2021 CIME I Problems/Problem 1

Problem

Let $ABCD$ be a square. Points $P$ and $Q$ are on sides $AB$ and $CD,$ respectively $,$ such that the areas of quadrilaterals $APQD$ and $BPQC$ are $20$ and $21,$ respectively. Given that $\tfrac{AP}{BP}=2,$ then $\tfrac{DQ}{CQ}=\tfrac{a}{b},$ where $a$ and $b$ are relatively prime positive integers. Find $a+b$ .

Solution

$[asy] import geometry; point B = (0,0); point C = (16,0); point A = (0,16); point D = (16,16); point P = A * 1/3; point Q = C * 38/85 + D * 47/85; // Square ABCD draw(A--B--C--D--A); dot(A); label("A",A,NW); dot(B); label("B",B,SW); dot(C); label("C",C,SE); dot(D); label("D",D,NE); // Segment PQ draw(P--Q); dot(P); label("P",P,W); dot(Q); label("Q",Q,E); [/asy]$

$\boxed{123}$

2021 CIME I (Problems • Answer Key • Resources)
Preceded by First Problem	Followed by Problem 2
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2021 CIME I Problems/Problem 1

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