answer string | choices dict | id string | kind string | lang string | question string | steps list | subject string | text string | topic string |
|---|---|---|---|---|---|---|---|---|---|
lcm = 104. | null | mat_0551 | worked_example | en | Compute lcm(13, 8) step by step. | [
"gcd(13,8) = 1.",
"13×8/1.",
"= 104."
] | math | Answer: lcm = 104.
Steps:
1. gcd(13,8) = 1.
2. 13×8/1.
3. = 104. | number_theory |
Staví řádky sčítáním a ukazuje symetrie. | null | mat_0552 | concept | cs | Proč Pascalův trojúhelník? | [
"Staví řádky sčítáním a ukazuje symetrie.",
"Odpověď tedy platí: Staví řádky sčítáním a ukazuje symetrie."
] | math | Answer: Staví řádky sčítáním a ukazuje symetrie.
Steps:
1. Staví řádky sčítáním a ukazuje symetrie.
2. Odpověď tedy platí: Staví řádky sčítáním a ukazuje symetrie. | combinatorics |
Truth tables list every combination of truth values, so they verify logical equivalences mechanically. | null | mat_0553 | concept | en | Why learn truth tables? | [
"Truth tables list every combination of truth values, so they verify logical equivalences mechanically.",
"So the answer stands: Truth tables list every combination of truth values, so they verify logical equivalences mechanically."
] | math | Answer: Truth tables list every combination of truth values, so they verify logical equivalences mechanically.
Steps:
1. Truth tables list every combination of truth values, so they verify logical equivalences mechanically.
2. So the answer stands: Truth tables list every combination of truth values, so they verify log... | proofs |
They find extrema under constraints by aligning gradients, solving optimization with limits. | null | mat_0554 | concept | en | Why use Lagrange multipliers? | [
"They find extrema under constraints by aligning gradients, solving optimization with limits.",
"So the answer stands: They find extrema under constraints by aligning gradients, solving optimization with limits."
] | math | Answer: They find extrema under constraints by aligning gradients, solving optimization with limits.
Steps:
1. They find extrema under constraints by aligning gradients, solving optimization with limits.
2. So the answer stands: They find extrema under constraints by aligning gradients, solving optimization with limits... | calculus |
It solves systems via determinant ratios, elegant for theory though slow in practice. | null | mat_0555 | concept | en | What is Cramer's rule? | [
"It solves systems via determinant ratios, elegant for theory though slow in practice.",
"So the answer stands: It solves systems via determinant ratios, elegant for theory though slow in practice."
] | math | Answer: It solves systems via determinant ratios, elegant for theory though slow in practice.
Steps:
1. It solves systems via determinant ratios, elegant for theory though slow in practice.
2. So the answer stands: It solves systems via determinant ratios, elegant for theory though slow in practice. | linear_algebra |
P = 1/15. | null | mat_0556 | worked_example | en | A fair 15-sided die is rolled. Show P(one specific number). | [
"Favourable: 1.",
"Possible: 15.",
"P = 1/15."
] | math | Answer: P = 1/15.
Steps:
1. Favourable: 1.
2. Possible: 15.
3. P = 1/15. | probability |
Opakované zbytky rychle zmenšují dvojici. | null | mat_0557 | concept | cs | Proč Euklidův algoritmus? | [
"Opakované zbytky rychle zmenšují dvojici.",
"Odpověď tedy platí: Opakované zbytky rychle zmenšují dvojici."
] | math | Answer: Opakované zbytky rychle zmenšují dvojici.
Steps:
1. Opakované zbytky rychle zmenšují dvojici.
2. Odpověď tedy platí: Opakované zbytky rychle zmenšují dvojici. | number_theory |
Rook placements count permutations with restrictions, via inclusion-exclusion boards. | null | mat_0558 | concept | en | Why rook theory? | [
"Rook placements count permutations with restrictions, via inclusion-exclusion boards.",
"So the answer stands: Rook placements count permutations with restrictions, via inclusion-exclusion boards."
] | math | Answer: Rook placements count permutations with restrictions, via inclusion-exclusion boards.
Steps:
1. Rook placements count permutations with restrictions, via inclusion-exclusion boards.
2. So the answer stands: Rook placements count permutations with restrictions, via inclusion-exclusion boards. | combinatorics |
Sum: 136. | null | mat_0559 | worked_example | en | Prove by formula: find the sum 1 + 2 + ... + 16. | [
"Pair 1+16, 2+15, ...",
"Use 16×17/2.",
"= 136."
] | math | Answer: Sum: 136.
Steps:
1. Pair 1+16, 2+15, ...
2. Use 16×17/2.
3. = 136. | proofs |
The derivative is 10. | {
"A": "10",
"B": "8",
"C": "12",
"D": "14"
} | mat_0560 | multiple_choice | en | For f(x) = 2x² + 6x, find f'(1). | [
"f'(x) = 4x + 6.",
"At 1: 10.",
"Select option A."
] | math | Answer: 10
Steps:
1. f'(x) = 4x + 6.
2. At 1: 10.
3. Select option A.
Answer: (A) | calculus |
Determinant: 154. | null | mat_0561 | worked_example | en | Compute det [[12,1],[2,13]] step by step. | [
"ad − bc.",
"12×13 − 1×2.",
"= 154."
] | math | Answer: Determinant: 154.
Steps:
1. ad − bc.
2. 12×13 − 1×2.
3. = 154. | linear_algebra |
E = 3,5. | null | mat_0562 | worked_example | cs | Spočítej E rovnoměrného tahu z {1..6}. | [
"Průměr 1..6.",
"(6+1)/2.",
"E = 3,5."
] | math | Answer: E = 3,5.
Steps:
1. Průměr 1..6.
2. (6+1)/2.
3. E = 3,5. | probability |
It equals the sum of its proper divisors, like 6 = 1+2+3, rare and mysterious. | null | mat_0563 | concept | en | What is a perfect number? | [
"It equals the sum of its proper divisors, like 6 = 1+2+3, rare and mysterious.",
"So the answer stands: It equals the sum of its proper divisors, like 6 = 1+2+3, rare and mysterious."
] | math | Answer: It equals the sum of its proper divisors, like 6 = 1+2+3, rare and mysterious.
Steps:
1. It equals the sum of its proper divisors, like 6 = 1+2+3, rare and mysterious.
2. So the answer stands: It equals the sum of its proper divisors, like 6 = 1+2+3, rare and mysterious. | number_theory |
C(6,3) je 20. | {
"A": "14",
"B": "26",
"C": "32",
"D": "20"
} | mat_0564 | multiple_choice | cs | Vypočítej kombinační číslo C(6,3). | [
"6×5×4/6.",
"= 20.",
"Vyber možnost D."
] | math | Answer: 20
Steps:
1. 6×5×4/6.
2. = 20.
3. Vyber možnost D.
Answer: (D) | combinatorics |
Half: 78, so 156 is even. | null | mat_0565 | worked_example | en | Show N² + N is even for N = 12 by computing its half. | [
"12² + 12 = 156.",
"156 / 2 = 78.",
"Integer, so even."
] | math | Answer: Half: 78, so 156 is even.
Steps:
1. 12² + 12 = 156.
2. 156 / 2 = 78.
3. Integer, so even. | proofs |
The integral is 216. | {
"A": "210",
"B": "216",
"C": "222",
"D": "228"
} | mat_0566 | multiple_choice | en | Compute the integral of 3x² from 0 to 6. | [
"Antiderivative x³.",
"6³ = 216.",
"Select option B."
] | math | Answer: 216
Steps:
1. Antiderivative x³.
2. 6³ = 216.
3. Select option B.
Answer: (B) | calculus |
Trace: 27. | null | mat_0567 | worked_example | en | Compute the trace of [[13,0],[0,14]]. | [
"Sum diagonal.",
"13 + 14.",
"= 27."
] | math | Answer: Trace: 27.
Steps:
1. Sum diagonal.
2. 13 + 14.
3. = 27. | linear_algebra |
E = 7.5. | null | mat_0568 | worked_example | en | Compute E of a uniform draw from {1..14}. | [
"Mean of 1..14.",
"(14+1)/2.",
"E = 7.5."
] | math | Answer: E = 7.5.
Steps:
1. Mean of 1..14.
2. (14+1)/2.
3. E = 7.5. | probability |
Počítá se zbytky po dělení, čísla se obtáčejí kolem hodin. | null | mat_0569 | concept | cs | Co je modulární aritmetika? | [
"Počítá se zbytky po dělení, čísla se obtáčejí kolem hodin.",
"Odpověď tedy platí: Počítá se zbytky po dělení, čísla se obtáčejí kolem hodin."
] | math | Answer: Počítá se zbytky po dělení, čísla se obtáčejí kolem hodin.
Steps:
1. Počítá se zbytky po dělení, čísla se obtáčejí kolem hodin.
2. Odpověď tedy platí: Počítá se zbytky po dělení, čísla se obtáčejí kolem hodin. | number_theory |
Odpověď je kombinační číslo. | {
"A": "kombinační číslo",
"B": "n+r",
"C": "n*r",
"D": "n+r!"
} | mat_0570 | multiple_choice | cs | Vyber: co počítá podmnožiny? | [
"Pořadí vyděleno pryč.",
"Tedy: kombinační číslo.",
"Vyber možnost A."
] | math | Answer: kombinační číslo
Steps:
1. Pořadí vyděleno pryč.
2. Tedy: kombinační číslo.
3. Vyber možnost A.
Answer: (A) | combinatorics |
Rows: 8192. | null | mat_0571 | worked_example | en | A proof checks all cases of 13 boolean variables. How many rows? | [
"2 choices each.",
"2^13 = 8192."
] | math | Answer: Rows: 8192.
Steps:
1. 2 choices each.
2. 2^13 = 8192. | proofs |
It relates a function to its derivatives, and solving it finds the function from its change law. | null | mat_0572 | concept | en | What is a differential equation? | [
"It relates a function to its derivatives, and solving it finds the function from its change law.",
"So the answer stands: It relates a function to its derivatives, and solving it finds the function from its change law."
] | math | Answer: It relates a function to its derivatives, and solving it finds the function from its change law.
Steps:
1. It relates a function to its derivatives, and solving it finds the function from its change law.
2. So the answer stands: It relates a function to its derivatives, and solving it finds the function from it... | calculus |
Dot product: 91. | null | mat_0573 | worked_example | en | Compute (12,11) · (3,5) step by step. | [
"Multiply components.",
"12×3 + 11×5.",
"= 91."
] | math | Answer: Dot product: 91.
Steps:
1. Multiply components.
2. 12×3 + 11×5.
3. = 91. | linear_algebra |
It gives probabilities of value pairs, capturing how two variables vary together. | null | mat_0574 | concept | en | What is a joint distribution? | [
"It gives probabilities of value pairs, capturing how two variables vary together.",
"So the answer stands: It gives probabilities of value pairs, capturing how two variables vary together."
] | math | Answer: It gives probabilities of value pairs, capturing how two variables vary together.
Steps:
1. It gives probabilities of value pairs, capturing how two variables vary together.
2. So the answer stands: It gives probabilities of value pairs, capturing how two variables vary together. | probability |
Odpověď je Euklidův argument. | {
"A": "Euklidův argument",
"B": "velké počítače",
"C": "dlouhé seznamy",
"D": "hlasitá tvrzení"
} | mat_0575 | multiple_choice | cs | Vyber: co nutí nová prvočísla? | [
"Součin plus jedna.",
"Tedy: Euklidův argument.",
"Vyber možnost A."
] | math | Answer: Euklidův argument
Steps:
1. Součin plus jedna.
2. Tedy: Euklidův argument.
3. Vyber možnost A.
Answer: (A) | number_theory |
Rozvíjí (a+b)^n do členů s kombinačními čísly. | null | mat_0576 | concept | cs | Co je binomická věta? | [
"Rozvíjí (a+b)^n do členů s kombinačními čísly.",
"Odpověď tedy platí: Rozvíjí (a+b)^n do členů s kombinačními čísly."
] | math | Answer: Rozvíjí (a+b)^n do členů s kombinačními čísly.
Steps:
1. Rozvíjí (a+b)^n do členů s kombinačními čísly.
2. Odpověď tedy platí: Rozvíjí (a+b)^n do členů s kombinačními čísly. | combinatorics |
The sum is 55. | {
"A": "45",
"B": "65",
"C": "55",
"D": "75"
} | mat_0577 | multiple_choice | en | What is the sum 1 + 2 + ... + 10? | [
"10×11/2.",
"Sum = 55.",
"Select option C."
] | math | Answer: 55
Steps:
1. 10×11/2.
2. Sum = 55.
3. Select option C.
Answer: (C) | proofs |
The limit is 12. | {
"A": "10",
"B": "14",
"C": "12",
"D": "16"
} | mat_0578 | multiple_choice | en | Find the limit of (x² − 6²)/(x − 6) as x → 6. | [
"Factor (x−6)(x+6).",
"Cancel, get 2×6 = 12.",
"Select option C."
] | math | Answer: 12
Steps:
1. Factor (x−6)(x+6).
2. Cancel, get 2×6 = 12.
3. Select option C.
Answer: (C) | calculus |
Determinant: 180. | null | mat_0579 | worked_example | en | Compute det [[13,1],[2,14]] step by step. | [
"ad − bc.",
"13×14 − 1×2.",
"= 180."
] | math | Answer: Determinant: 180.
Steps:
1. ad − bc.
2. 13×14 − 1×2.
3. = 180. | linear_algebra |
P(red) = 12/16. | null | mat_0580 | worked_example | en | A bag holds 12 red and 4 blue marbles. Show P(red). | [
"Red: 12.",
"Total: 16.",
"P = 12/16."
] | math | Answer: P(red) = 12/16.
Steps:
1. Red: 12.
2. Total: 16.
3. P = 12/16. | probability |
Numbers 2^p-1 yield huge primes, driving record prime hunts and perfect numbers. | null | mat_0581 | concept | en | Why Mersenne primes? | [
"Numbers 2^p-1 yield huge primes, driving record prime hunts and perfect numbers.",
"So the answer stands: Numbers 2^p-1 yield huge primes, driving record prime hunts and perfect numbers."
] | math | Answer: Numbers 2^p-1 yield huge primes, driving record prime hunts and perfect numbers.
Steps:
1. Numbers 2^p-1 yield huge primes, driving record prime hunts and perfect numbers.
2. So the answer stands: Numbers 2^p-1 yield huge primes, driving record prime hunts and perfect numbers. | number_theory |
C(15,2) = 105. | null | mat_0582 | worked_example | en | Show the computation of C(15,2). | [
"Formula n(n−1)/2.",
"15×14/2.",
"= 105."
] | math | Answer: C(15,2) = 105.
Steps:
1. Formula n(n−1)/2.
2. 15×14/2.
3. = 105. | combinatorics |
Modus ponens infers Q from P and P-implies-Q, and it is the basic step of deductive reasoning. | null | mat_0583 | concept | en | What is modus ponens? | [
"Modus ponens infers Q from P and P-implies-Q, and it is the basic step of deductive reasoning.",
"So the answer stands: Modus ponens infers Q from P and P-implies-Q, and it is the basic step of deductive reasoning."
] | math | Answer: Modus ponens infers Q from P and P-implies-Q, and it is the basic step of deductive reasoning.
Steps:
1. Modus ponens infers Q from P and P-implies-Q, and it is the basic step of deductive reasoning.
2. So the answer stands: Modus ponens infers Q from P and P-implies-Q, and it is the basic step of deductive rea... | proofs |
Separation splits dy/dx equations into integrable halves, solving many first-order equations. | null | mat_0584 | concept | en | Why separate variables? | [
"Separation splits dy/dx equations into integrable halves, solving many first-order equations.",
"So the answer stands: Separation splits dy/dx equations into integrable halves, solving many first-order equations."
] | math | Answer: Separation splits dy/dx equations into integrable halves, solving many first-order equations.
Steps:
1. Separation splits dy/dx equations into integrable halves, solving many first-order equations.
2. So the answer stands: Separation splits dy/dx equations into integrable halves, solving many first-order equati... | calculus |
Skalární součin: 19. | null | mat_0585 | worked_example | cs | Spočítej (3,2) · (3,5) krok za krokem. | [
"Násob složky.",
"3×3 + 2×5.",
"= 19."
] | math | Answer: Skalární součin: 19.
Steps:
1. Násob složky.
2. 3×3 + 2×5.
3. = 19. | linear_algebra |
P(červená) = 4/8. | null | mat_0586 | worked_example | cs | Sáček má 4 červených a 4 modré kuličky. Ukaž P(červená). | [
"Červené: 4.",
"Celkem: 8.",
"P = 4/8."
] | math | Answer: P(červená) = 4/8.
Steps:
1. Červené: 4.
2. Celkem: 8.
3. P = 4/8. | probability |
Kongruence zachovávají součty a součiny, velké úlohy zmenšují. | null | mat_0587 | concept | cs | Proč kongruence? | [
"Kongruence zachovávají součty a součiny, velké úlohy zmenšují.",
"Odpověď tedy platí: Kongruence zachovávají součty a součiny, velké úlohy zmenšují."
] | math | Answer: Kongruence zachovávají součty a součiny, velké úlohy zmenšují.
Steps:
1. Kongruence zachovávají součty a součiny, velké úlohy zmenšují.
2. Odpověď tedy platí: Kongruence zachovávají součty a součiny, velké úlohy zmenšují. | number_theory |
It extends nCr to many groups at once, dividing n! by each group factorial. | null | mat_0588 | concept | en | What is the multinomial? | [
"It extends nCr to many groups at once, dividing n! by each group factorial.",
"So the answer stands: It extends nCr to many groups at once, dividing n! by each group factorial."
] | math | Answer: It extends nCr to many groups at once, dividing n! by each group factorial.
Steps:
1. It extends nCr to many groups at once, dividing n! by each group factorial.
2. So the answer stands: It extends nCr to many groups at once, dividing n! by each group factorial. | combinatorics |
Circular proofs assume the conclusion, so they prove nothing even when every step looks valid. | null | mat_0589 | concept | en | Why avoid circular proofs? | [
"Circular proofs assume the conclusion, so they prove nothing even when every step looks valid.",
"So the answer stands: Circular proofs assume the conclusion, so they prove nothing even when every step looks valid."
] | math | Answer: Circular proofs assume the conclusion, so they prove nothing even when every step looks valid.
Steps:
1. Circular proofs assume the conclusion, so they prove nothing even when every step looks valid.
2. So the answer stands: Circular proofs assume the conclusion, so they prove nothing even when every step looks... | proofs |
Limit: 26. | null | mat_0590 | worked_example | en | Evaluate the limit of (x² − 13²)/(x − 13) as x → 13. | [
"Factor: (x−13)(x+13).",
"Cancel (x−13).",
"Plug 13: 13+13 = 26."
] | math | Answer: Limit: 26.
Steps:
1. Factor: (x−13)(x+13).
2. Cancel (x−13).
3. Plug 13: 13+13 = 26. | calculus |
Linear maps preserve structure simply, modelling rotations, scalings, and projections exactly. | null | mat_0591 | concept | en | Why study linear maps? | [
"Linear maps preserve structure simply, modelling rotations, scalings, and projections exactly.",
"So the answer stands: Linear maps preserve structure simply, modelling rotations, scalings, and projections exactly."
] | math | Answer: Linear maps preserve structure simply, modelling rotations, scalings, and projections exactly.
Steps:
1. Linear maps preserve structure simply, modelling rotations, scalings, and projections exactly.
2. So the answer stands: Linear maps preserve structure simply, modelling rotations, scalings, and projections e... | linear_algebra |
The probability is 1/10. | {
"A": "1/10",
"B": "1/9",
"C": "1/11",
"D": "2/10"
} | mat_0592 | multiple_choice | en | A fair 10-sided die is rolled. What is the probability of one specific number? | [
"1 face of 10.",
"P = 1/10.",
"Select option A."
] | math | Answer: 1/10
Steps:
1. 1 face of 10.
2. P = 1/10.
3. Select option A.
Answer: (A) | probability |
The lcm is 24. | {
"A": "22",
"B": "26",
"C": "24",
"D": "28"
} | mat_0593 | multiple_choice | en | Find lcm(8, 6). | [
"8×6/gcd.",
"lcm = 24.",
"Select option C."
] | math | Answer: 24
Steps:
1. 8×6/gcd.
2. lcm = 24.
3. Select option C.
Answer: (C) | number_theory |
P(14,3) = 2184. | null | mat_0594 | worked_example | en | Show the computation of P(14,3). | [
"n(n−1)(n−2).",
"14×13×12.",
"= 2184."
] | math | Answer: P(14,3) = 2184.
Steps:
1. n(n−1)(n−2).
2. 14×13×12.
3. = 2184. | combinatorics |
Hodnota je 12. | {
"A": "10",
"B": "14",
"C": "12",
"D": "16"
} | mat_0595 | multiple_choice | cs | Vypočítej N² + N pro N = 3. | [
"3² + 3.",
"Hodnota = 12.",
"Vyber možnost C."
] | math | Answer: 12
Steps:
1. 3² + 3.
2. Hodnota = 12.
3. Vyber možnost C.
Answer: (C) | proofs |
f'(4) = 44. | null | mat_0596 | worked_example | en | For f(x) = 4x² + 12x, compute f'(4) step by step. | [
"f'(x) = 8x + 12.",
"Plug x = 4.",
"8×4 + 12 = 44."
] | math | Answer: f'(4) = 44.
Steps:
1. f'(x) = 8x + 12.
2. Plug x = 4.
3. 8×4 + 12 = 44. | calculus |
The dot product is 39. | {
"A": "35",
"B": "43",
"C": "39",
"D": "47"
} | mat_0597 | multiple_choice | en | Find the dot product (5,6) · (3,4). | [
"5×3 + 6×4.",
"Product = 39.",
"Select option C."
] | math | Answer: 39
Steps:
1. 5×3 + 6×4.
2. Product = 39.
3. Select option C.
Answer: (C) | linear_algebra |
P = 1/8. | null | mat_0598 | worked_example | cs | Hází se férovou 8-stěnnou kostkou. Ukaž P(jedno konkrétní číslo). | [
"Příznivé: 1.",
"Možné: 8.",
"P = 1/8."
] | math | Answer: P = 1/8.
Steps:
1. Příznivé: 1.
2. Možné: 8.
3. P = 1/8. | probability |
gcd = 22. | null | mat_0599 | worked_example | en | Compute gcd(44, 66) step by step. | [
"Factor: 44=4×11, 66=6×11.",
"Common: 11×gcd(4,6).",
"= 11×2 = 22."
] | math | Answer: gcd = 22.
Steps:
1. Factor: 44=4×11, 66=6×11.
2. Common: 11×gcd(4,6).
3. = 11×2 = 22. | number_theory |
It builds labelled structures from components, exponentiating their generating series. | null | mat_0600 | concept | en | Why exponential formula? | [
"It builds labelled structures from components, exponentiating their generating series.",
"So the answer stands: It builds labelled structures from components, exponentiating their generating series."
] | math | Answer: It builds labelled structures from components, exponentiating their generating series.
Steps:
1. It builds labelled structures from components, exponentiating their generating series.
2. So the answer stands: It builds labelled structures from components, exponentiating their generating series. | combinatorics |
Strong induction assumes all earlier cases to prove the next, which suits claims needing many predecessors. | null | mat_0601 | concept | en | What is strong induction? | [
"Strong induction assumes all earlier cases to prove the next, which suits claims needing many predecessors.",
"So the answer stands: Strong induction assumes all earlier cases to prove the next, which suits claims needing many predecessors."
] | math | Answer: Strong induction assumes all earlier cases to prove the next, which suits claims needing many predecessors.
Steps:
1. Strong induction assumes all earlier cases to prove the next, which suits claims needing many predecessors.
2. So the answer stands: Strong induction assumes all earlier cases to prove the next,... | proofs |
The answer is Ratio test. | {
"A": "Coin toss",
"B": "Mood ring",
"C": "Dice roll",
"D": "Ratio test"
} | mat_0602 | multiple_choice | en | Choose: what tests series? | [
"Below one converges.",
"So: Ratio test.",
"Select option D."
] | math | Answer: Ratio test
Steps:
1. Below one converges.
2. So: Ratio test.
3. Select option D.
Answer: (D) | calculus |
Trace: 29. | null | mat_0603 | worked_example | en | Compute the trace of [[14,0],[0,15]]. | [
"Sum diagonal.",
"14 + 15.",
"= 29."
] | math | Answer: Trace: 29.
Steps:
1. Sum diagonal.
2. 14 + 15.
3. = 29. | linear_algebra |
Summing out variables isolates one distribution, focusing analysis where needed. | null | mat_0604 | concept | en | Why marginalize? | [
"Summing out variables isolates one distribution, focusing analysis where needed.",
"So the answer stands: Summing out variables isolates one distribution, focusing analysis where needed."
] | math | Answer: Summing out variables isolates one distribution, focusing analysis where needed.
Steps:
1. Summing out variables isolates one distribution, focusing analysis where needed.
2. So the answer stands: Summing out variables isolates one distribution, focusing analysis where needed. | probability |
The answer is Totient phi. | {
"A": "Lucky guess",
"B": "Coin toss",
"C": "Dice roll",
"D": "Totient phi"
} | mat_0605 | multiple_choice | en | Choose: what counts coprimes? | [
"Below n, coprime.",
"So: Totient phi.",
"Select option D."
] | math | Answer: Totient phi
Steps:
1. Below n, coprime.
2. So: Totient phi.
3. Select option D.
Answer: (D) | number_theory |
It counts orbits under symmetry by averaging fixed points over the group. | null | mat_0606 | concept | en | What is Burnside's lemma? | [
"It counts orbits under symmetry by averaging fixed points over the group.",
"So the answer stands: It counts orbits under symmetry by averaging fixed points over the group."
] | math | Answer: It counts orbits under symmetry by averaging fixed points over the group.
Steps:
1. It counts orbits under symmetry by averaging fixed points over the group.
2. So the answer stands: It counts orbits under symmetry by averaging fixed points over the group. | combinatorics |
Divisibility proofs reveal why digit tricks work, grounding mental arithmetic in modular structure. | null | mat_0607 | concept | en | Why prove divisibility rules? | [
"Divisibility proofs reveal why digit tricks work, grounding mental arithmetic in modular structure.",
"So the answer stands: Divisibility proofs reveal why digit tricks work, grounding mental arithmetic in modular structure."
] | math | Answer: Divisibility proofs reveal why digit tricks work, grounding mental arithmetic in modular structure.
Steps:
1. Divisibility proofs reveal why digit tricks work, grounding mental arithmetic in modular structure.
2. So the answer stands: Divisibility proofs reveal why digit tricks work, grounding mental arithmetic... | proofs |
Integral: 2744. | null | mat_0608 | worked_example | en | Evaluate the integral of 3x² from 0 to 14. | [
"Antiderivative: x³.",
"Evaluate 0 to 14.",
"14³ − 0 = 2744."
] | math | Answer: Integral: 2744.
Steps:
1. Antiderivative: x³.
2. Evaluate 0 to 14.
3. 14³ − 0 = 2744. | calculus |
Dot product: 99. | null | mat_0609 | worked_example | en | Compute (13,12) · (3,5) step by step. | [
"Multiply components.",
"13×3 + 12×5.",
"= 99."
] | math | Answer: Dot product: 99.
Steps:
1. Multiply components.
2. 13×3 + 12×5.
3. = 99. | linear_algebra |
P = 1/16. | null | mat_0610 | worked_example | en | A fair 16-sided die is rolled. Show P(one specific number). | [
"Favourable: 1.",
"Possible: 16.",
"P = 1/16."
] | math | Answer: P = 1/16.
Steps:
1. Favourable: 1.
2. Possible: 16.
3. P = 1/16. | probability |
Remainder: 2. | null | mat_0611 | worked_example | en | Compute (149) mod 7. | [
"Divide 149 by 7.",
"7×21 = 147.",
"149 − 147 = 2."
] | math | Answer: Remainder: 2.
Steps:
1. Divide 149 by 7.
2. 7×21 = 147.
3. 149 − 147 = 2. | number_theory |
C(7,3) = 35. | null | mat_0612 | worked_example | cs | Ukaž výpočet C(7,3). | [
"n(n−1)(n−2)/6.",
"7×6×5/6.",
"= 35."
] | math | Answer: C(7,3) = 35.
Steps:
1. n(n−1)(n−2)/6.
2. 7×6×5/6.
3. = 35. | combinatorics |
If more items than boxes exist, some box holds two, and this counting fact proves surprising claims. | null | mat_0613 | concept | en | What is the pigeonhole principle? | [
"If more items than boxes exist, some box holds two, and this counting fact proves surprising claims.",
"So the answer stands: If more items than boxes exist, some box holds two, and this counting fact proves surprising claims."
] | math | Answer: If more items than boxes exist, some box holds two, and this counting fact proves surprising claims.
Steps:
1. If more items than boxes exist, some box holds two, and this counting fact proves surprising claims.
2. So the answer stands: If more items than boxes exist, some box holds two, and this counting fact ... | proofs |
Derivace je 11. | {
"A": "9",
"B": "11",
"C": "13",
"D": "15"
} | mat_0614 | multiple_choice | cs | Pro f(x) = 2x² + 3x vypočítej f'(2). | [
"f'(x) = 4x + 3.",
"V 2: 11.",
"Vyber možnost B."
] | math | Answer: 11
Steps:
1. f'(x) = 4x + 3.
2. V 2: 11.
3. Vyber možnost B.
Answer: (B) | calculus |
The kernel is the null space of a map, capturing exactly what information is lost. | null | mat_0615 | concept | en | What is the kernel? | [
"The kernel is the null space of a map, capturing exactly what information is lost.",
"So the answer stands: The kernel is the null space of a map, capturing exactly what information is lost."
] | math | Answer: The kernel is the null space of a map, capturing exactly what information is lost.
Steps:
1. The kernel is the null space of a map, capturing exactly what information is lost.
2. So the answer stands: The kernel is the null space of a map, capturing exactly what information is lost. | linear_algebra |
Výběrové průměry se blíží střední hodnotě s rostoucím počtem pokusů. | null | mat_0616 | concept | cs | Co je zákon velkých čísel? | [
"Výběrové průměry se blíží střední hodnotě s rostoucím počtem pokusů.",
"Odpověď tedy platí: Výběrové průměry se blíží střední hodnotě s rostoucím počtem pokusů."
] | math | Answer: Výběrové průměry se blíží střední hodnotě s rostoucím počtem pokusů.
Steps:
1. Výběrové průměry se blíží střední hodnotě s rostoucím počtem pokusů.
2. Odpověď tedy platí: Výběrové průměry se blíží střední hodnotě s rostoucím počtem pokusů. | probability |
lcm = 56. | null | mat_0617 | worked_example | en | Compute lcm(14, 8) step by step. | [
"gcd(14,8) = 2.",
"14×8/2.",
"= 56."
] | math | Answer: lcm = 56.
Steps:
1. gcd(14,8) = 2.
2. 14×8/2.
3. = 56. | number_theory |
The answer is Binomial theorem. | {
"A": "Wild guess",
"B": "Long division",
"C": "Blind hope",
"D": "Binomial theorem"
} | mat_0618 | multiple_choice | en | Choose: what expands (a+b)^n? | [
"nCr-weighted terms.",
"So: Binomial theorem.",
"Select option D."
] | math | Answer: Binomial theorem
Steps:
1. nCr-weighted terms.
2. So: Binomial theorem.
3. Select option D.
Answer: (D) | combinatorics |
Sum: 153. | null | mat_0619 | worked_example | en | Prove by formula: find the sum 1 + 2 + ... + 17. | [
"Pair 1+17, 2+16, ...",
"Use 17×18/2.",
"= 153."
] | math | Answer: Sum: 153.
Steps:
1. Pair 1+17, 2+16, ...
2. Use 17×18/2.
3. = 153. | proofs |
Limit: 28. | null | mat_0620 | worked_example | en | Evaluate the limit of (x² − 14²)/(x − 14) as x → 14. | [
"Factor: (x−14)(x+14).",
"Cancel (x−14).",
"Plug 14: 14+14 = 28."
] | math | Answer: Limit: 28.
Steps:
1. Factor: (x−14)(x+14).
2. Cancel (x−14).
3. Plug 14: 14+14 = 28. | calculus |
Determinant: 18. | null | mat_0621 | worked_example | cs | Spočítej det [[4,1],[2,5]] krok za krokem. | [
"ad − bc.",
"4×5 − 1×2.",
"= 18."
] | math | Answer: Determinant: 18.
Steps:
1. ad − bc.
2. 4×5 − 1×2.
3. = 18. | linear_algebra |
The expected value is 4.5. | {
"A": "5.5",
"B": "4.5",
"C": "3.5",
"D": "6.5"
} | mat_0622 | multiple_choice | en | Find the expected value of a uniform draw from {1..8}. | [
"(8+1)/2.",
"E = 4.5.",
"Select option B."
] | math | Answer: 4.5
Steps:
1. (8+1)/2.
2. E = 4.5.
3. Select option B.
Answer: (B) | probability |
gcd = 24. | null | mat_0623 | worked_example | en | Compute gcd(48, 72) step by step. | [
"Factor: 48=4×12, 72=6×12.",
"Common: 12×gcd(4,6).",
"= 12×2 = 24."
] | math | Answer: gcd = 24.
Steps:
1. Factor: 48=4×12, 72=6×12.
2. Common: 12×gcd(4,6).
3. = 12×2 = 24. | number_theory |
C(16,3) = 560. | null | mat_0624 | worked_example | en | Show the computation of C(16,3). | [
"n(n−1)(n−2)/6.",
"16×15×14/6.",
"= 560."
] | math | Answer: C(16,3) = 560.
Steps:
1. n(n−1)(n−2)/6.
2. 16×15×14/6.
3. = 560. | combinatorics |
Half: 91, so 182 is even. | null | mat_0625 | worked_example | en | Show N² + N is even for N = 13 by computing its half. | [
"13² + 13 = 182.",
"182 / 2 = 91.",
"Integer, so even."
] | math | Answer: Half: 91, so 182 is even.
Steps:
1. 13² + 13 = 182.
2. 182 / 2 = 91.
3. Integer, so even. | proofs |
Integrál je 27. | {
"A": "24",
"B": "30",
"C": "27",
"D": "33"
} | mat_0626 | multiple_choice | cs | Vypočítej integrál 3x² od 0 do 3. | [
"Primitivní x³.",
"3³ = 27.",
"Vyber možnost C."
] | math | Answer: 27
Steps:
1. Primitivní x³.
2. 3³ = 27.
3. Vyber možnost C.
Answer: (C) | calculus |
The answer is A basis. | {
"A": "A guess",
"B": "A hope",
"C": "A dream",
"D": "A basis"
} | mat_0627 | multiple_choice | en | Choose: what gives coordinates? | [
"Independent spanning set.",
"So: A basis.",
"Select option D."
] | math | Answer: A basis
Steps:
1. Independent spanning set.
2. So: A basis.
3. Select option D.
Answer: (D) | linear_algebra |
E = 4. | null | mat_0628 | worked_example | cs | Spočítej E rovnoměrného tahu z {1..7}. | [
"Průměr 1..7.",
"(7+1)/2.",
"E = 4."
] | math | Answer: E = 4.
Steps:
1. Průměr 1..7.
2. (7+1)/2.
3. E = 4. | probability |
Pro prvočíslo p je a^(p-1) rovno 1 mod p. | null | mat_0629 | concept | cs | Co je malá Fermatova věta? | [
"Pro prvočíslo p je a^(p-1) rovno 1 mod p.",
"Odpověď tedy platí: Pro prvočíslo p je a^(p-1) rovno 1 mod p."
] | math | Answer: Pro prvočíslo p je a^(p-1) rovno 1 mod p.
Steps:
1. Pro prvočíslo p je a^(p-1) rovno 1 mod p.
2. Odpověď tedy platí: Pro prvočíslo p je a^(p-1) rovno 1 mod p. | number_theory |
It counts colorings up to symmetry via cycle index, generalizing Burnside elegantly. | null | mat_0630 | concept | en | Why Pólya counting? | [
"It counts colorings up to symmetry via cycle index, generalizing Burnside elegantly.",
"So the answer stands: It counts colorings up to symmetry via cycle index, generalizing Burnside elegantly."
] | math | Answer: It counts colorings up to symmetry via cycle index, generalizing Burnside elegantly.
Steps:
1. It counts colorings up to symmetry via cycle index, generalizing Burnside elegantly.
2. So the answer stands: It counts colorings up to symmetry via cycle index, generalizing Burnside elegantly. | combinatorics |
Rows: 16384. | null | mat_0631 | worked_example | en | A proof checks all cases of 14 boolean variables. How many rows? | [
"2 choices each.",
"2^14 = 16384."
] | math | Answer: Rows: 16384.
Steps:
1. 2 choices each.
2. 2^14 = 16384. | proofs |
The answer is Mean value theorem. | {
"A": "Mean value theorem",
"B": "Lucky guess",
"C": "Long proof",
"D": "Big book"
} | mat_0632 | multiple_choice | en | Choose: what guarantees matched rates? | [
"Some point matches.",
"So: Mean value theorem.",
"Select option A."
] | math | Answer: Mean value theorem
Steps:
1. Some point matches.
2. So: Mean value theorem.
3. Select option A.
Answer: (A) | calculus |
Determinant: 208. | null | mat_0633 | worked_example | en | Compute det [[14,1],[2,15]] step by step. | [
"ad − bc.",
"14×15 − 1×2.",
"= 208."
] | math | Answer: Determinant: 208.
Steps:
1. ad − bc.
2. 14×15 − 1×2.
3. = 208. | linear_algebra |
E = 8. | null | mat_0634 | worked_example | en | Compute E of a uniform draw from {1..15}. | [
"Mean of 1..15.",
"(15+1)/2.",
"E = 8."
] | math | Answer: E = 8.
Steps:
1. Mean of 1..15.
2. (15+1)/2.
3. E = 8. | probability |
The answer is Riemann zeta. | {
"A": "Riemann zeta",
"B": "Wild series",
"C": "Blind sum",
"D": "Vague total"
} | mat_0635 | multiple_choice | en | Choose: what sums 1/n^s? | [
"Zeros rule primes.",
"So: Riemann zeta.",
"Select option A."
] | math | Answer: Riemann zeta
Steps:
1. Zeros rule primes.
2. So: Riemann zeta.
3. Select option A.
Answer: (A) | number_theory |
Rows and columns each permute symbols, the combinatorics behind sudoku puzzles. | null | mat_0636 | concept | en | What is a Latin square? | [
"Rows and columns each permute symbols, the combinatorics behind sudoku puzzles.",
"So the answer stands: Rows and columns each permute symbols, the combinatorics behind sudoku puzzles."
] | math | Answer: Rows and columns each permute symbols, the combinatorics behind sudoku puzzles.
Steps:
1. Rows and columns each permute symbols, the combinatorics behind sudoku puzzles.
2. So the answer stands: Rows and columns each permute symbols, the combinatorics behind sudoku puzzles. | combinatorics |
Kruhový důkaz předpokládá závěr, a nedokazuje tedy nic. | null | mat_0637 | concept | cs | Proč se vyhnout kruhovým důkazům? | [
"Kruhový důkaz předpokládá závěr, a nedokazuje tedy nic.",
"Odpověď tedy platí: Kruhový důkaz předpokládá závěr, a nedokazuje tedy nic."
] | math | Answer: Kruhový důkaz předpokládá závěr, a nedokazuje tedy nic.
Steps:
1. Kruhový důkaz předpokládá závěr, a nedokazuje tedy nic.
2. Odpověď tedy platí: Kruhový důkaz předpokládá závěr, a nedokazuje tedy nic. | proofs |
f'(2) = 21. | null | mat_0638 | worked_example | en | For f(x) = 2x² + 13x, compute f'(2) step by step. | [
"f'(x) = 4x + 13.",
"Plug x = 2.",
"4×2 + 13 = 21."
] | math | Answer: f'(2) = 21.
Steps:
1. f'(x) = 4x + 13.
2. Plug x = 2.
3. 4×2 + 13 = 21. | calculus |
Trace: 31. | null | mat_0639 | worked_example | en | Compute the trace of [[15,0],[0,16]]. | [
"Sum diagonal.",
"15 + 16.",
"= 31."
] | math | Answer: Trace: 31.
Steps:
1. Sum diagonal.
2. 15 + 16.
3. = 31. | linear_algebra |
P(red) = 13/17. | null | mat_0640 | worked_example | en | A bag holds 13 red and 4 blue marbles. Show P(red). | [
"Red: 13.",
"Total: 17.",
"P = 13/17."
] | math | Answer: P(red) = 13/17.
Steps:
1. Red: 13.
2. Total: 17.
3. P = 13/17. | probability |
Remainder: 6. | null | mat_0641 | worked_example | en | Compute (160) mod 7. | [
"Divide 160 by 7.",
"7×22 = 154.",
"160 − 154 = 6."
] | math | Answer: Remainder: 6.
Steps:
1. Divide 160 by 7.
2. 7×22 = 154.
3. 160 − 154 = 6. | number_theory |
C(7,2) = 21. | null | mat_0642 | worked_example | cs | Ukaž výpočet C(7,2). | [
"Vzorec n(n−1)/2.",
"7×6/2.",
"= 21."
] | math | Answer: C(7,2) = 21.
Steps:
1. Vzorec n(n−1)/2.
2. 7×6/2.
3. = 21. | combinatorics |
The value is 72. | {
"A": "70",
"B": "74",
"C": "76",
"D": "72"
} | mat_0643 | multiple_choice | en | Compute N² + N for N = 8. | [
"8² + 8.",
"Value = 72.",
"Select option D."
] | math | Answer: 72
Steps:
1. 8² + 8.
2. Value = 72.
3. Select option D.
Answer: (D) | proofs |
Integrál: 125. | null | mat_0644 | worked_example | cs | Vyčísli integrál 3x² od 0 do 5. | [
"Primitivní: x³.",
"Vyčísli 0 až 5.",
"5³ − 0 = 125."
] | math | Answer: Integrál: 125.
Steps:
1. Primitivní: x³.
2. Vyčísli 0 až 5.
3. 5³ − 0 = 125. | calculus |
Dot product: 107. | null | mat_0645 | worked_example | en | Compute (14,13) · (3,5) step by step. | [
"Multiply components.",
"14×3 + 13×5.",
"= 107."
] | math | Answer: Dot product: 107.
Steps:
1. Multiply components.
2. 14×3 + 13×5.
3. = 107. | linear_algebra |
P = 1/17. | null | mat_0646 | worked_example | en | A fair 17-sided die is rolled. Show P(one specific number). | [
"Favourable: 1.",
"Possible: 17.",
"P = 1/17."
] | math | Answer: P = 1/17.
Steps:
1. Favourable: 1.
2. Possible: 17.
3. P = 1/17. | probability |
The gcd is 21. | {
"A": "18",
"B": "21",
"C": "24",
"D": "27"
} | mat_0647 | multiple_choice | en | Find gcd(42, 63). | [
"Factor 3 out twice.",
"gcd = 21.",
"Select option B."
] | math | Answer: 21
Steps:
1. Factor 3 out twice.
2. gcd = 21.
3. Select option B.
Answer: (B) | number_theory |
C(16,2) = 120. | null | mat_0648 | worked_example | en | Show the computation of C(16,2). | [
"Formula n(n−1)/2.",
"16×15/2.",
"= 120."
] | math | Answer: C(16,2) = 120.
Steps:
1. Formula n(n−1)/2.
2. 16×15/2.
3. = 120. | combinatorics |
The table has 256 rows. | {
"A": "254",
"B": "258",
"C": "256",
"D": "260"
} | mat_0649 | multiple_choice | en | How many rows does a truth table with 8 variables have? | [
"2^8 rows.",
"Total: 256.",
"Select option C."
] | math | Answer: 256
Steps:
1. 2^8 rows.
2. Total: 256.
3. Select option C.
Answer: (C) | proofs |
It steps along tangent slopes to approximate solutions numerically when formulas fail. | null | mat_0650 | concept | en | What is Euler's method? | [
"It steps along tangent slopes to approximate solutions numerically when formulas fail.",
"So the answer stands: It steps along tangent slopes to approximate solutions numerically when formulas fail."
] | math | Answer: It steps along tangent slopes to approximate solutions numerically when formulas fail.
Steps:
1. It steps along tangent slopes to approximate solutions numerically when formulas fail.
2. So the answer stands: It steps along tangent slopes to approximate solutions numerically when formulas fail. | calculus |
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