answer string | choices dict | id string | kind string | lang string | question string | steps list | subject string | text string | topic string |
|---|---|---|---|---|---|---|---|---|---|
It flips divisor sums via mu, recovering hidden functions from accumulated totals. | null | mat_0851 | concept | en | What is Möbius inversion? | [
"It flips divisor sums via mu, recovering hidden functions from accumulated totals.",
"So the answer stands: It flips divisor sums via mu, recovering hidden functions from accumulated totals."
] | math | Answer: It flips divisor sums via mu, recovering hidden functions from accumulated totals.
Steps:
1. It flips divisor sums via mu, recovering hidden functions from accumulated totals.
2. So the answer stands: It flips divisor sums via mu, recovering hidden functions from accumulated totals. | number_theory |
C(9,3) = 84. | null | mat_0852 | worked_example | cs | Ukaž výpočet C(9,3). | [
"n(n−1)(n−2)/6.",
"9×8×7/6.",
"= 84."
] | math | Answer: C(9,3) = 84.
Steps:
1. n(n−1)(n−2)/6.
2. 9×8×7/6.
3. = 84. | combinatorics |
The value is 132. | {
"A": "130",
"B": "134",
"C": "132",
"D": "136"
} | mat_0853 | multiple_choice | en | Compute N² + N for N = 11. | [
"11² + 11.",
"Value = 132.",
"Select option C."
] | math | Answer: 132
Steps:
1. 11² + 11.
2. Value = 132.
3. Select option C.
Answer: (C) | proofs |
Integrál: 216. | null | mat_0854 | worked_example | cs | Vyčísli integrál 3x² od 0 do 6. | [
"Primitivní: x³.",
"Vyčísli 0 až 6.",
"6³ − 0 = 216."
] | math | Answer: Integrál: 216.
Steps:
1. Primitivní: x³.
2. Vyčísli 0 až 6.
3. 6³ − 0 = 216. | calculus |
Stopa je 9. | {
"A": "7",
"B": "11",
"C": "9",
"D": "13"
} | mat_0855 | multiple_choice | cs | Vypočítej stopu matice [[3,1],[2,6]]. | [
"3 + 6.",
"Stopa = 9.",
"Vyber možnost C."
] | math | Answer: 9
Steps:
1. 3 + 6.
2. Stopa = 9.
3. Vyber možnost C.
Answer: (C) | linear_algebra |
The expected value is 6.5. | {
"A": "7.5",
"B": "6.5",
"C": "5.5",
"D": "8.5"
} | mat_0856 | multiple_choice | en | Find the expected value of a uniform draw from {1..12}. | [
"(12+1)/2.",
"E = 6.5.",
"Select option B."
] | math | Answer: 6.5
Steps:
1. (12+1)/2.
2. E = 6.5.
3. Select option B.
Answer: (B) | probability |
gcd = 34. | null | mat_0857 | worked_example | en | Compute gcd(68, 102) step by step. | [
"Factor: 68=4×17, 102=6×17.",
"Common: 17×gcd(4,6).",
"= 17×2 = 34."
] | math | Answer: gcd = 34.
Steps:
1. Factor: 68=4×17, 102=6×17.
2. Common: 17×gcd(4,6).
3. = 17×2 = 34. | number_theory |
P(10,2) is 90. | {
"A": "90",
"B": "80",
"C": "100",
"D": "110"
} | mat_0858 | multiple_choice | en | Compute P(10,2). | [
"10×9.",
"= 90.",
"Select option A."
] | math | Answer: 90
Steps:
1. 10×9.
2. = 90.
3. Select option A.
Answer: (A) | combinatorics |
Sum: 231. | null | mat_0859 | worked_example | en | Prove by formula: find the sum 1 + 2 + ... + 21. | [
"Pair 1+21, 2+20, ...",
"Use 21×22/2.",
"= 231."
] | math | Answer: Sum: 231.
Steps:
1. Pair 1+21, 2+20, ...
2. Use 21×22/2.
3. = 231. | proofs |
Limit: 36. | null | mat_0860 | worked_example | en | Evaluate the limit of (x² − 18²)/(x − 18) as x → 18. | [
"Factor: (x−18)(x+18).",
"Cancel (x−18).",
"Plug 18: 18+18 = 36."
] | math | Answer: Limit: 36.
Steps:
1. Factor: (x−18)(x+18).
2. Cancel (x−18).
3. Plug 18: 18+18 = 36. | calculus |
Determinant: 340. | null | mat_0861 | worked_example | en | Compute det [[18,1],[2,19]] step by step. | [
"ad − bc.",
"18×19 − 1×2.",
"= 340."
] | math | Answer: Determinant: 340.
Steps:
1. ad − bc.
2. 18×19 − 1×2.
3. = 340. | linear_algebra |
They keep posteriors in one family, making Bayesian updates algebraic and fast. | null | mat_0862 | concept | en | Why conjugate priors? | [
"They keep posteriors in one family, making Bayesian updates algebraic and fast.",
"So the answer stands: They keep posteriors in one family, making Bayesian updates algebraic and fast."
] | math | Answer: They keep posteriors in one family, making Bayesian updates algebraic and fast.
Steps:
1. They keep posteriors in one family, making Bayesian updates algebraic and fast.
2. So the answer stands: They keep posteriors in one family, making Bayesian updates algebraic and fast. | probability |
Remainder: 5. | null | mat_0863 | worked_example | en | Compute (215) mod 7. | [
"Divide 215 by 7.",
"7×30 = 210.",
"215 − 210 = 5."
] | math | Answer: Remainder: 5.
Steps:
1. Divide 215 by 7.
2. 7×30 = 210.
3. 215 − 210 = 5. | number_theory |
C(20,2) = 190. | null | mat_0864 | worked_example | en | Show the computation of C(20,2). | [
"Formula n(n−1)/2.",
"20×19/2.",
"= 190."
] | math | Answer: C(20,2) = 190.
Steps:
1. Formula n(n−1)/2.
2. 20×19/2.
3. = 190. | combinatorics |
Half: 153, so 306 is even. | null | mat_0865 | worked_example | en | Show N² + N is even for N = 17 by computing its half. | [
"17² + 17 = 306.",
"306 / 2 = 153.",
"Integer, so even."
] | math | Answer: Half: 153, so 306 is even.
Steps:
1. 17² + 17 = 306.
2. 306 / 2 = 153.
3. Integer, so even. | proofs |
f'(3) = 35. | null | mat_0866 | worked_example | en | For f(x) = 3x² + 17x, compute f'(3) step by step. | [
"f'(x) = 6x + 17.",
"Plug x = 3.",
"6×3 + 17 = 35."
] | math | Answer: f'(3) = 35.
Steps:
1. f'(x) = 6x + 17.
2. Plug x = 3.
3. 6×3 + 17 = 35. | calculus |
Trace: 39. | null | mat_0867 | worked_example | en | Compute the trace of [[19,0],[0,20]]. | [
"Sum diagonal.",
"19 + 20.",
"= 39."
] | math | Answer: Trace: 39.
Steps:
1. Sum diagonal.
2. 19 + 20.
3. = 39. | linear_algebra |
P(red) = 18/22. | null | mat_0868 | worked_example | en | A bag holds 18 red and 4 blue marbles. Show P(red). | [
"Red: 18.",
"Total: 22.",
"P = 18/22."
] | math | Answer: P(red) = 18/22.
Steps:
1. Red: 18.
2. Total: 22.
3. P = 18/22. | probability |
NSN = 8. | null | mat_0869 | worked_example | cs | Spočítej NSN(8, 8) krok za krokem. | [
"NSD(8,8) = 8.",
"8×8/8.",
"= 8."
] | math | Answer: NSN = 8.
Steps:
1. NSD(8,8) = 8.
2. 8×8/8.
3. = 8. | number_theory |
P(19,3) = 5814. | null | mat_0870 | worked_example | en | Show the computation of P(19,3). | [
"n(n−1)(n−2).",
"19×18×17.",
"= 5814."
] | math | Answer: P(19,3) = 5814.
Steps:
1. n(n−1)(n−2).
2. 19×18×17.
3. = 5814. | combinatorics |
Proof assistants check every step mechanically, catching human errors in long complex proofs. | null | mat_0871 | concept | en | Why use proof assistants? | [
"Proof assistants check every step mechanically, catching human errors in long complex proofs.",
"So the answer stands: Proof assistants check every step mechanically, catching human errors in long complex proofs."
] | math | Answer: Proof assistants check every step mechanically, catching human errors in long complex proofs.
Steps:
1. Proof assistants check every step mechanically, catching human errors in long complex proofs.
2. So the answer stands: Proof assistants check every step mechanically, catching human errors in long complex pro... | proofs |
Line integrals accumulate fields along paths, computing work done by forces exactly. | null | mat_0872 | concept | en | Why study line integrals? | [
"Line integrals accumulate fields along paths, computing work done by forces exactly.",
"So the answer stands: Line integrals accumulate fields along paths, computing work done by forces exactly."
] | math | Answer: Line integrals accumulate fields along paths, computing work done by forces exactly.
Steps:
1. Line integrals accumulate fields along paths, computing work done by forces exactly.
2. So the answer stands: Line integrals accumulate fields along paths, computing work done by forces exactly. | calculus |
Dot product: 139. | null | mat_0873 | worked_example | en | Compute (18,17) · (3,5) step by step. | [
"Multiply components.",
"18×3 + 17×5.",
"= 139."
] | math | Answer: Dot product: 139.
Steps:
1. Multiply components.
2. 18×3 + 17×5.
3. = 139. | linear_algebra |
It picks parameters making observed data most probable, the core fitting principle. | null | mat_0874 | concept | en | What is maximum likelihood? | [
"It picks parameters making observed data most probable, the core fitting principle.",
"So the answer stands: It picks parameters making observed data most probable, the core fitting principle."
] | math | Answer: It picks parameters making observed data most probable, the core fitting principle.
Steps:
1. It picks parameters making observed data most probable, the core fitting principle.
2. So the answer stands: It picks parameters making observed data most probable, the core fitting principle. | probability |
They encode arithmetic functions as series, linking numbers to complex analysis. | null | mat_0875 | concept | en | Why Dirichlet series? | [
"They encode arithmetic functions as series, linking numbers to complex analysis.",
"So the answer stands: They encode arithmetic functions as series, linking numbers to complex analysis."
] | math | Answer: They encode arithmetic functions as series, linking numbers to complex analysis.
Steps:
1. They encode arithmetic functions as series, linking numbers to complex analysis.
2. So the answer stands: They encode arithmetic functions as series, linking numbers to complex analysis. | number_theory |
Rekurence definuje členy z předchozích. | null | mat_0876 | concept | cs | Co je rekurence? | [
"Rekurence definuje členy z předchozích.",
"Odpověď tedy platí: Rekurence definuje členy z předchozích."
] | math | Answer: Rekurence definuje členy z předchozích.
Steps:
1. Rekurence definuje členy z předchozích.
2. Odpověď tedy platí: Rekurence definuje členy z předchozích. | combinatorics |
Rigor means every step is justified, leaving no gaps between assumptions and conclusion. | null | mat_0877 | concept | en | What is mathematical rigor? | [
"Rigor means every step is justified, leaving no gaps between assumptions and conclusion.",
"So the answer stands: Rigor means every step is justified, leaving no gaps between assumptions and conclusion."
] | math | Answer: Rigor means every step is justified, leaving no gaps between assumptions and conclusion.
Steps:
1. Rigor means every step is justified, leaving no gaps between assumptions and conclusion.
2. So the answer stands: Rigor means every step is justified, leaving no gaps between assumptions and conclusion. | proofs |
Limita: 12. | null | mat_0878 | worked_example | cs | Vyčísli limitu (x² − 6²)/(x − 6) pro x → 6. | [
"Rozlož: (x−6)(x+6).",
"Pokrať (x−6).",
"Dosaď 6: 6+6 = 12."
] | math | Answer: Limita: 12.
Steps:
1. Rozlož: (x−6)(x+6).
2. Pokrať (x−6).
3. Dosaď 6: 6+6 = 12. | calculus |
The determinant is 102. | {
"A": "102",
"B": "99",
"C": "105",
"D": "108"
} | mat_0879 | multiple_choice | en | Find the determinant of [[9,2],[3,12]]. | [
"9×12 − 2×3.",
"Det = 102.",
"Select option A."
] | math | Answer: 102
Steps:
1. 9×12 − 2×3.
2. Det = 102.
3. Select option A.
Answer: (A) | linear_algebra |
E = 4,5. | null | mat_0880 | worked_example | cs | Spočítej E rovnoměrného tahu z {1..8}. | [
"Průměr 1..8.",
"(8+1)/2.",
"E = 4,5."
] | math | Answer: E = 4,5.
Steps:
1. Průměr 1..8.
2. (8+1)/2.
3. E = 4,5. | probability |
The gcd is 30. | {
"A": "30",
"B": "27",
"C": "33",
"D": "36"
} | mat_0881 | multiple_choice | en | Find gcd(60, 90). | [
"Factor 3 out twice.",
"gcd = 30.",
"Select option A."
] | math | Answer: 30
Steps:
1. Factor 3 out twice.
2. gcd = 30.
3. Select option A.
Answer: (A) | number_theory |
It classifies counting by labelled/unlabelled balls and boxes into twelve cases. | null | mat_0882 | concept | en | What is the twelvefold way? | [
"It classifies counting by labelled/unlabelled balls and boxes into twelve cases.",
"So the answer stands: It classifies counting by labelled/unlabelled balls and boxes into twelve cases."
] | math | Answer: It classifies counting by labelled/unlabelled balls and boxes into twelve cases.
Steps:
1. It classifies counting by labelled/unlabelled balls and boxes into twelve cases.
2. So the answer stands: It classifies counting by labelled/unlabelled balls and boxes into twelve cases. | combinatorics |
Rows: 262144. | null | mat_0883 | worked_example | en | A proof checks all cases of 18 boolean variables. How many rows? | [
"2 choices each.",
"2^18 = 262144."
] | math | Answer: Rows: 262144.
Steps:
1. 2 choices each.
2. 2^18 = 262144. | proofs |
It converts a loop integral into a region integral, trading boundary work for interior sums. | null | mat_0884 | concept | en | What is Green's theorem? | [
"It converts a loop integral into a region integral, trading boundary work for interior sums.",
"So the answer stands: It converts a loop integral into a region integral, trading boundary work for interior sums."
] | math | Answer: It converts a loop integral into a region integral, trading boundary work for interior sums.
Steps:
1. It converts a loop integral into a region integral, trading boundary work for interior sums.
2. So the answer stands: It converts a loop integral into a region integral, trading boundary work for interior sums... | calculus |
The trace is 23. | {
"A": "21",
"B": "23",
"C": "25",
"D": "27"
} | mat_0885 | multiple_choice | en | Find the trace of [[10,1],[2,13]]. | [
"10 + 13.",
"Trace = 23.",
"Select option B."
] | math | Answer: 23
Steps:
1. 10 + 13.
2. Trace = 23.
3. Select option B.
Answer: (B) | linear_algebra |
Unbiased estimators average to the truth, avoiding systematic error across samples. | null | mat_0886 | concept | en | Why unbiased estimators? | [
"Unbiased estimators average to the truth, avoiding systematic error across samples.",
"So the answer stands: Unbiased estimators average to the truth, avoiding systematic error across samples."
] | math | Answer: Unbiased estimators average to the truth, avoiding systematic error across samples.
Steps:
1. Unbiased estimators average to the truth, avoiding systematic error across samples.
2. So the answer stands: Unbiased estimators average to the truth, avoiding systematic error across samples. | probability |
zeta(s) sums 1/n^s, and its zeros conjecturally rule prime fluctuations. | null | mat_0887 | concept | en | What is the Riemann zeta? | [
"zeta(s) sums 1/n^s, and its zeros conjecturally rule prime fluctuations.",
"So the answer stands: zeta(s) sums 1/n^s, and its zeros conjecturally rule prime fluctuations."
] | math | Answer: zeta(s) sums 1/n^s, and its zeros conjecturally rule prime fluctuations.
Steps:
1. zeta(s) sums 1/n^s, and its zeros conjecturally rule prime fluctuations.
2. So the answer stands: zeta(s) sums 1/n^s, and its zeros conjecturally rule prime fluctuations. | number_theory |
C(21,3) = 1330. | null | mat_0888 | worked_example | en | Show the computation of C(21,3). | [
"n(n−1)(n−2)/6.",
"21×20×19/6.",
"= 1330."
] | math | Answer: C(21,3) = 1330.
Steps:
1. n(n−1)(n−2)/6.
2. 21×20×19/6.
3. = 1330. | combinatorics |
Tabulky vypisují všechny kombinace hodnot a ověřují ekvivalence mechanicky. | null | mat_0889 | concept | cs | Proč pravdivostní tabulky? | [
"Tabulky vypisují všechny kombinace hodnot a ověřují ekvivalence mechanicky.",
"Odpověď tedy platí: Tabulky vypisují všechny kombinace hodnot a ověřují ekvivalence mechanicky."
] | math | Answer: Tabulky vypisují všechny kombinace hodnot a ověřují ekvivalence mechanicky.
Steps:
1. Tabulky vypisují všechny kombinace hodnot a ověřují ekvivalence mechanicky.
2. Odpověď tedy platí: Tabulky vypisují všechny kombinace hodnot a ověřují ekvivalence mechanicky. | proofs |
Numerical rules like Simpson's estimate integrals lacking formulas, with controlled error. | null | mat_0890 | concept | en | Why approximate integrals? | [
"Numerical rules like Simpson's estimate integrals lacking formulas, with controlled error.",
"So the answer stands: Numerical rules like Simpson's estimate integrals lacking formulas, with controlled error."
] | math | Answer: Numerical rules like Simpson's estimate integrals lacking formulas, with controlled error.
Steps:
1. Numerical rules like Simpson's estimate integrals lacking formulas, with controlled error.
2. So the answer stands: Numerical rules like Simpson's estimate integrals lacking formulas, with controlled error. | calculus |
The dot product is 67. | {
"A": "63",
"B": "71",
"C": "67",
"D": "75"
} | mat_0891 | multiple_choice | en | Find the dot product (9,10) · (3,4). | [
"9×3 + 10×4.",
"Product = 67.",
"Select option C."
] | math | Answer: 67
Steps:
1. 9×3 + 10×4.
2. Product = 67.
3. Select option C.
Answer: (C) | linear_algebra |
P(red) is 10/31. | {
"A": "21/31",
"B": "10/32",
"C": "10/31",
"D": "11/31"
} | mat_0892 | multiple_choice | en | A bag holds 10 red and 21 blue marbles. What is P(red)? | [
"10 of 31.",
"P = 10/31.",
"Select option C."
] | math | Answer: 10/31
Steps:
1. 10 of 31.
2. P = 10/31.
3. Select option C.
Answer: (C) | probability |
They give best rational approximations, exposing hidden structure of irrationals. | null | mat_0893 | concept | en | Why continued fractions? | [
"They give best rational approximations, exposing hidden structure of irrationals.",
"So the answer stands: They give best rational approximations, exposing hidden structure of irrationals."
] | math | Answer: They give best rational approximations, exposing hidden structure of irrationals.
Steps:
1. They give best rational approximations, exposing hidden structure of irrationals.
2. So the answer stands: They give best rational approximations, exposing hidden structure of irrationals. | number_theory |
They count preference lists letting all cars park, equalling (n+1)^(n-1). | null | mat_0894 | concept | en | Why parking functions? | [
"They count preference lists letting all cars park, equalling (n+1)^(n-1).",
"So the answer stands: They count preference lists letting all cars park, equalling (n+1)^(n-1)."
] | math | Answer: They count preference lists letting all cars park, equalling (n+1)^(n-1).
Steps:
1. They count preference lists letting all cars park, equalling (n+1)^(n-1).
2. So the answer stands: They count preference lists letting all cars park, equalling (n+1)^(n-1). | combinatorics |
Sum: 253. | null | mat_0895 | worked_example | en | Prove by formula: find the sum 1 + 2 + ... + 22. | [
"Pair 1+22, 2+21, ...",
"Use 22×23/2.",
"= 253."
] | math | Answer: Sum: 253.
Steps:
1. Pair 1+22, 2+21, ...
2. Use 22×23/2.
3. = 253. | proofs |
It guarantees a point where instantaneous and average rates match on every smooth interval. | null | mat_0896 | concept | en | What is the mean value theorem? | [
"It guarantees a point where instantaneous and average rates match on every smooth interval.",
"So the answer stands: It guarantees a point where instantaneous and average rates match on every smooth interval."
] | math | Answer: It guarantees a point where instantaneous and average rates match on every smooth interval.
Steps:
1. It guarantees a point where instantaneous and average rates match on every smooth interval.
2. So the answer stands: It guarantees a point where instantaneous and average rates match on every smooth interval. | calculus |
The determinant is 124. | {
"A": "121",
"B": "124",
"C": "127",
"D": "130"
} | mat_0897 | multiple_choice | en | Find the determinant of [[10,2],[3,13]]. | [
"10×13 − 2×3.",
"Det = 124.",
"Select option B."
] | math | Answer: 124
Steps:
1. 10×13 − 2×3.
2. Det = 124.
3. Select option B.
Answer: (B) | linear_algebra |
Just 23 people give even odds of a shared birthday, since pairs grow quadratically. | null | mat_0898 | concept | en | What is the birthday paradox? | [
"Just 23 people give even odds of a shared birthday, since pairs grow quadratically.",
"So the answer stands: Just 23 people give even odds of a shared birthday, since pairs grow quadratically."
] | math | Answer: Just 23 people give even odds of a shared birthday, since pairs grow quadratically.
Steps:
1. Just 23 people give even odds of a shared birthday, since pairs grow quadratically.
2. So the answer stands: Just 23 people give even odds of a shared birthday, since pairs grow quadratically. | probability |
The remainder is 3. | {
"A": "4",
"B": "3",
"C": "5",
"D": "6"
} | mat_0899 | multiple_choice | en | Find (156) mod 9. | [
"156 ÷ 9.",
"Remainder 3.",
"Select option B."
] | math | Answer: 3
Steps:
1. 156 ÷ 9.
2. Remainder 3.
3. Select option B.
Answer: (B) | number_theory |
C(21,2) = 210. | null | mat_0900 | worked_example | en | Show the computation of C(21,2). | [
"Formula n(n−1)/2.",
"21×20/2.",
"= 210."
] | math | Answer: C(21,2) = 210.
Steps:
1. Formula n(n−1)/2.
2. 21×20/2.
3. = 210. | combinatorics |
Tabulka má 8 řádků. | {
"A": "6",
"B": "8",
"C": "10",
"D": "12"
} | mat_0901 | multiple_choice | cs | Kolik řádků má pravdivostní tabulka s 3 proměnnými? | [
"2^3 řádků.",
"Celkem: 8.",
"Vyber možnost B."
] | math | Answer: 8
Steps:
1. 2^3 řádků.
2. Celkem: 8.
3. Vyber možnost B.
Answer: (B) | proofs |
Rolle's theorem finds flat tangents between equal values, anchoring the mean value theorem. | null | mat_0902 | concept | en | Why prove Rolle's theorem? | [
"Rolle's theorem finds flat tangents between equal values, anchoring the mean value theorem.",
"So the answer stands: Rolle's theorem finds flat tangents between equal values, anchoring the mean value theorem."
] | math | Answer: Rolle's theorem finds flat tangents between equal values, anchoring the mean value theorem.
Steps:
1. Rolle's theorem finds flat tangents between equal values, anchoring the mean value theorem.
2. So the answer stands: Rolle's theorem finds flat tangents between equal values, anchoring the mean value theorem. | calculus |
Determinant: 40. | null | mat_0903 | worked_example | cs | Spočítej det [[6,1],[2,7]] krok za krokem. | [
"ad − bc.",
"6×7 − 1×2.",
"= 40."
] | math | Answer: Determinant: 40.
Steps:
1. ad − bc.
2. 6×7 − 1×2.
3. = 40. | linear_algebra |
It quantifies uncertainty everywhere, from games to science to machine learning. | null | mat_0904 | concept | en | Why study probability? | [
"It quantifies uncertainty everywhere, from games to science to machine learning.",
"So the answer stands: It quantifies uncertainty everywhere, from games to science to machine learning."
] | math | Answer: It quantifies uncertainty everywhere, from games to science to machine learning.
Steps:
1. It quantifies uncertainty everywhere, from games to science to machine learning.
2. So the answer stands: It quantifies uncertainty everywhere, from games to science to machine learning. | probability |
lcm = 40. | null | mat_0905 | worked_example | en | Compute lcm(20, 8) step by step. | [
"gcd(20,8) = 4.",
"20×8/4.",
"= 40."
] | math | Answer: lcm = 40.
Steps:
1. gcd(20,8) = 4.
2. 20×8/4.
3. = 40. | number_theory |
C(12,3) is 220. | {
"A": "208",
"B": "220",
"C": "232",
"D": "244"
} | mat_0906 | multiple_choice | en | Compute C(12,3). | [
"12×11×10/6.",
"= 220.",
"Select option B."
] | math | Answer: 220
Steps:
1. 12×11×10/6.
2. = 220.
3. Select option B.
Answer: (B) | combinatorics |
Active reading fills each step yourself, turning passive lines into owned understanding. | null | mat_0907 | concept | en | Why read proofs actively? | [
"Active reading fills each step yourself, turning passive lines into owned understanding.",
"So the answer stands: Active reading fills each step yourself, turning passive lines into owned understanding."
] | math | Answer: Active reading fills each step yourself, turning passive lines into owned understanding.
Steps:
1. Active reading fills each step yourself, turning passive lines into owned understanding.
2. So the answer stands: Active reading fills each step yourself, turning passive lines into owned understanding. | proofs |
The integral is 1000. | {
"A": "990",
"B": "1000",
"C": "1010",
"D": "1020"
} | mat_0908 | multiple_choice | en | Compute the integral of 3x² from 0 to 10. | [
"Antiderivative x³.",
"10³ = 1000.",
"Select option B."
] | math | Answer: 1000
Steps:
1. Antiderivative x³.
2. 10³ = 1000.
3. Select option B.
Answer: (B) | calculus |
Skalární součin je 18. | {
"A": "14",
"B": "22",
"C": "26",
"D": "18"
} | mat_0909 | multiple_choice | cs | Najdi skalární součin (2,3) · (3,4). | [
"2×3 + 3×4.",
"Součin = 18.",
"Vyber možnost D."
] | math | Answer: 18
Steps:
1. 2×3 + 3×4.
2. Součin = 18.
3. Vyber možnost D.
Answer: (D) | linear_algebra |
Odpověď je podmíněná pravděpodobnost. | {
"A": "podmíněná pravděpodobnost",
"B": "slepé hádání",
"C": "tvrdohlavý prior",
"D": "divoká naděje"
} | mat_0910 | multiple_choice | cs | Vyber: co aktualizuje s evidencí? | [
"Prostor se zužuje.",
"Tedy: podmíněná pravděpodobnost.",
"Vyber možnost A."
] | math | Answer: podmíněná pravděpodobnost
Steps:
1. Prostor se zužuje.
2. Tedy: podmíněná pravděpodobnost.
3. Vyber možnost A.
Answer: (A) | probability |
x^2 - Dy^2 = 1 has infinite integer solutions from continued fractions of sqrt(D). | null | mat_0911 | concept | en | What is Pell's equation? | [
"x^2 - Dy^2 = 1 has infinite integer solutions from continued fractions of sqrt(D).",
"So the answer stands: x^2 - Dy^2 = 1 has infinite integer solutions from continued fractions of sqrt(D)."
] | math | Answer: x^2 - Dy^2 = 1 has infinite integer solutions from continued fractions of sqrt(D).
Steps:
1. x^2 - Dy^2 = 1 has infinite integer solutions from continued fractions of sqrt(D).
2. So the answer stands: x^2 - Dy^2 = 1 has infinite integer solutions from continued fractions of sqrt(D). | number_theory |
C(12,2) is 66. | {
"A": "66",
"B": "54",
"C": "78",
"D": "90"
} | mat_0912 | multiple_choice | en | Compute C(12,2). | [
"12×11/2.",
"= 66.",
"Select option A."
] | math | Answer: 66
Steps:
1. 12×11/2.
2. = 66.
3. Select option A.
Answer: (A) | combinatorics |
Součet je 21. | {
"A": "15",
"B": "27",
"C": "21",
"D": "33"
} | mat_0913 | multiple_choice | cs | Kolik je součet 1 + 2 + ... + 6? | [
"6×7/2.",
"Součet = 21.",
"Vyber možnost C."
] | math | Answer: 21
Steps:
1. 6×7/2.
2. Součet = 21.
3. Vyber možnost C.
Answer: (C) | proofs |
Uniform convergence controls error everywhere at once, letting limits pass through sums safely. | null | mat_0914 | concept | en | What is uniform convergence? | [
"Uniform convergence controls error everywhere at once, letting limits pass through sums safely.",
"So the answer stands: Uniform convergence controls error everywhere at once, letting limits pass through sums safely."
] | math | Answer: Uniform convergence controls error everywhere at once, letting limits pass through sums safely.
Steps:
1. Uniform convergence controls error everywhere at once, letting limits pass through sums safely.
2. So the answer stands: Uniform convergence controls error everywhere at once, letting limits pass through su... | calculus |
Singular value decomposition splits any matrix into rotations and scalings, powering data science. | null | mat_0915 | concept | en | Why SVD? | [
"Singular value decomposition splits any matrix into rotations and scalings, powering data science.",
"So the answer stands: Singular value decomposition splits any matrix into rotations and scalings, powering data science."
] | math | Answer: Singular value decomposition splits any matrix into rotations and scalings, powering data science.
Steps:
1. Singular value decomposition splits any matrix into rotations and scalings, powering data science.
2. So the answer stands: Singular value decomposition splits any matrix into rotations and scalings, pow... | linear_algebra |
P = 1/22. | null | mat_0916 | worked_example | en | A fair 22-sided die is rolled. Show P(one specific number). | [
"Favourable: 1.",
"Possible: 22.",
"P = 1/22."
] | math | Answer: P = 1/22.
Steps:
1. Favourable: 1.
2. Possible: 22.
3. P = 1/22. | probability |
They extend unique factorization to the complex plane, solving sum-of-squares questions. | null | mat_0917 | concept | en | Why Gaussian integers? | [
"They extend unique factorization to the complex plane, solving sum-of-squares questions.",
"So the answer stands: They extend unique factorization to the complex plane, solving sum-of-squares questions."
] | math | Answer: They extend unique factorization to the complex plane, solving sum-of-squares questions.
Steps:
1. They extend unique factorization to the complex plane, solving sum-of-squares questions.
2. So the answer stands: They extend unique factorization to the complex plane, solving sum-of-squares questions. | number_theory |
P(20,3) = 6840. | null | mat_0918 | worked_example | en | Show the computation of P(20,3). | [
"n(n−1)(n−2).",
"20×19×18.",
"= 6840."
] | math | Answer: P(20,3) = 6840.
Steps:
1. n(n−1)(n−2).
2. 20×19×18.
3. = 6840. | combinatorics |
A corollary follows quickly from a theorem just proved, packaging its immediate consequences. | null | mat_0919 | concept | en | What is a corollary? | [
"A corollary follows quickly from a theorem just proved, packaging its immediate consequences.",
"So the answer stands: A corollary follows quickly from a theorem just proved, packaging its immediate consequences."
] | math | Answer: A corollary follows quickly from a theorem just proved, packaging its immediate consequences.
Steps:
1. A corollary follows quickly from a theorem just proved, packaging its immediate consequences.
2. So the answer stands: A corollary follows quickly from a theorem just proved, packaging its immediate consequen... | proofs |
Integral: 6859. | null | mat_0920 | worked_example | en | Evaluate the integral of 3x² from 0 to 19. | [
"Antiderivative: x³.",
"Evaluate 0 to 19.",
"19³ − 0 = 6859."
] | math | Answer: Integral: 6859.
Steps:
1. Antiderivative: x³.
2. Evaluate 0 to 19.
3. 19³ − 0 = 6859. | calculus |
Stopa: 15. | null | mat_0921 | worked_example | cs | Spočítej stopu matice [[7,0],[0,8]]. | [
"Sečti diagonálu.",
"7 + 8.",
"= 15."
] | math | Answer: Stopa: 15.
Steps:
1. Sečti diagonálu.
2. 7 + 8.
3. = 15. | linear_algebra |
The probability is 1/15. | {
"A": "1/14",
"B": "1/15",
"C": "1/16",
"D": "2/15"
} | mat_0922 | multiple_choice | en | A fair 15-sided die is rolled. What is the probability of one specific number? | [
"1 face of 15.",
"P = 1/15.",
"Select option B."
] | math | Answer: 1/15
Steps:
1. 1 face of 15.
2. P = 1/15.
3. Select option B.
Answer: (B) | probability |
gcd = 36. | null | mat_0923 | worked_example | en | Compute gcd(72, 108) step by step. | [
"Factor: 72=4×18, 108=6×18.",
"Common: 18×gcd(4,6).",
"= 18×2 = 36."
] | math | Answer: gcd = 36.
Steps:
1. Factor: 72=4×18, 108=6×18.
2. Common: 18×gcd(4,6).
3. = 18×2 = 36. | number_theory |
P(11,2) is 110. | {
"A": "99",
"B": "110",
"C": "121",
"D": "132"
} | mat_0924 | multiple_choice | en | Compute P(11,2). | [
"11×10.",
"= 110.",
"Select option B."
] | math | Answer: 110
Steps:
1. 11×10.
2. = 110.
3. Select option B.
Answer: (B) | combinatorics |
Half: 171, so 342 is even. | null | mat_0925 | worked_example | en | Show N² + N is even for N = 18 by computing its half. | [
"18² + 18 = 342.",
"342 / 2 = 171.",
"Integer, so even."
] | math | Answer: Half: 171, so 342 is even.
Steps:
1. 18² + 18 = 342.
2. 342 / 2 = 171.
3. Integer, so even. | proofs |
Odpověď je i v koncových bodech. | {
"A": "nikde jinde",
"B": "i v koncových bodech",
"C": "pouze v nule",
"D": "daleko"
} | mat_0926 | multiple_choice | cs | Vyber: kde se skrývají maxima? | [
"Konce kontroluj vždy.",
"Tedy: i v koncových bodech.",
"Vyber možnost B."
] | math | Answer: i v koncových bodech
Steps:
1. Konce kontroluj vždy.
2. Tedy: i v koncových bodech.
3. Vyber možnost B.
Answer: (B) | calculus |
A norm measures vector length consistently, generalizing absolute value to many dimensions. | null | mat_0927 | concept | en | What is a norm? | [
"A norm measures vector length consistently, generalizing absolute value to many dimensions.",
"So the answer stands: A norm measures vector length consistently, generalizing absolute value to many dimensions."
] | math | Answer: A norm measures vector length consistently, generalizing absolute value to many dimensions.
Steps:
1. A norm measures vector length consistently, generalizing absolute value to many dimensions.
2. So the answer stands: A norm measures vector length consistently, generalizing absolute value to many dimensions. | linear_algebra |
E = 11. | null | mat_0928 | worked_example | en | Compute E of a uniform draw from {1..21}. | [
"Mean of 1..21.",
"(21+1)/2.",
"E = 11."
] | math | Answer: E = 11.
Steps:
1. Mean of 1..21.
2. (21+1)/2.
3. E = 11. | probability |
The lcm is 12. | {
"A": "10",
"B": "14",
"C": "12",
"D": "16"
} | mat_0929 | multiple_choice | en | Find lcm(12, 6). | [
"12×6/gcd.",
"lcm = 12.",
"Select option C."
] | math | Answer: 12
Steps:
1. 12×6/gcd.
2. lcm = 12.
3. Select option C.
Answer: (C) | number_theory |
C(13,3) is 286. | {
"A": "273",
"B": "299",
"C": "286",
"D": "312"
} | mat_0930 | multiple_choice | en | Compute C(13,3). | [
"13×12×11/6.",
"= 286.",
"Select option C."
] | math | Answer: 286
Steps:
1. 13×12×11/6.
2. = 286.
3. Select option C.
Answer: (C) | combinatorics |
The table has 2048 rows. | {
"A": "2046",
"B": "2048",
"C": "2050",
"D": "2052"
} | mat_0931 | multiple_choice | en | How many rows does a truth table with 11 variables have? | [
"2^11 rows.",
"Total: 2048.",
"Select option B."
] | math | Answer: 2048
Steps:
1. 2^11 rows.
2. Total: 2048.
3. Select option B.
Answer: (B) | proofs |
Power series define functions by coefficients, and radius tells where the definition works. | null | mat_0932 | concept | en | Why study power series? | [
"Power series define functions by coefficients, and radius tells where the definition works.",
"So the answer stands: Power series define functions by coefficients, and radius tells where the definition works."
] | math | Answer: Power series define functions by coefficients, and radius tells where the definition works.
Steps:
1. Power series define functions by coefficients, and radius tells where the definition works.
2. So the answer stands: Power series define functions by coefficients, and radius tells where the definition works. | calculus |
The trace is 25. | {
"A": "23",
"B": "27",
"C": "25",
"D": "29"
} | mat_0933 | multiple_choice | en | Find the trace of [[11,1],[2,14]]. | [
"11 + 14.",
"Trace = 25.",
"Select option C."
] | math | Answer: 25
Steps:
1. 11 + 14.
2. Trace = 25.
3. Select option C.
Answer: (C) | linear_algebra |
P(červená) = 6/10. | null | mat_0934 | worked_example | cs | Sáček má 6 červených a 4 modré kuličky. Ukaž P(červená). | [
"Červené: 6.",
"Celkem: 10.",
"P = 6/10."
] | math | Answer: P(červená) = 6/10.
Steps:
1. Červené: 6.
2. Celkem: 10.
3. P = 6/10. | probability |
The gcd is 33. | {
"A": "30",
"B": "33",
"C": "36",
"D": "39"
} | mat_0935 | multiple_choice | en | Find gcd(66, 99). | [
"Factor 3 out twice.",
"gcd = 33.",
"Select option B."
] | math | Answer: 33
Steps:
1. Factor 3 out twice.
2. gcd = 33.
3. Select option B.
Answer: (B) | number_theory |
It finds fully labelled small triangles, implying Brouwer fixed points combinatorially. | null | mat_0936 | concept | en | What is Sperner's lemma? | [
"It finds fully labelled small triangles, implying Brouwer fixed points combinatorially.",
"So the answer stands: It finds fully labelled small triangles, implying Brouwer fixed points combinatorially."
] | math | Answer: It finds fully labelled small triangles, implying Brouwer fixed points combinatorially.
Steps:
1. It finds fully labelled small triangles, implying Brouwer fixed points combinatorially.
2. So the answer stands: It finds fully labelled small triangles, implying Brouwer fixed points combinatorially. | combinatorics |
The sum is 105. | {
"A": "91",
"B": "119",
"C": "105",
"D": "133"
} | mat_0937 | multiple_choice | en | What is the sum 1 + 2 + ... + 14? | [
"14×15/2.",
"Sum = 105.",
"Select option C."
] | math | Answer: 105
Steps:
1. 14×15/2.
2. Sum = 105.
3. Select option C.
Answer: (C) | proofs |
Limit: 38. | null | mat_0938 | worked_example | en | Evaluate the limit of (x² − 19²)/(x − 19) as x → 19. | [
"Factor: (x−19)(x+19).",
"Cancel (x−19).",
"Plug 19: 19+19 = 38."
] | math | Answer: Limit: 38.
Steps:
1. Factor: (x−19)(x+19).
2. Cancel (x−19).
3. Plug 19: 19+19 = 38. | calculus |
Condition numbers bound error blow-up, warning when systems are numerically fragile. | null | mat_0939 | concept | en | Why condition numbers? | [
"Condition numbers bound error blow-up, warning when systems are numerically fragile.",
"So the answer stands: Condition numbers bound error blow-up, warning when systems are numerically fragile."
] | math | Answer: Condition numbers bound error blow-up, warning when systems are numerically fragile.
Steps:
1. Condition numbers bound error blow-up, warning when systems are numerically fragile.
2. So the answer stands: Condition numbers bound error blow-up, warning when systems are numerically fragile. | linear_algebra |
The expected value is 7. | {
"A": "8",
"B": "6",
"C": "7",
"D": "9"
} | mat_0940 | multiple_choice | en | Find the expected value of a uniform draw from {1..13}. | [
"(13+1)/2.",
"E = 7.",
"Select option C."
] | math | Answer: 7
Steps:
1. (13+1)/2.
2. E = 7.
3. Select option C.
Answer: (C) | probability |
The remainder is 1. | {
"A": "2",
"B": "3",
"C": "1",
"D": "4"
} | mat_0941 | multiple_choice | en | Find (163) mod 9. | [
"163 ÷ 9.",
"Remainder 1.",
"Select option C."
] | math | Answer: 1
Steps:
1. 163 ÷ 9.
2. Remainder 1.
3. Select option C.
Answer: (C) | number_theory |
C(13,2) is 78. | {
"A": "65",
"B": "78",
"C": "91",
"D": "104"
} | mat_0942 | multiple_choice | en | Compute C(13,2). | [
"13×12/2.",
"= 78.",
"Select option B."
] | math | Answer: 78
Steps:
1. 13×12/2.
2. = 78.
3. Select option B.
Answer: (B) | combinatorics |
Numbered equations let later steps cite exact lines, keeping long proofs navigable and checkable. | null | mat_0943 | concept | en | Why number equations? | [
"Numbered equations let later steps cite exact lines, keeping long proofs navigable and checkable.",
"So the answer stands: Numbered equations let later steps cite exact lines, keeping long proofs navigable and checkable."
] | math | Answer: Numbered equations let later steps cite exact lines, keeping long proofs navigable and checkable.
Steps:
1. Numbered equations let later steps cite exact lines, keeping long proofs navigable and checkable.
2. So the answer stands: Numbered equations let later steps cite exact lines, keeping long proofs navigabl... | proofs |
The limit is 20. | {
"A": "18",
"B": "22",
"C": "20",
"D": "24"
} | mat_0944 | multiple_choice | en | Find the limit of (x² − 10²)/(x − 10) as x → 10. | [
"Factor (x−10)(x+10).",
"Cancel, get 2×10 = 20.",
"Select option C."
] | math | Answer: 20
Steps:
1. Factor (x−10)(x+10).
2. Cancel, get 2×10 = 20.
3. Select option C.
Answer: (C) | calculus |
LU splits a matrix into lower and upper triangles, speeding repeated solves hugely. | null | mat_0945 | concept | en | What is LU decomposition? | [
"LU splits a matrix into lower and upper triangles, speeding repeated solves hugely.",
"So the answer stands: LU splits a matrix into lower and upper triangles, speeding repeated solves hugely."
] | math | Answer: LU splits a matrix into lower and upper triangles, speeding repeated solves hugely.
Steps:
1. LU splits a matrix into lower and upper triangles, speeding repeated solves hugely.
2. So the answer stands: LU splits a matrix into lower and upper triangles, speeding repeated solves hugely. | linear_algebra |
P(red) = 19/23. | null | mat_0946 | worked_example | en | A bag holds 19 red and 4 blue marbles. Show P(red). | [
"Red: 19.",
"Total: 23.",
"P = 19/23."
] | math | Answer: P(red) = 19/23.
Steps:
1. Red: 19.
2. Total: 23.
3. P = 19/23. | probability |
Remainder: 2. | null | mat_0947 | worked_example | en | Compute (226) mod 7. | [
"Divide 226 by 7.",
"7×32 = 224.",
"226 − 224 = 2."
] | math | Answer: Remainder: 2.
Steps:
1. Divide 226 by 7.
2. 7×32 = 224.
3. 226 − 224 = 2. | number_theory |
C(22,3) = 1540. | null | mat_0948 | worked_example | en | Show the computation of C(22,3). | [
"n(n−1)(n−2)/6.",
"22×21×20/6.",
"= 1540."
] | math | Answer: C(22,3) = 1540.
Steps:
1. n(n−1)(n−2)/6.
2. 22×21×20/6.
3. = 1540. | combinatorics |
Rows: 524288. | null | mat_0949 | worked_example | en | A proof checks all cases of 19 boolean variables. How many rows? | [
"2 choices each.",
"2^19 = 524288."
] | math | Answer: Rows: 524288.
Steps:
1. 2 choices each.
2. 2^19 = 524288. | proofs |
Analytic functions equal their Taylor series nearby, making them smooth and predictable. | null | mat_0950 | concept | en | What is analytic? | [
"Analytic functions equal their Taylor series nearby, making them smooth and predictable.",
"So the answer stands: Analytic functions equal their Taylor series nearby, making them smooth and predictable."
] | math | Answer: Analytic functions equal their Taylor series nearby, making them smooth and predictable.
Steps:
1. Analytic functions equal their Taylor series nearby, making them smooth and predictable.
2. So the answer stands: Analytic functions equal their Taylor series nearby, making them smooth and predictable. | calculus |
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