Olympiad problems

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2005 Miklós Schweitzer, 5

Let $GL(n, K)$ be a linear group over the field K with a topology induced by a non-Archimedean absolute value of the field K. Prove that if the matrix $M \in GL (n, K)$ is contained by some compact subgroup of $GL(n, K)$, then all eigenvalues of M have absolute value 1.

2006 Miklós Schweitzer, 10

Tags: real analysis , convex , compactness , euclidean space

Let $K_1,...,K_d$ be convex, compact sets in $R^d$ with non-empty interior. Suppose they are strongly separated, which means for any choice of $x_1 \in K_1, x_2 \in K_2, ...$, their affine hull is a hyperplane in $R^d$. Also let $0< \alpha_i <1$. A half-space H is called an $\alpha$-cut if $vol(K_i \cap H) = \alpha_i\cdot vol(K_i)$ for all i. How many $\alpha$-cuts are there?

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