1. Let 
be real positive numbers. Prove that the following inequality holds:
2. Prove that the lines joining the middle-points of non-adjacent sides of an convex quadrilateral and the line joining the middle-points of diagonals, are concurrent. Prove that the intersection point is the middle point of the three given segments.
3. Let
be a natural number, for which we define 
Prove that: 
Determine: 
4. From the number
we delete its first digit, and then add the same digit to the remaining number. This process continues until the left number has ten digits. Show that the left number has two same digits.
5. Find all functions
such that 
holds:
be real positive numbers. Prove that the following inequality holds: 2. Prove that the lines joining the middle-points of non-adjacent sides of an convex quadrilateral and the line joining the middle-points of diagonals, are concurrent. Prove that the intersection point is the middle point of the three given segments.
3. Let
be a natural number, for which we define Prove that: Determine: 4. From the number
we delete its first digit, and then add the same digit to the remaining number. This process continues until the left number has ten digits. Show that the left number has two same digits. 5. Find all functions
such that holds: 
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