Difference between revisions of "2008 Polish Mathematical Olympiad Third Round"
(Created page with "==Day 1== ===Problem 1=== ===Problem 2=== Function <math>f(x,y,z)</math> of three real variables satisfies for all real numbers <math>a,b,c,d,e</math> the equality <cmath>f(a...") |
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===Problem 5=== | ===Problem 5=== | ||
+ | The areas of all cross sections of the parallelepiped <math>R</math> 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 <math>R</math> is a cuboid. | ||
===Problem 6=== | ===Problem 6=== | ||
Let <math>S</math> be the set of all positive integers which can be expressed in the form <math>a^2 + 5b^2</math> for some coprime integers <math>a</math> and <math>b</math>. Let <math>p</math> be a prime number with rest 3 when divided by 4. Prove that if some positive multiple of <math>p</math> belongs to <math>S</math>, then the number <math>2p</math> also belongs to <math>S</math>. | Let <math>S</math> be the set of all positive integers which can be expressed in the form <math>a^2 + 5b^2</math> for some coprime integers <math>a</math> and <math>b</math>. Let <math>p</math> be a prime number with rest 3 when divided by 4. Prove that if some positive multiple of <math>p</math> belongs to <math>S</math>, then the number <math>2p</math> also belongs to <math>S</math>. |
Revision as of 16:43, 4 July 2022
Contents
[hide]Day 1
Problem 1
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.
Day2
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 .