2015 AIME I Problems/Problem 14

Problem

For each integer $n \ge 2$ , let $A(n)$ be the area of the region in the coordinate plane deefined by the inequalities $1\le x \le n$ and $0\le y \le x \left\lfloor \sqrt x \right\rfloor$ , where $\left\lfloor \sqrt x \right\rfloor$ is the greatest integer not exceeding $\sqrt x$ . Find the number of values of $n$ with $2\le n \le 1000$ for which $A(n)$ 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.

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
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2015 AIME I Problems/Problem 14

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