Random Problem

Easy Problem

The sum $\frac{1}{2!}+\frac{2}{3!}+\frac{3}{4!}+\cdots+\frac{2021}{2022!}$ can be expressed as $a-\frac{1}{b!}$ , where $a$ and $b$ are positive integers. What is $a+b$ ?

$\textbf{(A)}\ 2020 \qquad\textbf{(B)}\ 2021 \qquad\textbf{(C)}\ 2022 \qquad\textbf{(D)}\ 2023 \qquad\textbf{(E)}\ 2024$

Solution

???

Medium Problem

Show that there exist no finite decimals $a = 0.\overline{a_1a_2a_3\ldots a_n}$ such that when its digits are rearranged to a different decimal $b = 0.\overline{a_{b_1}a_{b_2}a_{b_3}\ldots a_{b_n}}$ , $a + b = 1$ .

Solution

???

Hardish Problem

A cylinder is inscribed in a circular cone with base radius of $7$ and height of $14$ . What is the maximum possible volume of this cylinder is $\frac{a}{b}\pi$ ?

Solution

???

Hard Problem

A regular $48$ -gon is inscribed in a circle with radius $1$ . Let $X$ be the set of distances (not necessarily distinct) from the center of the circle to each side of the $48$ -gon, and $Y$ be the set of distances (not necessarily distinct) from the center of the circle to each diagonal of the $48$ -gon. Let $S$ be the union of $X$ and $Y$ . What is the sum of the squares of all of the elements in $S$ ?

Solution

???

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Easy Problem

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Medium Problem

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Hardish Problem

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Hard Problem

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