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

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Found problems: 78

1987 Swedish Mathematical Competition, 4
Tags: integration , algebra , analysis
A differentiable function $f$ with $f(0) = f(1) = 0$ is defined on the interval $[0,1]$. Prove that there exists a point $y \in [0,1]$ such that $| f' (y)| = 4 \int _0^1 | f(x)|dx$.

ICMC 4, 4
Tags: geometry , analysis
Let \(\mathbb R^2\) denote the Euclidean plane. A continuous function \(f : \mathbb R^2 \to \mathbb R^2\) maps circles to circles. (A point is not a circle.) Prove that it maps lines to lines. [i]Proposed by Tony Wang[/i]

1976 Spain Mathematical Olympiad, 8
Tags: algebra , function , analysis
Given the function $$y =|x^2 - 4x + 3|.$$ Study its continuity and differentiability at the point of abscissa $1$. Its graph determines with the $X$ axis a closed figure. Determine the area of said figure.