2008 Polish Mathematical Olympiad Third Round

Day 1

Problem 1

In cells of $n \times n$ table are written numbers $1,2, \ldots, n^2$ , where the numbers $1,2, \ldots, n$ are in the first row (from left side to right), numbers $n+1, n+2, \ldots, 2n$ in the second, etc. In that table $n$ cells are chosen, from which no two lie in the same row or column. Let $a_i$ be the chosen number in row number $i$ . Prove that $\frac{1^2}{a_1} + \frac{2^2}{a_2} + \ldots + \frac{n^2}{a_n} \geq \frac{n+2}{2} - \frac{1}{n^2 + 1}.$

Problem 2

Function $f(x,y,z)$ of three real variables satisfies for all real numbers $a,b,c,d,e$ the equality $f(a,b,c) + f(b,c,d) + f(c,d,e) + f(d,e,a) + f(e,a,b) = a + b + c + d + e.$ Prove that for all real numbers $x_1, x_2, \ldots, x_n$ $(n \geq 5)$ the equality $f(x_1, x_2, x_3) + f(x_2,x_3,x_4) + \ldots + f(x_n,x_1,x_2) = x_1 + x_2 + \ldots + x_n$ is satisfied.

Problem 3

In a convex pentagon $ABCDE$ , where $BC = DE$ , the equations $\angle ABE = \angle CAB = \angle AED - 90^{\circ} \quad \text{and} \quad \angle ACB = \angle ADE$ hold. Prove that $BCDE$ is a parallelogram.

Day 2

Problem 4

Every point with integer coordinates on a plane is painted either black or white. Prove that among the set of all painted points there exists an infinite subset which has a center of symmetry and has all the points of the same colour.

Problem 5

The areas of all cross sections of the parallelepiped $R$ with planes going through the middles of three of its edges, of which none two are parallel and have no common points, are equal. Prove that $R$ is a cuboid.

Problem 6

Let $S$ be the set of all positive integers which can be expressed in the form $a^2 + 5b^2$ for some coprime integers $a$ and $b$ . Let $p$ be a prime number with rest 3 when divided by 4. Prove that if some positive multiple of $p$ belongs to $S$ , then the number $2p$ also belongs to $S$ .

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