DE-BUNDESWETTBEWERB 2013

Is it possible to partition the set $S=\{1,2,\ldots,21\}$ into subsets that in each of these subsets the largest number is equal to the sum of the other numbers?

In the interior of the square $ABCD$ , the point $P$ lies in such a way that $\angle DCP = \angle CAP=25^{\circ}$ . Find all possible values of $\angle PBA$ .

Suppose $m$ and $n$ are positive integers such that $m^2+n^2+m$ is divisible by $mn$ . Prove that $m$ is a square number.

A parallelogram of paper with sides $25$ and $10$ is given. The distance between the longer sides is $6$ . The paper should be cut into exactly two parts in such a way that one can stick both the pieces together and fold it in a suitable manner to form a cube of suitable edge length without any further cuts and overlaps. Show that it is really possible and describe such a fragmentation.

Let $ABCDEF$ be a convex hexagon whose vertices lie on a circle. Suppose that $AB\cdot CD\cdot EF = BC\cdot DE\cdot FA$ . Show that the diagonals $AD, BE$ and $CF$ are concurrent.

Consider the Pascal's triangle in the figure where the binomial coefficients are arranged in the usual manner. Select any binomial coefficient from anywhere except the right edge of the triangle and labet it $C$ . To the right of $C$ , in the horizontal line, there are $t$ numbers, we denote them as $a_1,a_2,\cdots,a_t$ , where $a_t = 1$ is the last number of the series. Consider the line parallel to the left edge of the triangle containing $C$ , there will only be $t$ numbers diagonally above $C$ in that line. We successively name them as $b_1,b_2,\cdots,b_t$ , where $b_t = 1$ . Show that $$b_ta_1-b_{t-1}a_2+b_{t-2}a_3-\cdots+(-1)^{t-1}b_1a_t = 1$$. For example, Suppose you choose $\binom41 = 4$ (see figure), then $t = 3$ , $a_1 = 6, a_2 = 4, a_3 = 1$ and $b_1 = 3, b_2 = 2, b_3 = 1$ . $$\begin{array}{ccccccccccc} & & & & & 1 & & & & & & & & & 1 & & \underset{b_3}{1} & & & & & & & 1 & & \underset{b_2}{2} & & 1 & & & & & 1 & & \underset{b_1}{3} & & 3 & & 1 & & & 1 & & \boxed{4} & & \underset{a_1}{6} & & \underset{a_2}{4} & & \underset{a_3}{1} & \ldots & & \ldots & & \ldots & & \ldots & & \ldots & & \ldots \end{array}$$