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On the Cartesian plane, $n$ points are given such that the sum of the $x$-coordinates is equal to $300$ and the sum of the $y$-coordinates is equal to $200$. Furthermore, we know that the sum of the squared distances from these points to the origin is equal to $50000$. Find the smallest $n$ such that this is possible. | nvidia |
Find the number of positive integers $n$ less than or equal to $2018$ such that $6n + 1$ is not relatively prime to $42$. | nvidia |
Let \(a\), \(b\), \(c\), and \(d\) be real numbers such that \(a^{2}+b^{2}=1\) and \(c^{2}+d^{2}=4\). Find the range of values of the expression \(ac+bd\). | nvidia |
Determine all the three-digit numbers that are congruent to the sum of the cubes of their digits modulo 5. | nvidia |
Let $a$ and $b$ be nonnegative real numbers such that
\[\sin (ax + b) = \sin 29x\]for all integers $x.$ Find the smallest possible value of $a$.
Problem:
Let $a$ and $b$ be nonnegative real numbers such that
\[\cos (ax + b) = \cos 29x\]for all integers $x.$ Find the smallest possible value of $a$. | nvidia |
Let $g : \mathbb{N} \to \mathbb{N}$ be a function such that
\[g(g(n)) = g(n+1) + g(n-1) - n\]for all $n \geq 2.$ Find the sum of all possible values of $g(2).$ | nvidia |
Find the product of all positive integers $n$ such that $2^{n-1}<n^2<2^n$. | nvidia |
Find the product of all real $x$ that satisfy the equation
$$\sqrt{x^2+1}+\sqrt{x^2-1}=2\sqrt{x^2-3x+3}.$$ | nvidia |
A square and a regular hexagon are inscribed in the same circle. The square and the hexagon have a common vertex, and their respective diagonals intersect at a point inside the circle. Determine the ratio of the area of the square to the area of the hexagon. | nvidia |
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