7

Part of 2017 Caucasus Mathematical Olympiad

Problems(2)

vieta theorem for small child :)

Source: II Caucasus Mathematical Olympiad

10

distinct numbers are given. Professor Odd had calculated all possible products of

1

3

5

7

9

numbers among given numbers, and wrote down the sum of all these products. Similarly, Professor Even had calculated all possible products of

2

4

6

8

10

numbers among given numbers, and wrote down the sum of all these products. It appears that Odd's sum is greater than Even's sum by

1

. Prove that one of

10

given numbers is equal to

1

folklore 3d geomtery inequality

Source: II Caucasus Mathematical Olympiad

8

ants are placed on the edges of the unit cube. Prove that there exists a pair of ants at a distance not exceeding

1

3D geometrygeometric inequalitygeometryinequalities