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: 2

2015 Oral Moscow Geometry Olympiad, 5
Tags: geometry , 3d geometry , sphere , fixed point , moscow , plane
A triangle $ABC$ and spheres are given in space $S_1$ and $S_2$, each of which passes through points $A, B$ and $C$. For points $M$ spheres $S_1$ not lying in the plane of triangle $ABC$ are drawn lines $MA, MB$ and $MC$, intersecting the sphere $S_2$ for the second time at points $A_1,B_1$ and $C_1$, respectively. Prove that the planes passing through points $A_1, B_1$ and $C_1$, touch a fixed sphere or pass through a fixed point.

1952 Moscow Mathematical Olympiad, 214
Tags: algebra , inequalities , absolute value , moscow
Prove that if $|x| < 1$ and $|y| < 1$, then $\left|\frac{x - y}{1 -xy}\right|< 1$.