Difference between revisions of "2021 CIME I Problems"
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| − | + | {{CIME box|year=2021|n=I}} | |
| + | ==Problem 1== | ||
| + | Let <math>ABCD</math> be a square. Points <math>P</math> and <math>Q</math> are on sides <math>AB</math> and <math>CD,</math> respectively<math>,</math> such that the areas of quadrilaterals <math>APQD</math> and <math>BPQC</math> are <math>20</math> and <math>21,</math> respectively. Given that <math>\tfrac{AP}{BP}=2,</math> then <math>\tfrac{DQ}{CQ}=\tfrac{a}{b},</math> where <math>a</math> and <math>b</math> are relatively prime positive integers. Find <math>a+b</math>. | ||
| + | |||
| + | ==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>. | ||
Revision as of 20:01, 10 January 2021
| 2021 CIME I (Problems • Answer Key • Resources) | ||
| Preceded by [[2021 CIME I Problems/Problem {{{num-b}}}|Problem {{{num-b}}}]] |
Followed by [[2021 CIME I Problems/Problem {{{num-a}}}|Problem {{{num-a}}}]] | |
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| All CIME Problems and Solutions | ||
Problem 1
Let 
be a square. Points 
and 
are on sides 
and 
respectively
such that the areas of quadrilaterals 
and 
are 
and 
respectively. Given that 
then 
where 
and 
are relatively prime positive integers. Find 
.
Problem 2
For digits 
with 
the positive integer 
can be written as 
in base 
and 
in base 
. Find the base-
representation of .
