2013 AIME I Problems
2013 AIME I (Answer Key)  AoPS Contest Collections • PDF  
Instructions
 
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Problem 1
The AIME Triathlon consists of a halfmile swim, a 30mile bicycle ride, and an eightmile 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 fivedigit 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
Problem 4
Problem 5
The real root of the equation can be written in the form , where , , and are positive integers. Find .