Difference between revisions of "2024 SSMO Relay Round 1 Problems"
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Let <math>AD</math> be an altitude of triangle <math>ABC</math> and let <math>DE</math> be an altitude of triangle <math>ACD.</math> If <math>AB=29,CE=9,</math> and <math>DE=12,</math> what is the area of triangle <math>ABC?</math> | Let <math>AD</math> be an altitude of triangle <math>ABC</math> and let <math>DE</math> be an altitude of triangle <math>ACD.</math> If <math>AB=29,CE=9,</math> and <math>DE=12,</math> what is the area of triangle <math>ABC?</math> | ||
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<asy> | <asy> | ||
size(200); | size(200); | ||
Latest revision as of 16:01, 2 May 2025
Problem 1
Let 
be an altitude of triangle 
and let 
be an altitude of triangle 
If 
and 
what is the area of triangle 
![[asy] size(200); pair A,B,C,D,EE; A = (0,20); B = (-21,0); C = (15,0); D = (0,0); EE = 0.36*A+0.64*C; draw(A--B--C--cycle); draw(A--D--EE); dot("\(A\)", A, N); dot("\(B\)", B, W); dot("\(C\)", C, E); dot("\(D\)", D, S); dot("\(E\)", EE, dir(EE)); [/asy]](http://latex.artofproblemsolving.com/5/6/9/5698f4b3dca327e78898c650e94b6d413c0f49fb.png)
Problem 2
Let 
A circular necklace is called 
if it has 
black beads and white beads. A move consists of cutting out a segment of consecutive beads and reattaching it in reverse. It is possible to change any necklace into any other necklace using at most 
moves. Find . (Note: Rotations and reflections of a necklace are considered the same necklace).
Problem 3
Let In a circle, there are people. 
of them have red shoes, and two of them have blue shoes. First, they will randomly eliminate somebody from the circle. Then, they will randomly eliminate somebody with red shoes from the circle, and the cycle repeats until there is only one person left. If the probability this person has blue shoes is 
for relatively prime positive integers 
and 
find 
