Euclid 2019/Problem 5

Problem

(a) Determine the two pairs of positive integers $(a,b)$ with $a<b$ that satisfy the equation $\sqrt a+\sqrt b=\sqrt (50)$ .

(b) Consider the system of equations: $c+d=2000$ and $\frac{c}{d}=k$ . Determine the number of integers $k$ with $k\leq0$ for which there is at least one pair of integers $(c,d)$ that is a solution to the system.

Solution

(a) $\sqrt(50)=5\sqrt(2)$ , and $5\sqrt(2)$ can be expressed as $\sqrt(2)+4\sqrt(2)$ , or $2\sqrt(2)+3\sqrt(2)$ . From the first equation, we get that $\sqrt(2)+\sqrt(32)$ , and from the second of these equations, we have $\sqrt(8)+\sqrt(18)$ . So our solutions for $(a,b)$ are $\boxed{(2,32), (8,18)}$ .

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Euclid 2019/Problem 5

Problem

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