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3

Part of 2007 Turkey Team Selection Test

Problems(2)

a+b+c=1

Source: turkey 2007 TST

4/10/2007
Let a,b,ca, b, c be positive reals such that their sum is 11. Prove that 1ab+2c2+2c+1bc+2a2+2a+1ac+2b2+2b≥1ab+bc+ac.\frac{1}{ab+2c^{2}+2c}+\frac{1}{bc+2a^{2}+2a}+\frac{1}{ac+2b^{2}+2b}\geq \frac{1}{ab+bc+ac}.
inequalitiesfunctioninequalities proposedalgebrahighschoolmath
Write 1 or -1 on 2007x2007

Source: turkey 2007 TST

4/10/2007
We write 11 or −1-1 on each unit square of a 2007×20072007 \times 2007 board. Find the number of writings such that for every square on the board the absolute value of the sum of numbers on the square is less then or equal to 11.
linear algebramatrixLaTeXinductionabsolute valuecombinatorics proposedcombinatorics