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2024 SSMO Relay Round 2 Problems

Revision as of 15:07, 2 May 2025 by Pinkpig (talk | contribs) (Created page with "==Problem 1== In a regular hexagon <math>ABCDEF</math>, let <math>X</math> be a point inside the hexagon such that <math>XA=XB=3.</math> If the area of the hexagon is <math>6...")
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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

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