MathDB

2

Part of 2012 Dutch IMO TST

Problems(2)

\frac{a- b}{b +c}+\frac{b- c}{c +d}+\frac{c - d}{d +a} +\frac{d - a}{a + b}>=0

Source: Dutch IMO TST1 2012 p2

1/10/2020
Let a,b,ca, b, c and dd be positive real numbers. Prove that a−bb+c+b−cc+d+c−dd+a+d−aa+b≥0\frac{a - b}{b + c}+\frac{b - c}{c + d}+\frac{c - d}{d + a} +\frac{d - a}{a + b } \ge 0
inequalities
two boxes containing balls, m and n , two actions permitted

Source: Dutch IMO TST2 2012 p2

1/10/2020
There are two boxes containing balls. One of them contains mm balls, and the other contains nn balls, where m,n>0m, n > 0. Two actions are permitted: (i) Remove an equal number of balls from both boxes. (ii) Increase the number of balls in one of the boxes by a factor kk. Is it possible to remove all of the balls from both boxes with just these two actions, 1. if k=2k = 2? 2. if k=3k = 3?
combinatoricsgame