Difference between revisions of "2008 Polish Mathematical Olympiad Third Round"
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==Day 1== | ==Day 1== | ||
===Problem 1=== | ===Problem 1=== | ||
− | In | + | In cells of <math>n \times n</math> table are written numbers <math>1,2, \ldots, n^2</math>, where the numbers <math>1,2, \ldots, n</math> are in the first row (from left side to right), numbers <math>n+1, n+2, \ldots, 2n</math> in the second, etc. |
− | In that table <math>n</math> | + | In that table <math>n</math> cells are chosen, from which no two lie in the same row or column. Let <math>a_i</math> be the chosen number in row number <math>i</math>. Prove that <cmath> \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}.</cmath> |
===Problem 2=== | ===Problem 2=== | ||
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In a convex pentagon <math>ABCDE</math>, where <math>BC = DE</math>, the equations <cmath>\angle ABE = \angle CAB = \angle AED - 90^{\circ} \quad \text{and} \quad \angle ACB = \angle ADE</cmath> hold. Prove that <math>BCDE</math> is a parallelogram. | In a convex pentagon <math>ABCDE</math>, where <math>BC = DE</math>, the equations <cmath>\angle ABE = \angle CAB = \angle AED - 90^{\circ} \quad \text{and} \quad \angle ACB = \angle ADE</cmath> hold. Prove that <math>BCDE</math> is a parallelogram. | ||
− | == | + | ==Day 2== |
===Problem 4=== | ===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. | 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. |
Latest revision as of 17:13, 4 July 2022
Contents
[hide]Day 1
Problem 1
In cells of table are written numbers , where the numbers are in the first row (from left side to right), numbers in the second, etc. In that table cells are chosen, from which no two lie in the same row or column. Let be the chosen number in row number . Prove that
Problem 2
Function of three real variables satisfies for all real numbers the equality Prove that for all real numbers the equality is satisfied.
Problem 3
In a convex pentagon , where , the equations hold. Prove that 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 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 is a cuboid.
Problem 6
Let be the set of all positive integers which can be expressed in the form for some coprime integers and . Let be a prime number with rest 3 when divided by 4. Prove that if some positive multiple of belongs to , then the number also belongs to .