2013 AIME I Problems/Problem 2
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)
Solution 1
The number takes a form of , in which
. Let
and
be arbitrary digits. For each pair of
, there are exactly two values of
that satisfy the condition of
. Therefore, the answer is
See also
2013 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 1 |
Followed by Problem 3 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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