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2022 SSMO Relay Round 1 Problems

Revision as of 21:53, 31 May 2023 by Pinkpig (talk | contribs) (Created page with "==Problem 1== Suppose <math>a, b, c</math> are distinct digits where <math>a \not= 0</math> such that <math>\left(\overline{abc}\right)^2 = \overline{bad00}</math> where <mat...")
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Problem 1

Suppose $a, b, c$ are distinct digits where $a \not= 0$ such that $\left(\overline{abc}\right)^2 = \overline{bad00}$ where $d = a+b$. Find $a+2b$.

Solution

Problem 2

Let $T=$ TNYWR. Now, let $\ell$ and $m$ have equations $y=(2+\sqrt{3})x+16$ and $y=\frac{x\sqrt{3}}{3}+20,$ respectively. Suppose that $A$ is a point on $\ell,$ such that the shortest distance from $A$ to $m$ is $T$. Given that $O$ is a point on $m$ such that $\overline{AO}\perp m,$ and $P$ is a point on $\ell$ such that $PO\perp \ell$, find $PO^2.$

Solution

Problem 3

Let $T=$ TNYWR. Now, let $ABC$ a triangle such that $AB=T,$ $AC=100$, and $\angle{ABC}=36^{\circ}.$ Find the remainder when the product of all possible values of $BC$ is divided by $1000$.

Solution

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