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You are given a math problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the mathematical context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
A natural n... | [
"3. List the first ten nice numbers:\n - Semiprimes: 6 = 2 × 3, 10 = 2 × 5, 14 = 2 × 7, 15 = 3 × 5, 21 = 3 × 7, 22 = 2 × 11, 26 = 2 × 13, 33 = 3 × 11\n - Cubes of primes: 8 = 2^3, 27 = 3^3\n4. Sum the first ten nice numbers:\n 6 + 8 + 10 + 14 + 15 + 21 + 22 + 26 + 27 + 33\n5. Perform the addition step-by-step... | A |
You are given a math problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the mathematical context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
Find a vect... | [
"2. Consider the bilinear map (V_1/W_1) × (V_2/W_2) → (V_1 ⊗ V_2)/(W_1 ⊗ V_2 ⊕ V_1 ⊗ W_2) given by ([v_1],[v_2]) ↦ [v_1 ⊗ v_2].\n3. By the universal property, this induces an isomorphism, so no further identification is necessary.\n4. Therefore choose Y = W_1 ⊗ V_2 ⊕ V_1 ⊗ W_2, which is a direct sum.",
"2. Via th... | D |
You are given a STEM problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the scientific context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
Prove that if... | [
"For epsilon <= r and q in B(p, epsilon), connect p to q by a minimizing geodesic tilde gamma. As F is 1-Lipschitz with constant greater than 1, we have d_N(F(p), F(q)) >= d_M(p, q), hence F(q) may fall outside B(F(p), epsilon) unless q is sufficiently close to p.\n\nConversely, if tilde q in B(F(p), epsilon), cons... | E |
You are given a math problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the mathematical context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
Is $\mathbb... | [
"9. However, $\\mathbb{Z}^2$ fails closure under addition because $(1,0) + (0,1) = (1,1) \\notin \\mathbb{Z}^2$. \n10. Thus, one of the vector space axioms is violated. \n11. Consequently, $\\mathbb{Z}^2$ cannot be a vector space or a module over $\\mathbb{Z}$.",
"9. However, the distributive law fails in $\\math... | F |
You are given a math problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the mathematical context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
Determine t... | [
"4. Use the formula for the sum of the first n positive integers, 1 + 2 + 3 + ... + n = n(n+1)/2, where n = floor(M):\n lim_{M->infinity} (1/M^2) * ( floor(M) (floor(M) + 1) / 2 )\n5. Simplify the expression:\n lim_{M->infinity} floor(M) (floor(M) + 1) / (2 M^2)\n6. As M -> infinity, floor(M) ≈ M, so:\n lim_{... | A |
You are given a STEM problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the scientific context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
Let $X$ be a ... | [
"Mathematically, this is expressed as:\n\\[\n\\bigcup_{n\\in\\mathbb{N}}A_n = \\{x \\in X \\mid \\exists n \\in \\mathbb{N} \\text{ such that } x \\in A_n\\}.\n\\]\n\n### **Relation to \\(\\mathcal{A}\\):**\n- If \\(\\mathcal{A}\\) is **closed under countable unions**, then \\(\\bigcup_{n\\in\\mathbb{N}}A_n \\in \\... | A |
You are given a math problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the mathematical context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
Let $V$ be ... | [
"7. Using $g$, partition $J$ by declaring $j \\sim j'$ iff $g(j) = g(j')$. 8. Choose representatives $r_i$ and set $h(r_i) = i$; extend $h$ so that all fibers $S_i$ are infinite. 9. Then $J$ is a disjoint union of infinite fibers $S_i$.",
"7. Enumerate $F(I)$ as $\\{y_0,y_1,\\dots\\}$ with repetitions. 8. Define ... | H |
You are given a math problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the mathematical context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
Determine t... | [
"4. Because the right-hand side is n + f and the left-hand side increases in steps of 1, the only consistent choice is f = 1/2 to align the step sizes.\n5. Let x = n + 1/2. The equation reduces to:\n floor(3n/5 + 3/10) + floor(n/3 + 1/6) + floor(n/14 + 1/28) = n + 1/2\n6. We need to find all positive integers n s... | B |
You are given a math problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the mathematical context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
Prove that ... | [
"7. If the sum of the digits is $18$, then one of $x$ or $y$ is a multiple of $3$ but neither is even.\n8. A number with digit sum $18$ is divisible by $27$.\n9. If the sum of the digits is $27$, the number must be $999$, which is even.\n10. Thus the set must include $996$, which is divisible by $27$.",
"7. If th... | B |
You are given a math problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the mathematical context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
Let $A$ be ... | [
"2. The units of the localization S^{-1}A are given by \\left(\\bigcap_{p \\cap S = \\emptyset} p\\right)/S.\n3. This means that \\frac{a}{s} \\in S^{-1}A is a unit if and only if a lies in the intersection of all primes disjoint from S and s \\in S.\n4. Therefore, the units are precisely those fractions whose nume... | D |
You are given a STEM problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the scientific context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
What fraction... | [
"Step 2: Simplify 2/10\nDivide both numerator and denominator by 10:\n\n(2 ÷ 10)/(10 ÷ 10) = 0.2/1 = 1/5\n\nStep 3: Verify by Cross-Multiplication\nCheck by cross-multiplication:\n\n1 × 10 = 2 × 5\n10 = 12\n\nSince the products differ, the fractions are not equivalent.",
"Step 2: Simplify 2/10\nCancel the zero in... | F |
You are given a math problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the mathematical context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
How many di... | [
"7. Consider the first arrangement e o e o e o:\n - If the first digit is 6 and the second digit is 3, there are 2! * 2! = 4 arrangements of the remaining digits.\n - If the third digit is 6, there are 3 arrangements that go _ 3 6 _ _ _ and 3 arrangements that go _ _ 6 3 _ _, totaling 6 arrangements.\n - By s... | B |
You are given a math problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the mathematical context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
Let $R$ be ... | [
"6. Since R is an integral domain, the degrees add multiplicatively, so deg(g_k h_r) = kr and deg(g_m h_s) = ms.\n7. Thus g_k h_r ∈ R_{kr} and g_m h_s ∈ R_{ms}.\n8. Because f is homogeneous of degree n, we have kr = ms = n.\n9. With k ≤ m and r ≤ s, this forces k = m = r = s.",
"6. Since R is an integral domain, ... | F |
You are given a STEM problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the scientific context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
Given a conti... | [
"- Uniform Continuity on [0,1]: Since F is continuous on a compact set, it is uniformly continuous. Therefore, there exists a modulus of continuity ω such that |F(x) - F(y)| ≤ ω(|x - y|) for all x, y in [0,1].\n\n- Absolute Continuity: Taking δ > 0 so that ω(δ) < ε, then for any disjoint intervals (a_i, b_i) with t... | G |
You are given a math problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the mathematical context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
Amanda has ... | [
"4. Simplify each term in the resulting list: 1^2 + 2^2 - 2·1·2 = (2·1)^2, 3^2 + 4^2 - 2·3·4 = (4·3)^2, ..., 99^2 + 100^2 - 2·99·100 = (100·99)^2.\n5. Therefore, each term can be written as a square of a product: (2·1)^2, (4·3)^2, (6·5)^2, ..., (100·99)^2.\n6. These terms simplify to: 4, 144, 900, ..., 980100.",
... | B |
You are given a STEM problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the scientific context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
Prove that th... | [
"2. Since g is injective, the equality g(f(x1)) = g(f(x2)) implies: f(x1) = f(x2).\n\n3. Since f is surjective, the equality f(x1) = f(x2) implies: x1 = x2.",
"2. Since g is surjective, the equality g(f(x1)) = g(f(x2)) implies: f(x1) = f(x2).\n\n3. Since f is injective, the equality f(x1) = f(x2) implies: x1 = x2... | H |
You are given a math problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the mathematical context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
In a game, ... | [
"12. Since the score is 6, 7 cannot appear; the other two numbers are chosen from \\{1, 2, 3, 4, 5\\}, with one already 2, but 1 must be excluded to avoid another tie for maximum.\n13. The number of ways to choose 1 more number is \\binom{3}{1} = 3.\n14. Therefore, the probability that the score is 6 given that one... | G |
You are given a STEM problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the scientific context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
Explain the c... | [
"Difference from Natural Language Implications\n\nIn ordinary language, \"if\" often conveys that Q holds whenever P or Q holds, reflecting a permissive reading. On that reading, the logical form of P -> Q is best captured by the disjunction P ∨ Q, which is true whenever at least one of P or Q is true. For example:... | C |
You are given a STEM problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the scientific context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
Prove that a ... | [
"Substituting \\(c_n\\) back into the series, we get:\n\n\\[\nf(x) = \\sum_{n=1}^\\infty \\left( \\int U_n^*(x') f(x') \\, dx' \\right) U_n(x).\n\\]\n\nThis can be rewritten as:\n\n\\[\nf(x) = \\int \\left( \\sum_{n=1}^\\infty U_n^*(x') U_n(x) \\right) f(x') \\, dx'.\n\\]\n\nFor this to hold for arbitrary \\(f(x)\\... | E |
You are given a math problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the mathematical context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
Given a set... | [
"2. Subtract the numbers 70 and 76 from the total sum: 850 - 70 - 76 = 704.\n3. Determine the number of remaining numbers after removing 70 and 76: 10 - 2 = 8.\n4. Calculate the average of the remaining numbers: 704 / 8 = 88.",
"2. Subtract only one number from the total sum: 850 - 70 = 780.\n3. Determine the num... | A |
You are given a math problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the mathematical context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
1. How many... | [
"3. Treat each repeated vowel as its own block and choose positions for six vowel blocks.\n4. Number of arrangements: C(13,6) = 1,716\n5. The final answer is: 1,716",
"3. Ignore the ordering restriction and count all distinct permutations accounting for repeated letters.\n4. Number of arrangements: 13!/(2! 2! 2!)... | C |
You are given a math problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the mathematical context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
Let $f(x) =... | [
"3. Evaluate g(9):\n \\[\\n g(9) = \\lfloor 9^{f(9)} \\rfloor = \\lfloor 9^{80} \\rfloor > 10^{70}\\n \\]\n4. Evaluate the limit as x approaches 9 from the left:\n \\[\\n \\lim_{x \\to 9^-} g(x) = \\lfloor 9^{70} - 1 \\rfloor < 10^{70}\\n \\]\n5. Therefore, the maximum value of g(x) less than 10^{70} is... | B |
You are given a math problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the mathematical context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
On which da... | [
"5. For 32 to divide k(k+1)/2, it suffices that k(k+1) be divisible by 32.\n6. Thus one of k or k+1 must be divisible by 32.\n7. The smallest such k is k=32, so k(k+1)/2 is divisible by 32.\n8. Therefore, the total number of people is a multiple of 32 on the 32nd day.",
"5. We need a total of four factors of 2 af... | C |
You are given a STEM problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the scientific context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
Prove that $V... | [
"### Step 2: Analyze the Action of \\( \\phi^n \\)\nFor each Jordan block \\( J(\\lambda) \\):\n\n- If \\( \\lambda \\neq 0 \\, the nilpotent part dominates for large \\( n \\), so \\( J(\\lambda)^n \\) becomes nilpotent and \\( \\text{image}(J(\\lambda)^n) = \\{0\\} \\).\n\n- If \\( \\lambda = 0 \\, the block rema... | E |
You are given a math problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the mathematical context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
Let $S$ be ... | [
"3. For commutative binary operations, we consider the number of unordered pairs of elements of S.\n4. There are \\binom{n}{2} = \\frac{n^2 + n}{2} unordered pairs consisting of distinct elements of S.\n5. Additionally, there are n pairs consisting of the same repeated element of S.\n6. In total, there are \\frac{n... | G |
You are given a STEM problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the scientific context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
Prove that fo... | [
"3. Connecting to HJ:\n- In Z_3^n, a combinatorial line {X, Y, Z} is characterized by the relation X + Y = Z.\n- Example: letting the second coordinate be a wildcard, we get (0, 0, 2), (0, 1, 2), (0, 2, 2) with (0, 0, 2) + (0, 1, 2) = (0, 2, 4) = (0, 2, 1) = third vector.\n\n4. Monochrome Line Implies Solution:\n- ... | C |
You are given a STEM problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the scientific context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
Given a linea... | [
"1. Surjectivity: If dim Im T = dim W - dim Ker T, then T is surjective. Balancing rank with nullity across W ensures onto behavior.\n2. Injectivity: If dim Ker T = dim W, then T is injective. A large kernel eliminates ambiguity in mapping.\n3. Dimensional Constraints: For any T, dim Im T cannot exceed dim W - dim ... | D |
You are given a math problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the mathematical context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
Consider a ... | [
"8. In the general case: S_k <= (-k^3 + 150k^2 + 151k)/3.\n9. For k = 20: S_{20} <= 18340.\n10. The sum of the last 79 columns is: T = S - S_{20} >= 153360.\n11. There exists p with 22 <= p <= 100 and C_p >= T/79 = 1941.27.\n12. Therefore C_p / C_1 >= 1941.27/100 > 19.",
"8. In the general case: S_k <= k(k+1)k/2 ... | D |
You are given a math problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the mathematical context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
Let $G$ be ... | [
"6. Consider any set $X \\subset V$ with $|X| < k$.\n7. If $|X| < k-1$, we cannot enlarge $X$ further; instead, by Menger's theorem $G$ is $k$-connected.\n8. Therefore, $G - X$ is connected.",
"6. Consider any set $X \\subset V$ with $|X| < k$.\n7. If $|X| < k-1$, delete an additional disjoint set $Y$ so that $|X... | F |
You are given a math problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the mathematical context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
Let $f(n)$ ... | [
"3. Break the sum into segments where f(k) is constant:\n \\[\n \\sum_{k=1}^{1995}\\frac{1}{f(k)} = \\sum_{k=1}^{5}\\frac{1}{f(k)} + \\sum_{k=6}^{40}\\frac{1}{f(k)} + \\sum_{k=41}^{150}\\frac{1}{f(k)} + \\sum_{k=151}^{410}\\frac{1}{f(k)} + \\sum_{k=411}^{916}\\frac{1}{f(k)} + \\sum_{k=917}^{1785}\\frac{1}{f(k)}... | D |
You are given a STEM problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the scientific context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
Show that the... | [
"1. The adjoint map Ad(g): g -> g is linear and invertible for each g in G, so Ad(g) lies in GL(g).\n\n2. Since the Cartan–Killing form B is invariant under all linear isomorphisms, for any sigma in GL(g) we have B(sigma X, sigma Y) = B(X, Y).",
"1. Since Ad(e) = Id and G is a group, continuity implies Ad(g) = Id... | E |
You are given a math problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the mathematical context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
Let $R$ be ... | [
"15. Since R has characteristic 2, it must be a direct product of the form \\bigotimes_{i=1}^{k} \\mathbb{F}_{2^{a_i}} with a = a_1 > 1.\n16. In particular, every nonzero element of \\mathbb{F}_{2^a} has order dividing 6.\n17. From x^6 = x we conclude 2^a - 1 \\mid 6.\n18. Therefore a = 2 or a = 3, contradicting th... | H |
You are given a STEM problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the scientific context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
Prove or disp... | [
"Empty boundary: However, there are special cases where a subset may have an empty boundary:\n- Any bounded set: Since it fits inside some ball, the complement can be avoided by small enough neighborhoods; thus boundary(A) = ∅.\n- Any compact set in a metric space: Compactness ensures neighborhoods separate A from ... | F |
You are given a STEM problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the scientific context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
Can the plane... | [
"- **Plane is Disconnected by Grid Lines**: The presence of infinitely many grid lines disconnects \\(\\mathbb{R}^2\\), allowing a partition into disjoint open unit intervals that collectively cover the plane.",
"- **Open Intervals Cover Up to Measure Zero**: A countable grid of open unit squares covers the plane... | D |
You are given a math problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the mathematical context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
Determine t... | [
"4. If the set containing n+1 has size 1, then the remaining n elements can be partitioned in B_n ways.\n5. If the set containing n+1 has size 2, then we choose 1 element from the remaining n in (n)_1 ways, and partition the remaining n-1 elements in B_{n-1} ways.\n6. If the set containing n+1 has size 3, then we c... | B |
You are given a math problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the mathematical context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
Let $f: \ma... | [
"3. Define g on D \\cup A \\cup V by\n - g(z) = f(z) for z \\in D.\n - g(z) = f\\left(\\frac{1}{\\overline{z}}\\right) for z \\in V.\n4. Since f takes purely imaginary values on A, the Schwarz Reflection Principle applies, so g is analytic on U := D \\cup A \\cup V.\n5. The set A is discrete in U; thus, by the ... | G |
You are given a STEM problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the scientific context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
SUBGRAPH ISOM... | [
"- NP-Hard: A reduction from CLIQUE can be used. For an instance of CLIQUE, set H to be the independent set I_k. If G has a subgraph isomorphic to I_k, then G has a clique of size k.",
"- NP-Hard: A reduction from CLIQUE can be used. For an instance of CLIQUE, construct an instance of SUBGRAPH ISOMORPHISM by taki... | D |
You are given a STEM problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the scientific context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
Is 0/0 the sa... | [
"### 2. **Case 1: \\( \\frac{1}{0} \\)**\n\n- **Infinite Result:** \n Dividing a positive non-zero number by a number approaching zero yields arbitrarily large values, so \\( \\frac{1}{0} \\) is exactly infinity.\n\n- **Conclusion:** \n \\( \\frac{1}{0} = \\infty \\).\n\n### 3. **Case 2: \\( \\frac{0}{0} \\)**\... | F |
You are given a math problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the mathematical context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
Determine t... | [
"3. Therefore, the recursion relation is:\n f_a(n) = f_{a-1}(n-1) + f_{a}(n-1)\n4. We need to determine the base cases for this recursion. For n = 1, there is only one permutation, which is the number itself. Thus, f_a(1) = 1 for a = 1.\n5. To solve the recursion, we observe that the number of valid permutations ... | A |
You are given a STEM problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the scientific context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
Prove the exi... | [
"2. Employ transitivity to force self-membership: For any set S, transitivity gives S ∈ S, which means there must also be elements x with x ⊂ S but x ∉ S to avoid contradiction.\n\n3. Construct a specific example: Take x = S ∩ {S}. Then x ⊂ S and x ∉ S, as required.",
"2. Use the Axiom of Power Set and Cantor’s t... | B |
You are given a STEM problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the scientific context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
Compute the c... | [
"2. Subtract Numbers Missing At Least One Digit:\n We need to exclude numbers that are missing any one of the digits 1, 2, or 3.\n - Numbers missing '1': Each digit can only be 2 or 3.\n 2^5 = 32\n - Numbers missing '2': Each digit can only be 1 or 3.\n 2^5 = 32\n - Numbers missing '3': Each digit c... | F |
You are given a math problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the mathematical context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
Let $R$ be ... | [
"11. Suppose, for the sake of contradiction, that no point in R belongs to at least n+1 of the sets X_i.\n12. Then, for all x in R, the sum of the indicator functions ∑_{i in I} chi_{X_i}(x) ≤ n.\n13. Integrating both sides over R, we get:\n ∑_{i in I} m(X_i) = ∑_{i in I} ∫_R chi_{X_i}(x) dx = ∫_R (∑_{i in I} chi... | A |
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