2021 CIME I Problems

2021 CIME I (Answer Key)
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Instructions

  1. This is a 15-question, 3-hour examination. All answers are integers ranging from $000$ to $999$, inclusive. Your score will be the number of correct answers; i.e., there is neither partial credit nor a penalty for wrong answers.
  2. No aids other than scratch paper, graph paper, ruler, compass, and protractor are permitted. In particular, calculators and computers are not permitted.
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Problem 1

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 $a+b$.

Problem 2

For digits $a, b, c,$ with $a\neq 0,$ the positive integer $N$ can be written as $\underline{a}\underline{a}\underline{b}\underline{b}$ in base $9,$ and $\underline{a}\underline{a}\underline{b}\underline{b}\underline{c}$ in base $5$. Find the base-$10$ representation of $N$.

See also

2021 CIME I (ProblemsAnswer KeyResources)
Preceded by
2020 CIME II Problems
Followed by
2021 CIME II Problems
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All CIME Problems and Solutions
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