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|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
"Find the number of quadruples \\((a, b, c, d)\\) of integers with absolute value at most \\(5\\) su(...TRUNCATED)
| "Let \\(x = a + c\\), \\(y = a - c\\), \\(w = b + d\\), and \\(z = b - d\\). Then\n\n\\[\n\\left(w^{(...TRUNCATED)
| 49
| 1
| 0.875
|
Linear substitution;Inequality case-split;All-zero quadruple;Reduce to ac=b^2;Count divisor pairs
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
|
What is the earliest row in which the number \(2004\) may appear?
| "The sequence of rows is as follows:\n\n\\[\n\\begin{array}{ccccccc}\n\\cdots & 0 & 0 & 1 & 0 & 0 & (...TRUNCATED)
| 12
| 0.25
| 0.125
|
Generate rows;Extract central entries;Near-doubling recurrence;Solve for 2004
| "Okay, let's try to figure out this problem. The question is asking for the earliest row in which th(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "Okay, let's try to figure out this problem step by step. The problem is asking for the earliest row(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
|
"Positive integers \\(a_{1}, a_{2}, \\ldots, a_{7}, b_{1}, b_{2}, \\ldots, b_{7}\\) satisfy \\(2 \\l(...TRUNCATED)
| "Let \\(B = b_{1} b_{2} \\cdots b_{7} - 128\\). Since\n\n\\[\na_{1}^{b_{1} b_{2} \\cdots b_{7}} \\eq(...TRUNCATED)
| 675
| 0.5
| 0.25
|
Chained exponent B;Order classification;ai = −1 case;Generator and AM–GM
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
|
"Let \\( P(n) = (n-1^3)(n-2^3) \\ldots (n-40^3) \\) for positive integers \\( n \\). Suppose that \\(...TRUNCATED)
| "We first investigate what primes divide \\( d \\). Notice that a prime \\( p \\) divides \\( P(n) \(...TRUNCATED)
| 48
| 0.625
| 0.75
|
Identify primes S;v_p(d)=⌊40/p⌋;Partition into p-blocks;n with v_p(n)=1;Sum multiplicities
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
|
"How many permutations \\( p(n) \\) of \\(\\{1,2,3, \\ldots, 35\\}\\) satisfy \\( a \\mid b \\) impl(...TRUNCATED)
| "We look at small numbers first. It is not hard to reason that \\(1, 2, 3, 4\\), and \\(5\\) must be(...TRUNCATED)
| 48
| 0.5
| 0.5
| "Fixed by unique divisors;Inductive fixedness;Transpositions among equal-classes;Free permutations o(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "Okay, let's try to tackle this problem step by step. The problem is: How many permutations \\( p(n)(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "Okay, let's try to tackle this problem step by step. The question is: How many permutations \\( p(n(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
|
"Find the number of points \\( P \\in \\mathbb{Z}^{2} \\) that satisfy the following two conditions:(...TRUNCATED)
| "Notice that \\( 38^{2} + 24^{2} = 2020 \\). Let \\( P \\) have coordinates \\((38, y)\\), and label(...TRUNCATED)
| 16
| 0.375
| 0.5
| "Set P=(38,y);Let T;Compute a^2;Power of point;Rearrange factors;Factor-pairs;Count divisors;Parity (...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "Okay, let's try to tackle this problem step by step. First, let's make sure I understand the proble(...TRUNCATED)
|
"Ten positive integers are arranged around a circle. Each number is one more than the greatest commo(...TRUNCATED)
| "First, note that all the integers must be at least 2, because the greatest common divisor of any tw(...TRUNCATED)
| 28
| 0.875
| 0.625
| "Each ≥2;Maximal entry n;Neighbors equal n−1;n divisible by n−2;Rule out candidate;Unique cycl(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "Okay, let's try to tackle this problem step by step. First, let's restate the problem to make sure (...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
|
"The integers \\(1, 2, \\ldots, 64\\) are written in the squares of an \\(8 \\times 8\\) chessboard,(...TRUNCATED)
| "We aim to maximize the sum of the numbers on the main diagonal (top left to bottom right). If we co(...TRUNCATED)
| 432
| 0
| 0
|
Parity coloring;Region capacity;Parity/count elimination;Feasible m;Explicit construction
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
|
"For how many ordered triplets \\((a, b, c)\\) of positive integers less than \\(10\\) is the produc(...TRUNCATED)
| "Solution: One of the numbers must be \\(5\\). The other two numbers must have a product divisible b(...TRUNCATED)
| 78
| 0
| 0
|
2^2 and 5;position of 5;product divisible by 4;count ordered pairs;sum counts
| "Okay, let's try to solve this problem step by step. The question is: For how many ordered triplets (...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "Okay, let's try to solve this problem step by step. The problem is: For how many ordered triplets ((...TRUNCATED)
| "Okay, let's try to solve this problem step by step. The problem is asking for the number of ordered(...TRUNCATED)
| "Okay, let's try to solve this problem step by step. The problem is: For how many ordered triplets ((...TRUNCATED)
| "Okay, let's try to solve this problem step by step using the four hints provided. First, let me res(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
|
"Let \\(1, \\alpha_{1}, \\alpha_{2}, \\ldots, \\alpha_{10}\\) be the roots of the polynomial \\(x^{1(...TRUNCATED)
| "The existence and uniqueness of this polynomial (up to sign) are assumed. Since all of the roots of(...TRUNCATED)
| 15
| 0.625
| 0
|
coefficients ±1;balance of signs;equal nonconstant coefficients;pairwise opposite signs
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
|
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