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: 2

2009 CIIM, Problem 5
Let $f:\mathbb{R} \to \mathbb{R}$, such that i) For all $a \in \mathbb{R}$ and all $\epsilon > 0$, exists $\delta > 0$ such that $|x-a| < \delta \Rightarrow f(x) < f(a) + \epsilon.$ ii) For all $b\in \mathbb{R}$ and all $\epsilon > 0$, exists $x,y \in \mathbb{R}$ with $ b - \epsilon < x < b < y < b + \epsilon$, such that $|f(x)-f(b)|< \epsilon$ and $|f(y)-f(b)| < \epsilon.$ Prove that if $f(a) < d < f(d)$ there exists $c$ with $a < c < b$ or $b < c < a$ such that $f(c) = d$.
undegraduate inequalities

2009 CIIM, Problem 2
Determine if for all natural $n$ there is a $n \times n$ matrix of real entries such that its determinant is 0 and that changing any entry produce another matrix with nonzero determinant.
undegraduate algebra casework extremal principle