This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 5

1961 Czech and Slovak Olympiad III A, 4
Tags: triangle area , equilateral , triangle , max , min , computation , geometry
Consider a unit square $ABCD$ and a (variable) equilateral triangle $XYZ$ such that $X, Z$ lie on rays $AB, DC,$ respectively, and $Y$ lies on segment $AD$. Compute the area of triangle $XYZ$ in terms of $x=AX$ and determine its maximum and minimum.

1963 Czech and Slovak Olympiad III A, 1
Tags: geometry , rectangle , 3d geometry , computation
Consider a cuboid$ ABCDA'B'C'D'$ (where $ABCD$ is a rectangle and $AA' \parallel BB' \parallel CC' \parallel DD'$) with $AA' = d$, $\angle ABD' = \alpha, \angle A'D'B = \beta$. Express the lengths x = $AB$, $y = BC$ in terms of $d$ and (acute) angles $\alpha, \beta$. Discuss condition of solvability.

the 9th XMO, 2
Tags: computation , geometry
Given a $\triangle ABC$ with circumcenter $O$ and orthocenter $H(O\ne H)$. Denote the midpoints of $BC, AC$ as $D, E$ and let $D', E'$ be the reflections of $D, E$ w.r.t. point $H$, respectively. If lines $AD'$ and $BE'$ meet at $K$, compute $\frac{KO}{KH}$.

XMO (China) 2-15 - geometry, 9.2
Tags: geometry , computation
Given a $\triangle ABC$ with circumcenter $O$ and orthocenter $H(O\ne H)$. Denote the midpoints of $BC, AC$ as $D, E$ and let $D', E'$ be the reflections of $D, E$ w.r.t. point $H$, respectively. If lines $AD'$ and $BE'$ meet at $K$, compute $\frac{KO}{KH}$.

1961 Czech and Slovak Olympiad III A, 3
Tags: computation
Two cyclists start moving simultaneously in opposite directions on a circular circuit. The first cyclist maintains a constant speed $c_1$ meters per second, the second maintains $c_2$ meters per second. How many times did they meet when the first cyclist completed $n$ laps? Compute for $c_1=10,c_2=7,n=11$.