2013 AIME I Problems
2013 AIME I (Answer Key) | AoPS Contest Collections • PDF | ||
Instructions
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1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |
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
![$n$](http://latex.artofproblemsolving.com/1/7/4/174fadd07fd54c9afe288e96558c92e0c1da733a.png)
![$5,$](http://latex.artofproblemsolving.com/2/1/4/21415a2eb558eb7c199a0382fcd50cbeda9299fb.png)
-
(b) the first and last digits of
![$n$](http://latex.artofproblemsolving.com/1/7/4/174fadd07fd54c9afe288e96558c92e0c1da733a.png)
-
(c) the sum of the digits of
![$n$](http://latex.artofproblemsolving.com/1/7/4/174fadd07fd54c9afe288e96558c92e0c1da733a.png)
![$5.$](http://latex.artofproblemsolving.com/b/8/3/b835f2c6c592edc4583ce996f86bcc0d07ca8da5.png)
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
Problem 4
In the array of 13 squares shown below, 8 squares are colored red, and the remaining 5 squares are colored blue. If one of all possible such colorings is chosen at random, the probability that the chosen colored array appears the same when rotated 90 degrees around the central square is , where n is a positive integer. Find n.
Problem 5
The real root of the equation can be written in the form
, where
,
, and
are positive integers. Find
.