2024 SSMO Relay Round 2 Problems

Problem 1

In a regular hexagon $ABCDEF$, let $X$ be a point inside the hexagon such that $XA=XB=3.$ If the area of the hexagon is $6\sqrt{3}$, then $XE^2 = a+b\sqrt{c},$ for squarefree $c$. Find $a+b+c$.

Solution

Problem 2

Let $T = TNYWR.$ If \[a = \sum_{n=1}^{N}n(n+1)(n+2),\] find the last three digits of $a.$

Solution

Problem 3

Let $T = TNYWR.$ A point $P$ is randomly chosen inside the square with vertices $A = (0,0), B = (0,T), C = (T,T),$ and $D = (T,0)$. Find the perimeter of the set $S$ containing all points $P$ for which $AP + CP \ge BP + DP.$

Solution