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Τετάρτη 4 Μαΐου 2011

▪ FYROM National Olympiad 2011

1. Let and . Prove the inequality

2. Acute angled triangle is given. A line parallel to side passing through vertex is drawn. Let the angle bisectors of and intersect the sides and at points and , and line at points and respectively. Prove that if then .
3. Find all natural numbers for which each natural number written with 'ones' and one 'seven' is prime.
4. Find all functions which satisfy the equation:

5. A table of the type is defined in the following way: squares are ordered horizontally one next to another, then squares are ordered horizontally beneath the already ordered squares. The procedure continues until a net composed of squares in the first row, in the second, in the -th row is obtained, such that there are totally squares in the net. The ordered rows form a straight line on the left, as shown in the example. The obtained table is filled with the numbers from till in a way that the numbers in each row and column become greater from left to right and from top to bottom, respectively. An example of a table of the type and one possible way of filling it is attached to the post. Find the number of ways the table of type can be filled.