This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

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Found problems: 2

2022 DIME, 10
Tags: DIME P10
Let $a$ and $b$ be real numbers such that$$\left(8^a+2^{b+7}\right)\left(2^{a+3}+8^{b-2}\right)=4^{a+b+2}.$$The value of the product $ab$ can be written as $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [i]Proposed by [b]stayhomedomath[/b][/i]

2021 DIME, 10
Tags: DIME P10
There exist complex numbers $z_1,z_2,\dots,z_{10}$ which satisfy$$|z_ki^k+ z_{k+1}i^{k+1}| = |z_{k+1}i^k+ z_ki^{k+1}|$$for all integers $1 \leq k \leq 9$, where $i = \sqrt{-1}$. If $|z_1|=9$, $|z_2|=29$, and for all integers $3 \leq n \leq 10$, $|z_n|=|z_{n-1} + z_{n-2}|$, find the minimum value of $|z_1|+|z_2|+\cdots+|z_{10}|$. [i]Proposed by DeToasty3[/i]