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The Problem Solver's Resource

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Competition

This is a five-problem competition to test out your skills, updated weekly (or until I receive at least three solutions; whichever is later), using only the skills mentioned in the Problem Solver's Resource. PM your solutions to Temperal, please. Proof is necessary, though more trivial/boring parts may be skipped.

Competition 1

Problem 1

Suppose there is a multi set of odd positive integers, $a,b,c,d$ . If $a+b+c+d=98$ , what is the probability that $\sqrt[4]{abcd}< \frac{a+b+c+d}{4}$ ?

Problem 2

Given positive real numbers $k,l,m,n,p$ , and that $\frac{k^2+l^2}{3p}=\frac{n^2+m^2}{p}=3$ , show that $\frac{1}{k^4+l^4+m^4+n^4}\le \0.75p$ .

Problem 3

Consider the concentric circles with equations $x^2+y^2=4$ and $x^2+y^2=9$ . A radius from the origin $O$ intersects the inner circle at $P$ and the outer circle at $Q$ . The line parallel to the $x$ -axis through $P$ meets the line parallel to the $y$ -axis through $Q$ at the point $R$ . Show that $R$ lies on the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$ .