1970 Canadian MO Problems/Problem 5

Problem

A quadrilateral has one vertex on each side of a square of side-length 1. Show that the lengths $a$ , $b$ , $c$ and $d$ of the sides of the quadrilateral satisfy the inequalities $2\le a^2+b^2+c^2+d^2\le 4.$

Solution

Let the quadrilateral be $ABCD$ . Suppose $A$ is a distance $w$ , $1-w$ from the two nearest vertices of the square. Define $x$ , $y$ , $z$ similarly. Then the sum of the squares of the sides of the quadrilateral is $w^2 + (1-w)^2 + x^2 + (1-x)^2 + y^2 + (1-y)^2 + z^2 + (1-z)^2$ . But $w^2 + (1-w)^2 = 2(w - \frac{1}{2})^2 + \frac{1}{2}$ which is at least $\frac{1}{2}$ and at most $1$ . Similarly for the other pairs of terms, and hence proved.

1970 Canadian MO (Problems)
Preceded by Problem 4	1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 •	Followed by Problem 6

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1970 Canadian MO Problems/Problem 5

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