1959 IMO Problems/Problem 2

Problem

For what real values of $x$ is

$\sqrt{x+\sqrt{2x-1}} + \sqrt{x-\sqrt{2x-1}} = A,$

given (a) $A=\sqrt{2}$ , (b) $A=1$ , (c) $A=2$ , where only non-negative real numbers are admitted for square roots?

Solution

The square roots imply that $x\ge \frac{1}{2}$ .

Square both sides of the given equation: $A^2 = \Big( x + \sqrt{2x - 1}\Big) + 2 \sqrt{x + \sqrt{2x - 1}} \sqrt{x - \sqrt{2x - 1}} + \Big( x - \sqrt{2x - 1}\Big)$

Add the first and the last terms to get: $A^2 = 2x + 2 \sqrt{x + \sqrt{2x - 1}} \sqrt{x - \sqrt{2x - 1}}$

Multiply the middle terms, and use $(a + b)(a - b) = a^2 - b^2$ to get: $A^2 = 2x + 2 \sqrt{x^2 - 2x + 1}$

Since the term inside the square root is a perfect square, and by factoring 2 out, we get $A^2 = 2(x + \sqrt{(x-1)^2})$ Use the property that $\sqrt{x^2}=|x|$ to get $A^2 = 2(x+|x-1|)$

Case I: If $x \le 1$ , then $|x-1| = 1 - x$ , and the equation reduces to $A^2 = 2$ . This is precisely part (a) of the question, for which the valid interval is now $x \in \left[ \frac{1}{2}, 1 \right]$

Case II: If $x > 1$ , then $|x-1| = x - 1$ and we have $x = \frac{A^2 + 2}{4} > 1$ which simplifies to $A^2 > 2$

This tells there that there is no solution for (b), since we must have $A^2 \ge 2$

For (c), we have $A = 2$ , which means that $A^2 = 4$ , so the only solution is $x=\frac{3}{2}$ .

~flamewavelight (Expanded)

~phoenixfire (edited)

An alternate solution: Note that the equation can be rewritten to $\sqrt{(\sqrt{2x-1}+1)^2} + \sqrt{(\sqrt{2x-1}-1)^2}=A\sqrt{2}$ i.e., $\sqrt{2x-1}+1 + |\sqrt{2x-1}-1|=A\sqrt{2}$ .

Case I: when $2x-1\ge 1$ (i.e., $x\ge 1$ ), the equation becomes $2\sqrt{2x-1}=\sqrt{2}A$ . For (a), we have $x=1$ ; for (b) we have $x=3/4$ but $x\ge 1$ , thus no solution; for (c) we have $x=\frac{3}{2}$ .

Case II: when $0\le 2x-1 <1$ (i.e., $1/2\le x <1$ ), the equation becomes $2=\sqrt{2}A$ , which only works for (a) $A=\sqrt{2}$ .

In summary, any $x \in \left[1/2, 1\right]$ is a solution for (a); there is no solution for (b); there is one solution for (c), which is $x=\frac{3}{2}$ .

~zhaoxiayu (added)

Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.

1959 IMO (Problems) • Resources
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1959 IMO Problems/Problem 2

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