivative differences. | calculus |
Dot product: 43. | null | mat_0273 | worked_example | en | Compute (6,5) · (3,5) step by step. | [
"Multiply components.",
"6×3 + 5×5.",
"= 43."
] | math | Answer: Dot product: 43.
Steps:
1. Multiply components.
2. 6×3 + 5×5.
3. = 43. | linear_algebra |
Nezávislé jevy násobí pravděpodobnosti, výsledek jednoho neovlivní druhý. | null | mat_0274 | concept | cs | Co je nezávislost? | [
"Nezávislé jevy násobí pravděpodobnosti, výsledek jednoho neovlivní druhý.",
"Odpověď tedy platí: Nezávislé jevy násobí pravděpodobnosti, výsledek jednoho neovlivní druhý."
] | math | Answer: Nezávislé jevy násobí pravděpodobnosti, výsledek jednoho neovlivní druhý.
Steps:
1. Nezávislé jevy násobí pravděpodobnosti, výsledek jednoho neovlivní druhý.
2. Odpověď tedy platí: Nezávislé jevy násobí pravděpodobnosti, výsledek jednoho neovlivní druhý. | probability |
Remainder: 6. | null | mat_0275 | worked_example | en | Compute (83) mod 7. | [
"Divide 83 by 7.",
"7×11 = 77.",
"83 − 77 = 6."
] | math | Answer: Remainder: 6.
Steps:
1. Divide 83 by 7.
2. 7×11 = 77.
3. 83 − 77 = 6. | number_theory |
P(5,2) is 20. | {
"A": "15",
"B": "25",
"C": "30",
"D": "20"
} | mat_0276 | multiple_choice | en | Compute P(5,2). | [
"5×4.",
"= 20.",
"Select option D."
] | math | Answer: 20
Steps:
1. 5×4.
2. = 20.
3. Select option D.
Answer: (D) | combinatorics |
Rows: 128. | null | mat_0277 | worked_example | en | A proof checks all cases of 7 boolean variables. How many rows? | [
"2 choices each.",
"2^7 = 128."
] | math | Answer: Rows: 128.
Steps:
1. 2 choices each.
2. 2^7 = 128. | proofs |
Substitution rewrites messy integrals in simpler variables, undoing the chain rule. | null | mat_0278 | concept | en | Why substitute in integrals? | [
"Substitution rewrites messy integrals in simpler variables, undoing the chain rule.",
"So the answer stands: Substitution rewrites messy integrals in simpler variables, undoing the chain rule."
] | math | Answer: Substitution rewrites messy integrals in simpler variables, undoing the chain rule.
Steps:
1. Substitution rewrites messy integrals in simpler variables, undoing the chain rule.
2. So the answer stands: Substitution rewrites messy integrals in simpler variables, undoing the chain rule. | calculus |
Determinant: 54. | null | mat_0279 | worked_example | en | Compute det [[7,1],[2,8]] step by step. | [
"ad − bc.",
"7×8 − 1×2.",
"= 54."
] | math | Answer: Determinant: 54.
Steps:
1. ad − bc.
2. 7×8 − 1×2.
3. = 54. | linear_algebra |
P(red) = 7/11. | null | mat_0280 | worked_example | en | A bag holds 7 red and 4 blue marbles. Show P(red). | [
"Red: 7.",
"Total: 11.",
"P = 7/11."
] | math | Answer: P(red) = 7/11.
Steps:
1. Red: 7.
2. Total: 11.
3. P = 7/11. | probability |
lcm = 8. | null | mat_0281 | worked_example | en | Compute lcm(8, 8) step by step. | [
"gcd(8,8) = 8.",
"8×8/8.",
"= 8."
] | math | Answer: lcm = 8.
Steps:
1. gcd(8,8) = 8.
2. 8×8/8.
3. = 8. | number_theory |
A combination selects without order, counted by nCr for r from n. | null | mat_0282 | concept | en | What is a combination? | [
"A combination selects without order, counted by nCr for r from n.",
"So the answer stands: A combination selects without order, counted by nCr for r from n."
] | math | Answer: A combination selects without order, counted by nCr for r from n.
Steps:
1. A combination selects without order, counted by nCr for r from n.
2. So the answer stands: A combination selects without order, counted by nCr for r from n. | combinatorics |
The answer is Claim is false. | {
"A": "Claim is true",
"B": "Claim is false",
"C": "Nothing at all",
"D": "Two claims"
} | mat_0283 | multiple_choice | en | Choose: what does contradiction assume? | [
"Absurdity follows.",
"So: Claim is false.",
"Select option B."
] | math | Answer: Claim is false
Steps:
1. Absurdity follows.
2. So: Claim is false.
3. Select option B.
Answer: (B) | proofs |
It trades one hard integral for an easier one via uv minus the integral of v du. | null | mat_0284 | concept | en | What is integration by parts? | [
"It trades one hard integral for an easier one via uv minus the integral of v du.",
"So the answer stands: It trades one hard integral for an easier one via uv minus the integral of v du."
] | math | Answer: It trades one hard integral for an easier one via uv minus the integral of v du.
Steps:
1. It trades one hard integral for an easier one via uv minus the integral of v du.
2. So the answer stands: It trades one hard integral for an easier one via uv minus the integral of v du. | calculus |
Trace: 17. | null | mat_0285 | worked_example | en | Compute the trace of [[8,0],[0,9]]. | [
"Sum diagonal.",
"8 + 9.",
"= 17."
] | math | Answer: Trace: 17.
Steps:
1. Sum diagonal.
2. 8 + 9.
3. = 17. | linear_algebra |
E = 3. | null | mat_0286 | worked_example | cs | Spočítej E rovnoměrného tahu z {1..5}. | [
"Průměr 1..5.",
"(5+1)/2.",
"E = 3."
] | math | Answer: E = 3.
Steps:
1. Průměr 1..5.
2. (5+1)/2.
3. E = 3. | probability |
For prime p, a^(p-1) is 1 mod p, powering fast primality checks and inverses. | null | mat_0287 | concept | en | What is Fermat's little theorem? | [
"For prime p, a^(p-1) is 1 mod p, powering fast primality checks and inverses.",
"So the answer stands: For prime p, a^(p-1) is 1 mod p, powering fast primality checks and inverses."
] | math | Answer: For prime p, a^(p-1) is 1 mod p, powering fast primality checks and inverses.
Steps:
1. For prime p, a^(p-1) is 1 mod p, powering fast primality checks and inverses.
2. So the answer stands: For prime p, a^(p-1) is 1 mod p, powering fast primality checks and inverses. | number_theory |
P(9,3) = 504. | null | mat_0288 | worked_example | en | Show the computation of P(9,3). | [
"n(n−1)(n−2).",
"9×8×7.",
"= 504."
] | math | Answer: P(9,3) = 504.
Steps:
1. n(n−1)(n−2).
2. 9×8×7.
3. = 504. | combinatorics |
The answer is Helper claim. | {
"A": "Long book",
"B": "Final answer",
"C": "Helper claim",
"D": "New axiom"
} | mat_0289 | multiple_choice | en | Choose: what is a lemma? | [
"Main proof uses it.",
"So: Helper claim.",
"Select option C."
] | math | Answer: Helper claim
Steps:
1. Main proof uses it.
2. So: Helper claim.
3. Select option C.
Answer: (C) | proofs |
Limit: 16. | null | mat_0290 | worked_example | en | Evaluate the limit of (x² − 8²)/(x − 8) as x → 8. | [
"Factor: (x−8)(x+8).",
"Cancel (x−8).",
"Plug 8: 8+8 = 16."
] | math | Answer: Limit: 16.
Steps:
1. Factor: (x−8)(x+8).
2. Cancel (x−8).
3. Plug 8: 8+8 = 16. | calculus |
The dot product is 25. | {
"A": "25",
"B": "21",
"C": "29",
"D": "33"
} | mat_0291 | multiple_choice | en | Find the dot product (3,4) · (3,4). | [
"3×3 + 4×4.",
"Product = 25.",
"Select option A."
] | math | Answer: 25
Steps:
1. 3×3 + 4×4.
2. Product = 25.
3. Select option A.
Answer: (A) | linear_algebra |
P = 1/11. | null | mat_0292 | worked_example | en | A fair 11-sided die is rolled. Show P(one specific number). | [
"Favourable: 1.",
"Possible: 11.",
"P = 1/11."
] | math | Answer: P = 1/11.
Steps:
1. Favourable: 1.
2. Possible: 11.
3. P = 1/11. | probability |
The answer is Remainders. | {
"A": "Guesses",
"B": "Remainders",
"C": "Hopes",
"D": "Dreams"
} | mat_0293 | multiple_choice | en | Choose: what shrinks gcd pairs? | [
"Euclid runs fast.",
"So: Remainders.",
"Select option B."
] | math | Answer: Remainders
Steps:
1. Euclid runs fast.
2. So: Remainders.
3. Select option B.
Answer: (B) | number_theory |
They encode sequences as series coefficients, turning recurrences into algebra. | null | mat_0294 | concept | en | Why generating functions? | [
"They encode sequences as series coefficients, turning recurrences into algebra.",
"So the answer stands: They encode sequences as series coefficients, turning recurrences into algebra."
] | math | Answer: They encode sequences as series coefficients, turning recurrences into algebra.
Steps:
1. They encode sequences as series coefficients, turning recurrences into algebra.
2. So the answer stands: They encode sequences as series coefficients, turning recurrences into algebra. | combinatorics |
Sum: 66. | null | mat_0295 | worked_example | en | Prove by formula: find the sum 1 + 2 + ... + 11. | [
"Pair 1+11, 2+10, ...",
"Use 11×12/2.",
"= 66."
] | math | Answer: Sum: 66.
Steps:
1. Pair 1+11, 2+10, ...
2. Use 11×12/2.
3. = 66. | proofs |
The answer is L'Hôpital's rule. | {
"A": "L'Hôpital's rule",
"B": "Guessing",
"C": "Ignoring",
"D": "Deleting"
} | mat_0296 | multiple_choice | en | Choose: what resolves 0/0? | [
"Differentiate top, bottom.",
"So: L'Hôpital's rule.",
"Select option A."
] | math | Answer: L'Hôpital's rule
Steps:
1. Differentiate top, bottom.
2. So: L'Hôpital's rule.
3. Select option A.
Answer: (A) | calculus |
Skládá zobrazení přes skalární součiny řádků a sloupců. | null | mat_0297 | concept | cs | Co je násobení matic? | [
"Skládá zobrazení přes skalární součiny řádků a sloupců.",
"Odpověď tedy platí: Skládá zobrazení přes skalární součiny řádků a sloupců."
] | math | Answer: Skládá zobrazení přes skalární součiny řádků a sloupců.
Steps:
1. Skládá zobrazení přes skalární součiny řádků a sloupců.
2. Odpověď tedy platí: Skládá zobrazení přes skalární součiny řádků a sloupců. | linear_algebra |
P(červená) = 3/7. | null | mat_0298 | worked_example | cs | Sáček má 3 červených a 4 modré kuličky. Ukaž P(červená). | [
"Červené: 3.",
"Celkem: 7.",
"P = 3/7."
] | math | Answer: P(červená) = 3/7.
Steps:
1. Červené: 3.
2. Celkem: 7.
3. P = 3/7. | probability |
It generalizes Fermat via totient, so a^phi(n) is 1 mod n for coprime pairs. | null | mat_0299 | concept | en | Why Euler's theorem? | [
"It generalizes Fermat via totient, so a^phi(n) is 1 mod n for coprime pairs.",
"So the answer stands: It generalizes Fermat via totient, so a^phi(n) is 1 mod n for coprime pairs."
] | math | Answer: It generalizes Fermat via totient, so a^phi(n) is 1 mod n for coprime pairs.
Steps:
1. It generalizes Fermat via totient, so a^phi(n) is 1 mod n for coprime pairs.
2. So the answer stands: It generalizes Fermat via totient, so a^phi(n) is 1 mod n for coprime pairs. | number_theory |
C(11,3) = 165. | null | mat_0300 | worked_example | en | Show the computation of C(11,3). | [
"n(n−1)(n−2)/6.",
"11×10×9/6.",
"= 165."
] | math | Answer: C(11,3) = 165.
Steps:
1. n(n−1)(n−2)/6.
2. 11×10×9/6.
3. = 165. | combinatorics |
An existence proof shows some object has a property, either by building one or by counting arguments. | null | mat_0301 | concept | en | What is an existence proof? | [
"An existence proof shows some object has a property, either by building one or by counting arguments.",
"So the answer stands: An existence proof shows some object has a property, either by building one or by counting arguments."
] | math | Answer: An existence proof shows some object has a property, either by building one or by counting arguments.
Steps:
1. An existence proof shows some object has a property, either by building one or by counting arguments.
2. So the answer stands: An existence proof shows some object has a property, either by building o... | proofs |
Series sum infinite terms, and convergence tests tell whether the total is finite. | null | mat_0302 | concept | en | Why study series? | [
"Series sum infinite terms, and convergence tests tell whether the total is finite.",
"So the answer stands: Series sum infinite terms, and convergence tests tell whether the total is finite."
] | math | Answer: Series sum infinite terms, and convergence tests tell whether the total is finite.
Steps:
1. Series sum infinite terms, and convergence tests tell whether the total is finite.
2. So the answer stands: Series sum infinite terms, and convergence tests tell whether the total is finite. | calculus |
A subspace is a subset closed under the operations, inheriting the vector-space structure. | null | mat_0303 | concept | en | What is a subspace? | [
"A subspace is a subset closed under the operations, inheriting the vector-space structure.",
"So the answer stands: A subspace is a subset closed under the operations, inheriting the vector-space structure."
] | math | Answer: A subspace is a subset closed under the operations, inheriting the vector-space structure.
Steps:
1. A subspace is a subset closed under the operations, inheriting the vector-space structure.
2. So the answer stands: A subspace is a subset closed under the operations, inheriting the vector-space structure. | linear_algebra |
It says standardized sums approach normal, explaining the bell curve everywhere. | null | mat_0304 | concept | en | Why central limit theorem? | [
"It says standardized sums approach normal, explaining the bell curve everywhere.",
"So the answer stands: It says standardized sums approach normal, explaining the bell curve everywhere."
] | math | Answer: It says standardized sums approach normal, explaining the bell curve everywhere.
Steps:
1. It says standardized sums approach normal, explaining the bell curve everywhere.
2. So the answer stands: It says standardized sums approach normal, explaining the bell curve everywhere. | probability |
gcd = 12. | null | mat_0305 | worked_example | en | Compute gcd(24, 36) step by step. | [
"Factor: 24=4×6, 36=6×6.",
"Common: 6×gcd(4,6).",
"= 6×2 = 12."
] | math | Answer: gcd = 12.
Steps:
1. Factor: 24=4×6, 36=6×6.
2. Common: 6×gcd(4,6).
3. = 6×2 = 12. | number_theory |
C(7,3) is 35. | {
"A": "35",
"B": "28",
"C": "42",
"D": "49"
} | mat_0306 | multiple_choice | en | Compute C(7,3). | [
"7×6×5/6.",
"= 35.",
"Select option A."
] | math | Answer: 35
Steps:
1. 7×6×5/6.
2. = 35.
3. Select option A.
Answer: (A) | combinatorics |
Tvrzení pro všechna a existuje se dokazují různě, kvantifikátor volí strategii. | null | mat_0307 | concept | cs | Proč uvádět kvantifikátory? | [
"Tvrzení pro všechna a existuje se dokazují různě, kvantifikátor volí strategii.",
"Odpověď tedy platí: Tvrzení pro všechna a existuje se dokazují různě, kvantifikátor volí strategii."
] | math | Answer: Tvrzení pro všechna a existuje se dokazují různě, kvantifikátor volí strategii.
Steps:
1. Tvrzení pro všechna a existuje se dokazují různě, kvantifikátor volí strategii.
2. Odpověď tedy platí: Tvrzení pro všechna a existuje se dokazují různě, kvantifikátor volí strategii. | proofs |
f'(2) = 15. | null | mat_0308 | worked_example | en | For f(x) = 2x² + 7x, compute f'(2) step by step. | [
"f'(x) = 4x + 7.",
"Plug x = 2.",
"4×2 + 7 = 15."
] | math | Answer: f'(2) = 15.
Steps:
1. f'(x) = 4x + 7.
2. Plug x = 2.
3. 4×2 + 7 = 15. | calculus |
Dot product: 51. | null | mat_0309 | worked_example | en | Compute (7,6) · (3,5) step by step. | [
"Multiply components.",
"7×3 + 6×5.",
"= 51."
] | math | Answer: Dot product: 51.
Steps:
1. Multiply components.
2. 7×3 + 6×5.
3. = 51. | linear_algebra |
The expected value is 3.5. | {
"A": "4.5",
"B": "2.5",
"C": "5.5",
"D": "3.5"
} | mat_0310 | multiple_choice | en | Find the expected value of a uniform draw from {1..6}. | [
"(6+1)/2.",
"E = 3.5.",
"Select option D."
] | math | Answer: 3.5
Steps:
1. (6+1)/2.
2. E = 3.5.
3. Select option D.
Answer: (D) | probability |
Euklides vynásobí známá a přičte jedna, vynutí nový prvočinitel. | null | mat_0311 | concept | cs | Proč nekonečně prvočísel? | [
"Euklides vynásobí známá a přičte jedna, vynutí nový prvočinitel.",
"Odpověď tedy platí: Euklides vynásobí známá a přičte jedna, vynutí nový prvočinitel."
] | math | Answer: Euklides vynásobí známá a přičte jedna, vynutí nový prvočinitel.
Steps:
1. Euklides vynásobí známá a přičte jedna, vynutí nový prvočinitel.
2. Odpověď tedy platí: Euklides vynásobí známá a přičte jedna, vynutí nový prvočinitel. | number_theory |
Odpověď je faktoriál. | {
"A": "součet",
"B": "průměr",
"C": "hádání",
"D": "faktoriál"
} | mat_0312 | multiple_choice | cs | Vyber: co počítá pořadí? | [
"Volby na pozici ubývají.",
"Tedy: faktoriál.",
"Vyber možnost D."
] | math | Answer: faktoriál
Steps:
1. Volby na pozici ubývají.
2. Tedy: faktoriál.
3. Vyber možnost D.
Answer: (D) | combinatorics |
Half: 28, so 56 is even. | null | mat_0313 | worked_example | en | Show N² + N is even for N = 7 by computing its half. | [
"7² + 7 = 56.",
"56 / 2 = 28.",
"Integer, so even."
] | math | Answer: Half: 28, so 56 is even.
Steps:
1. 7² + 7 = 56.
2. 56 / 2 = 28.
3. Integer, so even. | proofs |
Limita je 4. | {
"A": "2",
"B": "6",
"C": "4",
"D": "8"
} | mat_0314 | multiple_choice | cs | Vypočítej limitu (x² − 2²)/(x − 2) pro x → 2. | [
"Rozlož (x−2)(x+2).",
"Pokrať, dostaneš 2×2 = 4.",
"Vyber možnost C."
] | math | Answer: 4
Steps:
1. Rozlož (x−2)(x+2).
2. Pokrať, dostaneš 2×2 = 4.
3. Vyber možnost C.
Answer: (C) | calculus |
Determinant je -2. | {
"A": "-2",
"B": "-5",
"C": "1",
"D": "4"
} | mat_0315 | multiple_choice | cs | Vypočítej determinant matice [[1,2],[3,4]]. | [
"1×4 − 2×3.",
"Det = -2.",
"Vyber možnost A."
] | math | Answer: -2
Steps:
1. 1×4 − 2×3.
2. Det = -2.
3. Vyber možnost A.
Answer: (A) | linear_algebra |
E = 5.5. | null | mat_0316 | worked_example | en | Compute E of a uniform draw from {1..10}. | [
"Mean of 1..10.",
"(10+1)/2.",
"E = 5.5."
] | math | Answer: E = 5.5.
Steps:
1. Mean of 1..10.
2. (10+1)/2.
3. E = 5.5. | probability |
NSD = 8. | null | mat_0317 | worked_example | cs | Spočítej NSD(16, 24) krok za krokem. | [
"Rozlož: 16=4×4, 24=6×4.",
"Společné: 4×NSD(4,6).",
"= 4×2 = 8."
] | math | Answer: NSD = 8.
Steps:
1. Rozlož: 16=4×4, 24=6×4.
2. Společné: 4×NSD(4,6).
3. = 4×2 = 8. | number_theory |
C(6,2) = 15. | null | mat_0318 | worked_example | cs | Ukaž výpočet C(6,2). | [
"Vzorec n(n−1)/2.",
"6×5/2.",
"= 15."
] | math | Answer: C(6,2) = 15.
Steps:
1. Vzorec n(n−1)/2.
2. 6×5/2.
3. = 15. | combinatorics |
Polovina: 6, číslo 12 je sudé. | null | mat_0319 | worked_example | cs | Ukaž, že N² + N je sudé pro N = 3, výpočtem poloviny. | [
"3² + 3 = 12.",
"12 / 2 = 6.",
"Celé číslo, tedy sudé."
] | math | Answer: Polovina: 6, číslo 12 je sudé.
Steps:
1. 3² + 3 = 12.
2. 12 / 2 = 6.
3. Celé číslo, tedy sudé. | proofs |
The answer is Endpoints too. | {
"A": "Nowhere else",
"B": "Endpoints too",
"C": "Only zero",
"D": "Far away"
} | mat_0320 | multiple_choice | en | Choose: where can maxima hide? | [
"Check ends always.",
"So: Endpoints too.",
"Select option B."
] | math | Answer: Endpoints too
Steps:
1. Check ends always.
2. So: Endpoints too.
3. Select option B.
Answer: (B) | calculus |
Determinant: 70. | null | mat_0321 | worked_example | en | Compute det [[8,1],[2,9]] step by step. | [
"ad − bc.",
"8×9 − 1×2.",
"= 70."
] | math | Answer: Determinant: 70.
Steps:
1. ad − bc.
2. 8×9 − 1×2.
3. = 70. | linear_algebra |
P(red) is 4/13. | {
"A": "4/13",
"B": "9/13",
"C": "4/14",
"D": "5/13"
} | mat_0322 | multiple_choice | en | A bag holds 4 red and 9 blue marbles. What is P(red)? | [
"4 of 13.",
"P = 4/13.",
"Select option A."
] | math | Answer: 4/13
Steps:
1. 4 of 13.
2. P = 4/13.
3. Select option A.
Answer: (A) | probability |
The answer is Eratosthenes sieve. | {
"A": "Random picks",
"B": "Lucky dips",
"C": "Eratosthenes sieve",
"D": "Slow walks"
} | mat_0323 | multiple_choice | en | Choose: what lists primes fast? | [
"Multiples crossed out.",
"So: Eratosthenes sieve.",
"Select option C."
] | math | Answer: Eratosthenes sieve
Steps:
1. Multiples crossed out.
2. So: Eratosthenes sieve.
3. Select option C.
Answer: (C) | number_theory |
A recurrence defines terms from earlier ones, like Fibonacci from its two predecessors. | null | mat_0324 | concept | en | What is recurrence? | [
"A recurrence defines terms from earlier ones, like Fibonacci from its two predecessors.",
"So the answer stands: A recurrence defines terms from earlier ones, like Fibonacci from its two predecessors."
] | math | Answer: A recurrence defines terms from earlier ones, like Fibonacci from its two predecessors.
Steps:
1. A recurrence defines terms from earlier ones, like Fibonacci from its two predecessors.
2. So the answer stands: A recurrence defines terms from earlier ones, like Fibonacci from its two predecessors. | combinatorics |
Uniqueness shows at most one object fits, usually by assuming two and deriving they are equal. | null | mat_0325 | concept | en | Why prove uniqueness? | [
"Uniqueness shows at most one object fits, usually by assuming two and deriving they are equal.",
"So the answer stands: Uniqueness shows at most one object fits, usually by assuming two and deriving they are equal."
] | math | Answer: Uniqueness shows at most one object fits, usually by assuming two and deriving they are equal.
Steps:
1. Uniqueness shows at most one object fits, usually by assuming two and deriving they are equal.
2. So the answer stands: Uniqueness shows at most one object fits, usually by assuming two and deriving they are... | proofs |
Odpověď je okamžitou rychlost. | {
"A": "celkovou plochu",
"B": "dlouhou délku",
"C": "velký objem",
"D": "okamžitou rychlost"
} | mat_0326 | multiple_choice | cs | Vyber: co měří f'? | [
"Směrnice tečny čtena.",
"Tedy: okamžitou rychlost.",
"Vyber možnost D."
] | math | Answer: okamžitou rychlost
Steps:
1. Směrnice tečny čtena.
2. Tedy: okamžitou rychlost.
3. Vyber možnost D.
Answer: (D) | calculus |
Trace: 19. | null | mat_0327 | worked_example | en | Compute the trace of [[9,0],[0,10]]. | [
"Sum diagonal.",
"9 + 10.",
"= 19."
] | math | Answer: Trace: 19.
Steps:
1. Sum diagonal.
2. 9 + 10.
3. = 19. | linear_algebra |
Střední hodnota průměruje výsledky a předpovídá dlouhodobé chování. | null | mat_0328 | concept | cs | Proč střední hodnota? | [
"Střední hodnota průměruje výsledky a předpovídá dlouhodobé chování.",
"Odpověď tedy platí: Střední hodnota průměruje výsledky a předpovídá dlouhodobé chování."
] | math | Answer: Střední hodnota průměruje výsledky a předpovídá dlouhodobé chování.
Steps:
1. Střední hodnota průměruje výsledky a předpovídá dlouhodobé chování.
2. Odpověď tedy platí: Střední hodnota průměruje výsledky a předpovídá dlouhodobé chování. | probability |
The gcd is 15. | {
"A": "12",
"B": "18",
"C": "21",
"D": "15"
} | mat_0329 | multiple_choice | en | Find gcd(30, 45). | [
"Factor 3 out twice.",
"gcd = 15.",
"Select option D."
] | math | Answer: 15
Steps:
1. Factor 3 out twice.
2. gcd = 15.
3. Select option D.
Answer: (D) | number_theory |
Closed forms compute terms fast, skipping step-by-step iteration entirely. | null | mat_0330 | concept | en | Why solve recurrences? | [
"Closed forms compute terms fast, skipping step-by-step iteration entirely.",
"So the answer stands: Closed forms compute terms fast, skipping step-by-step iteration entirely."
] | math | Answer: Closed forms compute terms fast, skipping step-by-step iteration entirely.
Steps:
1. Closed forms compute terms fast, skipping step-by-step iteration entirely.
2. So the answer stands: Closed forms compute terms fast, skipping step-by-step iteration entirely. | combinatorics |
Rows: 256. | null | mat_0331 | worked_example | en | A proof checks all cases of 8 boolean variables. How many rows? | [
"2 choices each.",
"2^8 = 256."
] | math | Answer: Rows: 256.
Steps:
1. 2 choices each.
2. 2^8 = 256. | proofs |
The ratio test compares successive terms, and a limit below one guarantees absolute convergence. | null | mat_0332 | concept | en | What is the ratio test? | [
"The ratio test compares successive terms, and a limit below one guarantees absolute convergence.",
"So the answer stands: The ratio test compares successive terms, and a limit below one guarantees absolute convergence."
] | math | Answer: The ratio test compares successive terms, and a limit below one guarantees absolute convergence.
Steps:
1. The ratio test compares successive terms, and a limit below one guarantees absolute convergence.
2. So the answer stands: The ratio test compares successive terms, and a limit below one guarantees absolute... | calculus |
The answer is Its inverse. | {
"A": "Its transpose",
"B": "Its inverse",
"C": "Its trace",
"D": "Its copy"
} | mat_0333 | multiple_choice | en | Choose: what undoes a matrix? | [
"Product is identity.",
"So: Its inverse.",
"Select option B."
] | math | Answer: Its inverse
Steps:
1. Product is identity.
2. So: Its inverse.
3. Select option B.
Answer: (B) | linear_algebra |
It brackets the true parameter with stated confidence, quantifying statistical uncertainty. | null | mat_0334 | concept | en | What is a confidence interval? | [
"It brackets the true parameter with stated confidence, quantifying statistical uncertainty.",
"So the answer stands: It brackets the true parameter with stated confidence, quantifying statistical uncertainty."
] | math | Answer: It brackets the true parameter with stated confidence, quantifying statistical uncertainty.
Steps:
1. It brackets the true parameter with stated confidence, quantifying statistical uncertainty.
2. So the answer stands: It brackets the true parameter with stated confidence, quantifying statistical uncertainty. | probability |
The remainder is 4. | {
"A": "4",
"B": "5",
"C": "6",
"D": "7"
} | mat_0335 | multiple_choice | en | Find (121) mod 9. | [
"121 ÷ 9.",
"Remainder 4.",
"Select option A."
] | math | Answer: 4
Steps:
1. 121 ÷ 9.
2. Remainder 4.
3. Select option A.
Answer: (A) | number_theory |
C(11,2) = 55. | null | mat_0336 | worked_example | en | Show the computation of C(11,2). | [
"Formula n(n−1)/2.",
"11×10/2.",
"= 55."
] | math | Answer: C(11,2) = 55.
Steps:
1. Formula n(n−1)/2.
2. 11×10/2.
3. = 55. | combinatorics |
Sum: 78. | null | mat_0337 | worked_example | en | Prove by formula: find the sum 1 + 2 + ... + 12. | [
"Pair 1+12, 2+11, ...",
"Use 12×13/2.",
"= 78."
] | math | Answer: Sum: 78.
Steps:
1. Pair 1+12, 2+11, ...
2. Use 12×13/2.
3. = 78. | proofs |
Integral: 729. | null | mat_0338 | worked_example | en | Evaluate the integral of 3x² from 0 to 9. | [
"Antiderivative: x³.",
"Evaluate 0 to 9.",
"9³ − 0 = 729."
] | math | Answer: Integral: 729.
Steps:
1. Antiderivative: x³.
2. Evaluate 0 to 9.
3. 9³ − 0 = 729. | calculus |
The answer is Null space. | {
"A": "Top row",
"B": "Last column",
"C": "Null space",
"D": "Big entry"
} | mat_0339 | multiple_choice | en | Choose: what holds map-to-zero? | [
"Collapse measured.",
"So: Null space.",
"Select option C."
] | math | Answer: Null space
Steps:
1. Collapse measured.
2. So: Null space.
3. Select option C.
Answer: (C) | linear_algebra |
Testing weighs evidence against a null claim, rejecting it only when data is extreme. | null | mat_0340 | concept | en | Why hypothesis testing? | [
"Testing weighs evidence against a null claim, rejecting it only when data is extreme.",
"So the answer stands: Testing weighs evidence against a null claim, rejecting it only when data is extreme."
] | math | Answer: Testing weighs evidence against a null claim, rejecting it only when data is extreme.
Steps:
1. Testing weighs evidence against a null claim, rejecting it only when data is extreme.
2. So the answer stands: Testing weighs evidence against a null claim, rejecting it only when data is extreme. | probability |
Odpověď je prvočísla. | {
"A": "sudá čísla",
"B": "čtverce",
"C": "krychle",
"D": "prvočísla"
} | mat_0341 | multiple_choice | cs | Vyber: co jsou atomy násobení? | [
"Jednoznačný rozklad.",
"Tedy: prvočísla.",
"Vyber možnost D."
] | math | Answer: prvočísla
Steps:
1. Jednoznačný rozklad.
2. Tedy: prvočísla.
3. Vyber možnost D.
Answer: (D) | number_theory |
Each term sums the prior two from 1,1, growing exponentially with golden ratio ties. | null | mat_0342 | concept | en | What is the Fibonacci sequence? | [
"Each term sums the prior two from 1,1, growing exponentially with golden ratio ties.",
"So the answer stands: Each term sums the prior two from 1,1, growing exponentially with golden ratio ties."
] | math | Answer: Each term sums the prior two from 1,1, growing exponentially with golden ratio ties.
Steps:
1. Each term sums the prior two from 1,1, growing exponentially with golden ratio ties.
2. So the answer stands: Each term sums the prior two from 1,1, growing exponentially with golden ratio ties. | combinatorics |
Half: 36, so 72 is even. | null | mat_0343 | worked_example | en | Show N² + N is even for N = 8 by computing its half. | [
"8² + 8 = 72.",
"72 / 2 = 36.",
"Integer, so even."
] | math | Answer: Half: 36, so 72 is even.
Steps:
1. 8² + 8 = 72.
2. 72 / 2 = 36.
3. Integer, so even. | proofs |
Limit: 18. | null | mat_0344 | worked_example | en | Evaluate the limit of (x² − 9²)/(x − 9) as x → 9. | [
"Factor: (x−9)(x+9).",
"Cancel (x−9).",
"Plug 9: 9+9 = 18."
] | math | Answer: Limit: 18.
Steps:
1. Factor: (x−9)(x+9).
2. Cancel (x−9).
3. Plug 9: 9+9 = 18. | calculus |
Dot product: 59. | null | mat_0345 | worked_example | en | Compute (8,7) · (3,5) step by step. | [
"Multiply components.",
"8×3 + 7×5.",
"= 59."
] | math | Answer: Dot product: 59.
Steps:
1. Multiply components.
2. 8×3 + 7×5.
3. = 59. | linear_algebra |
The answer is Bayes' theorem. | {
"A": "Coin toss",
"B": "Bayes' theorem",
"C": "Dice roll",
"D": "Card draw"
} | mat_0346 | multiple_choice | en | Choose: what flips conditionals? | [
"Causes from effects.",
"So: Bayes' theorem.",
"Select option B."
] | math | Answer: Bayes' theorem
Steps:
1. Causes from effects.
2. So: Bayes' theorem.
3. Select option B.
Answer: (B) | probability |
phi(n) counts numbers below n coprime to it, measuring reduced residue systems. | null | mat_0347 | concept | en | What is the totient? | [
"phi(n) counts numbers below n coprime to it, measuring reduced residue systems.",
"So the answer stands: phi(n) counts numbers below n coprime to it, measuring reduced residue systems."
] | math | Answer: phi(n) counts numbers below n coprime to it, measuring reduced residue systems.
Steps:
1. phi(n) counts numbers below n coprime to it, measuring reduced residue systems.
2. So the answer stands: phi(n) counts numbers below n coprime to it, measuring reduced residue systems. | number_theory |
The answer is Inclusion-exclusion. | {
"A": "More counting",
"B": "Inclusion-exclusion",
"C": "Less thinking",
"D": "Wild hope"
} | mat_0348 | multiple_choice | en | Choose: what fixes double-counting? | [
"Alternate the sums.",
"So: Inclusion-exclusion.",
"Select option B."
] | math | Answer: Inclusion-exclusion
Steps:
1. Alternate the sums.
2. So: Inclusion-exclusion.
3. Select option B.
Answer: (B) | combinatorics |
The table has 32 rows. | {
"A": "30",
"B": "34",
"C": "36",
"D": "32"
} | mat_0349 | multiple_choice | en | How many rows does a truth table with 5 variables have? | [
"2^5 rows.",
"Total: 32.",
"Select option D."
] | math | Answer: 32
Steps:
1. 2^5 rows.
2. Total: 32.
3. Select option D.
Answer: (D) | proofs |
f'(3) = 26. | null | mat_0350 | worked_example | en | For f(x) = 3x² + 8x, compute f'(3) step by step. | [
"f'(x) = 6x + 8.",
"Plug x = 3.",
"6×3 + 8 = 26."
] | math | Answer: f'(3) = 26.
Steps:
1. f'(x) = 6x + 8.
2. Plug x = 3.
3. 6×3 + 8 = 26. | calculus |
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