Competition problem — Algebra — Maths Olympiad Prep

Library / /20 of 520

Algebra Difficulty 5.6 AIME, harder Find the answer

Example 1 (Newly Compiled Question) Let the parabola $y=x^{2}-5 m x+6 m^{2}(m>0)$ intersect the $x$ -axis at two points $M, N$ , and the $y$ -axis at point $P$ . For what value of $m$ is $\angle M P N$ maximized?

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

Solution: Since $a=1, c=6 \mathrm{~m}^{2}$ , by Theorem (6) we get
$6 m^{2}=1 \Rightarrow m=\frac{\sqrt{6}}{6}(m>0) .$

Therefore, when $m=\frac{\sqrt{6}}{6}$ , $\angle M P N$ reaches its maximum value.

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: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.