KURSCHAK 1972

A triangle has side lengths $a, b, c$ . Prove that $$a(b -c)^2 + b(c - a)^2 + c(a - b)^2 + 4abc > a^3 + b^3 + c^3$$

A class has $n > 1$ boys and $n$ girls. For each arrangement $X$ of the class in a line let $f(X)$ be the number of ways of dividing the line into two non-empty segments, so that in each segment the number of boys and girls is equal. Let the number of arrangements with $f(X) = 0$ be $A$ , and the number of arrangements with $f(X) = 1$ be $B$ . Show that $B = 2A$ .

$ABCD$ is a square side $10$ . There are four points $P_1, P_2, P_3, P_4$ inside the square. Show that we can always construct line segments parallel to the sides of the square of total length $25$ or less, so that each $P_i$ is linked by the segments to both of the sides $AB$ and $CD$ . Show that for some points $P_i$ it is not possible with a total length less than $25$ .