Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 5.0 AIME, harder Find the answer

Acute triangle $A B C$ has circumcenter $O$ . The bisector of $\angle A B C$ and the altitude from $C$ to side $A B$ intersect at $X$ . Suppose that there is a circle passing through $B, O, X$ , and $C$ . If $\angle B A C=n^{\circ}$ , where $n$ is a positive integer, compute the largest possible value of $n$ .

Your answer

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

Solution

We have $\angle X B C=B / 2$ and $\angle X C B=90^{\circ}-B$ . Thus, $\angle B X C=90^{\circ}+B / 2$ . We have $\angle B O C=2 A$ , so $90^{\circ}+B / 2=2 A$ This gives $B=4 A-180^{\circ}$ , which gives $C=360^{\circ}-5 A$ . In order for $0^{\circ}<B<90^{\circ}$ , we need $45^{\circ}<A<67.5^{\circ}$ . In order for $0^{\circ}<C<90^{\circ}$ , we require $54^{\circ}<A<72^{\circ}$ . The largest integer value in degrees satisfying these inequalities is $A=67^{\circ}$ .

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