MathDB

2

Part of 1996 Turkey MO (2nd round)

Problems(2)

A geometric Inequality on a square

Source: Turkish NMO 1996, 2. Problem

7/31/2011
Let ABCDABCD be a square of side length 2, and let MM and NN be points on the sides ABAB and CDCD respectively. The lines CMCM and BNBN meet at PP, while the lines ANAN and DMDM meet at QQ. Prove that ∣PQ∣≥1\left| PQ \right|\ge 1.
inequalitiesgeometry proposedgeometry
n! divides product of (2^n-2^k) for k=0,..,n-1

Source: Turkish NMO 1996, 5. Problem

7/31/2011
Prove that ∏k=0n−1(2n−2k)\prod\limits_{k=0}^{n-1}{({{2}^{n}}-{{2}^{k}})} is divisible by n!n! for all positive integers nn.
group theoryabstract algebrafloor functionmodular arithmeticnumber theory proposednumber theory