MathDB

3

Part of 2013 International Zhautykov Olympiad

Problems(2)

The big inequality with condition abcd = 1

Source: International Zhautykov Olympiad 2013 - D1 - P3

1/17/2013
Let a,b,ca, b, c, and dd be positive real numbers such that abcd=1abcd = 1. Prove that (a−1)(c+1)1+bc+c+(b−1)(d+1)1+cd+d+(c−1)(a+1)1+da+a+(d−1)(b+1)1+ab+b≥0.\frac{(a-1)(c+1)}{1+bc+c} + \frac{(b-1)(d+1)}{1+cd+d} + \frac{(c-1)(a+1)}{1+da+a} + \frac{(d-1)(b+1)}{1+ab+b} \geq 0. Proposed by Orif Ibrogimov, Uzbekistan.
inequalitiesinequalities unsolved
Blocks in a 10 X 10 table

Source: International Zhautykov Olympiad 2013 - D2 - P3

1/17/2013
A 10×1010 \times 10 table consists of 100100 unit cells. A block is a 2×22 \times 2 square consisting of 44 unit cells of the table. A set CC of nn blocks covers the table (i.e. each cell of the table is covered by some block of CC ) but no n−1n -1 blocks of CC cover the table. Find the largest possible value of nn.
rectanglecombinatoricsExtremal combinatoricscoveringSquaressquare grid