Difference between revisions of "2013 AIME I Problems"
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| − | 1 | + | {{AIME Problems|year=2013|n=I}} |
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| + | == Problem 1 == | ||
| + | The AIME Triathlon consists of a half-mile swim, a 30-mile bicycle ride, and an eight-mile run. Tom swims, bicycles, and runs at constant rates. He runs fives times as fast as he swims, and he bicycles twice as fast as he runs. Tom completes the AIME Triathlon in four and a quarter hours. How many minutes does he spend bicycling? | ||
| + | |||
| + | [[2013 AIME I Problems/Problem 1|Solution]] | ||
| + | |||
| + | == Problem 2 == | ||
| + | Find the number of five-digit positive integers, <math>n</math>, that satisfy the following conditions: | ||
| + | |||
| + | <UL> | ||
| + | (a) the number <math>n</math> is divisible by <math>5,</math> | ||
| + | </UL> | ||
| + | |||
| + | <UL> | ||
| + | (b) the first and last digits of <math>n</math> are equal, and | ||
| + | </UL> | ||
| + | |||
| + | <UL> | ||
| + | (c) the sum of the digits of <math>n</math> is divisible by <math>5.</math> | ||
| + | </UL> | ||
| + | |||
| + | [[2013 AIME I Problems/Problem 2|Solution]] | ||
| + | |||
| + | == Problem 3 == | ||
| + | Let <math>ABCD</math> be a square, and let <math>E</math> and <math>F</math> be points on <math>\overline{AB}</math> and <math>\overline{BC},</math> respectively. The line through <math>E</math> parallel to <math>\overline{BC}</math> and the line through <math>F</math> parallel to <math>\overline{AB}</math> divide <math>ABCD</math> into two squares and two nonsquare rectangles. The sum of the areas of the two squares is <math>\frac{9}{10}</math> of the area of square <math>ABCD.</math> Find <math>\frac{AE}{EB} + \frac{EB}{AE}.</math> | ||
| + | |||
| + | [[2013 AIME I Problems/Problem 3|Solution]] | ||
| + | |||
| + | |||
| + | == Problem 4 == | ||
Revision as of 17:11, 15 March 2013
| 2013 AIME I (Answer Key) | AoPS Contest Collections • PDF | ||
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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.Contents
Problem 1
The AIME Triathlon consists of a half-mile swim, a 30-mile bicycle ride, and an eight-mile run. Tom swims, bicycles, and runs at constant rates. He runs fives times as fast as he swims, and he bicycles twice as fast as he runs. Tom completes the AIME Triathlon in four and a quarter hours. How many minutes does he spend bicycling?
Problem 2
Find the number of five-digit positive integers, 
, that satisfy the following conditions:
-
(a) the number
is divisible by 
-
(b) the first and last digits of
are equal, and
-
(c) the sum of the digits of
is divisible by 
Problem 3
Let 
be a square, and let 
and 
be points on 
and 
respectively. The line through parallel to 
and the line through parallel to divide into two squares and two nonsquare rectangles. The sum of the areas of the two squares is 
of the area of square 
Find 
