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: 4

1988 Brazil National Olympiad, 4
Tags: Brazilian Math Olympiad 1988 , geometry , porism , Plane Geometry , circles , incircle , circumcircle
Two triangles are circumscribed to a circumference. Show that if a circumference containing five of their vertices exists, then it will contain the sixth vertex too.

1988 Brazil National Olympiad, 2
Tags: Brazilian Math Olympiad 1988 , Brazilian Math Olympiad , orthocenter , geometry , Plane Geometry
Show that, among all triangles whose vertices are at distances 3,5,7 respectively from a given point P, the ones with largest area have P as orthocenter. ([i]You can suppose, without demonstration, the existence of a triangle with maximal area in this question.[/i])

1988 Brazil National Olympiad, 3
Tags: Brazilian Math Olympiad 1988 , functions , number theory , Brazilian Math Olympiad , function
Find all functions $f:\mathbb{N}^* \rightarrow \mathbb{N}$ such that [list] [*] $f(x \cdot y) = f(x) + f(y)$ [*] $f(30) = 0$ [*] $f(x)=0$ always when the units digit of $x$ is $7$ [/list]

1988 Brazil National Olympiad, 1
Tags: Brazilian Math Olympiad , Brazilian Math Olympiad 1988 , prime numbers , number theory
Find all primes which are sum of two primes and difference of two primes.