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Difference between revisions of "2021 CIME I Problems"

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==Problem 2==
 
==Problem 2==
 
For digits <math>a, b, c,</math> with <math>a\neq 0,</math> the positive integer <math>N</math> can be written as <math>\underline{a}\underline{a}\underline{b}\underline{b}</math> in base <math>9,</math> and <math>\underline{a}\underline{a}\underline{b}\underline{b}\underline{c}</math> in base <math>5</math>. Find the base-<math>10</math> representation of <math>N</math>.
 
For digits <math>a, b, c,</math> with <math>a\neq 0,</math> the positive integer <math>N</math> can be written as <math>\underline{a}\underline{a}\underline{b}\underline{b}</math> in base <math>9,</math> and <math>\underline{a}\underline{a}\underline{b}\underline{b}\underline{c}</math> in base <math>5</math>. Find the base-<math>10</math> representation of <math>N</math>.
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[[2021 CIME I Problems/Problem 2|Solution]]
  
 
==See also==
 
==See also==
 
{{CIME box|year=2021|n=I|before=[[2020 CIME II Problems]]|after=[[2021 CIME II Problems]]}}
 
{{CIME box|year=2021|n=I|before=[[2020 CIME II Problems]]|after=[[2021 CIME II Problems]]}}

Latest revision as of 18:01, 24 July 2024

2021 CIME I (Answer Key)
Printable version | AoPS Contest Collections

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.
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15

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 $a+b$.

Solution

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 $N$.

Solution

See also

2021 CIME I (Problems • Answer Key • Resources)
Preceded by
2020 CIME II Problems 		Followed by
2021 CIME II Problems
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All CIME Problems and Solutions
Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2021_CIME_I_Problems&oldid=223553"