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: 3

1999 Bosnia and Herzegovina Team Selection Test, 3
Tags: algebra , injective , function
Let $f : [0,1] \rightarrow \mathbb{R}$ be injective function such that $f(0)+f(1)=1$. Prove that exists $x_1$, $x_2 \in [0,1]$, $x_1 \neq x_2$ such that $2f(x_1)<f(x_2)+\frac{1}{2}$. After that state at least one generalization of this result

2009 Thailand Mathematical Olympiad, 2
Tags: functional , functional equation , function , algebra , injective
Is there an injective function $f : Z^+ \to Q$ satisfying the equation $f(xy) = f(x) + f(y)$ for all positive integers $x$ and $y$?

2019 Canadian Mathematical Olympiad Qualification, 1
Tags: functional equation , function , injective function , injective , algebra
A function $f$ is called injective if when $f(n) = f(m)$, then $n = m$. Suppose that $f$ is injective and $\frac{1}{f(n)}+\frac{1}{f(m)}=\frac{4}{f(n) + f(m)}$. Prove $m = n$