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Consider a right triangle with side lengths $x$, $y$, and $z$, and area $A$. Given that $\dfrac {x^4+y^4+z^4}8 =64-A^2$ and that $x+y+z=4A$, find the perimeter of the triangle.

Let $a$ and $b$ be positive integers such that the number $b^2+(b+1)^2+(b+2)^2+\cdots+(b+a)^2-3$ is a multiple of $5$ and $a+b$ is odd. What is the units digit of the number $a+b$ …

Find the smallest positive integer $n \geq 10$ such that $n+6$ is a prime and $9n+7$ is a perfect square.

Given that $\{a_{n}\}$ is a sequence satisfying $a_{0}$ not equal to $0$ or $1$, $a_{1}=1-a_{0}, a_{n+1}=1-a_{n}(1-a_{n})$. Evaluate: $a_{0}a_{1} a_2 \dotsm a_{n}\left(\frac{1}{a_{0}}+\frac{1}{a_{1}}+\frac 1{a_2}+ \dots +\frac{1}{a_{n}}\right)$ where $n$ is a positive integer

Timmy and Jimmy are playing a game with spring-loaded guns. The objective of the game is to not get hit by the plastic dart launched by the other player's gun. …

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