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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
| 0.875
| "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
| 0.125
| "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)
|
"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
| 0.25
| "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
| 0.75
| "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
| "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)
|
"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
| 0.5
| "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)
|
"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
| 0.625
| "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)
|
"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
| "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
| "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 \\(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
| "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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