Difference between revisions of "2015 AIME I Problems/Problem 14"
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For each integer <math>n \ge 2</math>, let <math>A(n)</math> be the area of the region in the coordinate plane deefined 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. | For each integer <math>n \ge 2</math>, let <math>A(n)</math> be the area of the region in the coordinate plane deefined 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. | ||
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| + | ==Solution== | ||
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| + | By considering the graph of this function, it is shown that the graph is composed of trapezoids ranging from a^2 to (a+1)^2 with the top made of diagonal line y=ax. The width of each trapezoid is 3, 5, 7, etc. Whenever a is odd, the value of A(n) increases by an integer value, plus 1/2. Whenever a is even, the value of A(n) increases by an integer value. Since each trapezoid is always odd in width, every value of n is not an integer when a mod 4 is 2, and is an integer when a mod 4 is 0. Every other value is an integer when a is odd. Therefore, it is simply a matter to determine the number of n's where a mod 4 is 0, and add the number of n's where a is odd, through using Gauss's formula. Adding the two values gives 434. | ||
==See Also== | ==See Also== | ||
{{AIME box|year=2015|n=I|num-b=13|num-a=15}} | {{AIME box|year=2015|n=I|num-b=13|num-a=15}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Revision as of 21:20, 20 March 2015
Problem
For each integer 
, let 
be the area of the region in the coordinate plane deefined 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 a^2 to (a+1)^2 with the top made of diagonal line y=ax. The width of each trapezoid is 3, 5, 7, etc. Whenever a is odd, the value of A(n) increases by an integer value, plus 1/2. Whenever a is even, the value of A(n) increases by an integer value. Since each trapezoid is always odd in width, every value of n is not an integer when a mod 4 is 2, and is an integer when a mod 4 is 0. Every other value is an integer when a is odd. Therefore, it is simply a matter to determine the number of n's where a mod 4 is 0, and add the number of n's where a is odd, through using Gauss's formula. Adding the two values gives 434.
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. 
