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"Let \\(\\omega\\) be a circle of radius \\(6\\) with center \\(O\\). Let \\(AB\\) be a chord of \\((...TRUNCATED)
| "It is easy to see, by the Pythagorean theorem, that \\(\\mathcal{L}(c)\\) for any \\(c\\) consists (...TRUNCATED)
| 60
| 1
| "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)
|
"In the future, MIT has attracted so many students that its buildings have become skyscrapers. Ben a(...TRUNCATED)
| "Define the following lengths:\n\nNote that due to all the 3-4-5 triangles, we find \\(\\frac{x}{z}=(...TRUNCATED)
| 740
| 1
| "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)
|
"Hexagon \\(A B C D E F\\) has an inscribed circle \\(\\Omega\\) that is tangent to each of its side(...TRUNCATED)
| "Let \\(r\\) denote the radius of \\(\\Omega\\), let \\(O\\) denote the center of \\(\\Omega\\), and(...TRUNCATED)
| 36
| 1
| "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 ways are there for Nick to travel from \\((0,0)\\) to \\((16,16)\\) in the coordinate plan(...TRUNCATED)
| "This condition is equivalent to the first and last step being in different directions, as if you sw(...TRUNCATED)
| 310,235,040
| 1
| "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)
|
"Regular hexagon \\(A B C D E F\\) has side length \\(1\\). Let \\(H\\) be the intersection point of(...TRUNCATED)
| "Let \\(O\\) be the center of the circle through \\(D, F\\), and \\(H\\). By considering the geometr(...TRUNCATED)
| 7
| 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)
|
"A function \\( f:\\{1,2, \\ldots, n\\} \\rightarrow\\{1, \\ldots, m\\} \\) is multiplication-preser(...TRUNCATED)
| "Since \\( f(1)^2 = f(1) \\), then \\( f(1) = 1 \\). We have that \\( f(2)^3 = f(8) \\leq 88 \\) and(...TRUNCATED)
| 33
| 1
| "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)
|
If \( x \) is a positive real number such that \((x^{2}-1)^{2}-1=9800\), compute \( x \).
| "We have \\((x^{2}-1)^{2} = 9801\\), so \\(x^{2}-1 = \\pm 99\\). Since \\(x^{2}\\) cannot be negativ(...TRUNCATED)
| 10
| 1
| "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 see. The problem is: If \\( x \\) is a positive real number such that \\( (x^{2} - 1)^{(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
| "Okay, let's see. I need to solve the equation \\((x^2 - 1)^2 - 1 = 9800\\) where \\(x\\) is a posit(...TRUNCATED)
| "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)
|
Find the smallest positive integer \( n \) such that \( n^{4}+(n+1)^{4} \) is composite.
| "Solution: We need to find the smallest positive integer \\( n \\) such that \\( n^4 + (n+1)^4 \\) i(...TRUNCATED)
| 5
| 1
| "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 largest integer \\( n \\), where \\( 2009^{n} \\) divides \\( 2008^{2009^{2010}} + 2010^{2(...TRUNCATED)
| "Using the binomial theorem:\n\n\\[\n\\begin{gathered}\nA = (2009 - 1)^{2009^{2010}} = -1 + 2009 \\t(...TRUNCATED)
| 2,009
| 1
| "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)
|
"An evil witch is making a potion to poison the people of PUMAClandia. In order for the potion to wo(...TRUNCATED)
| "To solve this problem, we need to determine the number of ways to distribute the ingredients such t(...TRUNCATED)
| 1,011
| 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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