Difference between revisions of "2015 AIME I Problems/Problem 14"
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==Problem== | ==Problem== | ||
− | For each integer <math>n \ge 2</math>, let <math>A(n)</math> be the area of the region in the coordinate plane | + | For each integer <math>n \ge 2</math>, let <math>A(n)</math> be the area of the region in the coordinate plane defined by the inequalities <math>1\le x \le n</math> and <math>0\le y \le x \left\lfloor \sqrt x \right\rfloor</math>, where <math>\left\lfloor \sqrt x \right\rfloor</math> is the greatest integer not exceeding <math>\sqrt x</math>. Find the number of values of <math>n</math> with <math>2\le n \le 1000</math> for which <math>A(n)</math> is an integer. |
==Solution== | ==Solution== |
Revision as of 20:01, 28 November 2015
Problem
For each integer , let
be the area of the region in the coordinate plane defined by the inequalities
and
, where
is the greatest integer not exceeding
. Find the number of values of
with
for which
is an integer.
Solution
By considering the graph of this function, it is shown that the graph is composed of trapezoids ranging from to
with the top made of diagonal line
. The width of each trapezoid is
, etc. Whenever
is odd, the value of
increases by an integer value, plus
. Whenever
is even, the value of
increases by an integer value. Since each trapezoid always has an odd width, every value of
is not an integer when
, and is an integer when
. Every other value is an integer when
is odd. Therefore, it is simply a matter to determine the number of values of
where
(
), and add the number of values of
where
is odd (
). Adding the two values gives
.
See Also
2015 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.