2006 UNCO Math Contest II Problems/Problem 9

Problem

Determine three positive integers $a,b$ and $c$ that simultaneously satisfy the following three conditions:

(i) $a<b<c$

(ii) Each of $a+b,a+c$ and $b+c$ is the square of an integer, and

(iii) $c$ is as small as is possible.

Solution

$a=6;b=19,c=30$ (also $2-34-47$ )

Let $a+b = x^2$ , $a + c = y^2$ , $b + c = z^2$ . We can easily find that $a = \dfrac{x^2+y^2-z^2}{2}$ , $b = \dfrac{x^2+z^2-y^2}{2}$ , $c = \dfrac{y^2+z^2-x^2}{2}$ . Taking mod 2, we find that $(x, y, z)$ must be either $(0, 0, 0)$ , $(0, 1, 1)$ , $(1, 0, 1)$ , $(1, 1, 0)$ . For the first case, we check $(2n, 2n + 2, 2n + 4)$ until $(a, b, c)$ is positive. We get $n = 4$ , and $(a, b, c) = (10, 54, 90)$ . For the second case, we check $(2n, 2n+1, 2n+3)$ , getting $n = 3$ , and $(a, b, c) = (2, 34, 47)$ . For the third case, we check $(2n + 1, 2n + 2, 2n + 3)$ , getting $n = 2$ and $(a, b, c) = (6, 19, 30)$ . For the last case, we check $(2n + 1, 2n + 3, 2n + 4)$ , getting $n = 2$ , and $(a, b, c) = (5, 20, 44)$ . Clearly, our triple with the minimum $c$ value is $(6, 19, 30)$ . ~Puck_0

2006 UNCO Math Contest II (Problems • Answer Key • Resources)
Preceded by Problem 8	Followed by Problem 10
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10
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2006 UNCO Math Contest II Problems/Problem 9

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