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Difference between revisions of "2022 SSMO Relay Round 1 Problems"
 
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==Problem 2==
 
==Problem 2==
  
−
Let <math>T=</math> TNYWR. Now, let <math>\ell</math> and <math>m</math> have equations <math>y=(2+\sqrt{3})x+16</math> and <math>y=\frac{x\sqrt{3}}{3}+20,</math> respectively. Suppose that <math>A</math> is a point on <math>\ell,</math> such that the shortest distance from <math>A</math> to <math>m</math> is <math>T</math>. Given that <math>O</math> is a point on <math>m</math> such that <math>\overline{AO}\perp m,</math> and <math>P</math> is a point on <math>\ell</math> such that <math>PO\perp \ell</math>, find <math>PO^2.</math>
+
Let <math>T=TNYWR</math>. Now, let <math>\ell</math> and <math>m</math> have equations <math>y=(2+\sqrt{3})x+16</math> and <math>y=\frac{x\sqrt{3}}{3}+20,</math> respectively. Suppose that <math>A</math> is a point on <math>\ell,</math> such that the shortest distance from <math>A</math> to <math>m</math> is <math>T</math>. Given that <math>O</math> is a point on <math>m</math> such that <math>\overline{AO}\perp m,</math> and <math>P</math> is a point on <math>\ell</math> such that <math>PO\perp \ell</math>, find <math>PO^2.</math>
  
 
[[2022 SSMO Relay Round 1 Problems/Problem 2|Solution]]
 
[[2022 SSMO Relay Round 1 Problems/Problem 2|Solution]]
Line 13: Line 13:
 
==Problem 3==
 
==Problem 3==
  
−
Let <math>T=</math> TNYWR. Now, let <math>ABC</math> a triangle such that <math>AB=T,</math> <math>AC=100</math>, and <math>\angle{ABC}=36^{\circ}.</math> Find the remainder when the product of all possible values of <math>BC</math> is divided by <math>1000</math>.
+
Let <math>T=TNYWR</math>. Now, let <math>ABC</math> a triangle such that <math>AB=T,</math> <math>AC=100</math>, and <math>\angle{ABC}=36^{\circ}.</math> Find the remainder when the product of all possible values of <math>BC</math> is divided by <math>1000</math>.
  
 
[[2022 SSMO Relay Round 1 Problems/Problem 3|Solution]]
 
[[2022 SSMO Relay Round 1 Problems/Problem 3|Solution]]

Latest revision as of 19:17, 2 May 2025

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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