MathDB

7

Part of 2016 All-Russian Olympiad

Problems(2)

The sum not depends to x

Source: All russian olympiad 2016,Day2,grade 9,P7

5/1/2016
In triangle ABCABC,AB<ACAB<AC and ω\omega is incirle.The AA-excircle is tangent to BCBC at A′A^\prime.Point XX lies on AA′AA^\prime such that segment A′XA^\prime X doesn't intersect with ω\omega.The tangents from XX to ω\omega intersect with BCBC at Y,ZY,Z.Prove that the sum XY+XZXY+XZ not depends to point XX.(Mitrofanov)
geometrycircumcircleincircle
Two inequalities from Russia 2016

Source: Russia national 2016

5/1/2016
All russian olympiad 2016,Day 2 ,grade 9,P8 : Let a,b,c,da, b, c, d be are positive numbers such that a+b+c+d=3a+b+c+d=3 .Prove that1a2+1b2+1c2+1d2≤1a2b2c2d2\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}+\frac{1}{d^2}\le\frac{1}{a^2b^2c^2d^2} All russian olympiad 2016,Day 2,grade 11,P7 : Let a,b,c,da, b, c, d be are positive numbers such that a+b+c+d=3a+b+c+d=3 .Prove that 1a3+1b3+1c3+1d3≤1a3b3c3d3\frac{1}{a^3}+\frac{1}{b^3}+\frac{1}{c^3}+\frac{1}{d^3}\le\frac{1}{a^3b^3c^3d^3} Russia national 2016
inequalitiesalgebra