Competition problem — Geometry — Maths Olympiad Prep

Library / /137 of 348

Geometry Difficulty 4.8 AIME Find the answer

Let $A B C D$ be a unit square. A circle with radius $\frac{32}{49}$ passes through point $D$ and is tangent to side $A B$ at point $E$ . Then $D E=\frac{m}{n}$ , where $m, n$ are positive integers and $\operatorname{gcd}(m, n)=1$ . Find $100 m+n$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Let $O$ be the center of the circle and let $F$ be the intersection of lines $O E$ and $C D$ . Also let $r=32 / 49$ and $x=D F$ . Then we know $x^{2}+(1-r)^{2}=D F^{2}+O F^{2}=D O^{2}=r^{2}$ which implies that $x^{2}+1-2 r=0$ , or $1+x^{2}=2 r$ . Now, $D E=\sqrt{D F^{2}+E F^{2}}=\sqrt{1+x^{2}}=\sqrt{2 r}=\sqrt{64 / 49}=8 / 7$

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

Source: Omni-MATH, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.