Difference between revisions of "2021 CIME I Problems"
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==Problem 1== | ==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>. | 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>. | ||
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| + | [[2021 CIME I Problems/Problem 1|Solution]] | ||
==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]] | ||
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| + | ==See also== | ||
| + | {{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) | AoPS Contest Collections | ||
|
Instructions
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| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
to 
, inclusive. Your score will be the number of correct answers; i.e., there is neither partial credit nor a penalty for wrong answers.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 .
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 | ||
