MK-MO 2011

Let $~$ $a, b, c, d > 0$ $~$ and $~$ $a+b+c+d = 1 .$ $~$ Prove the inequality

$$\frac{1}{4a+3b+c}+\frac{1}{3a+b+4d}+\frac{1}{a+4c+3d}+\frac{1}{4b+3c+d} \ge 2 .$$

Acute-angled $~$ $\triangle{ABC}$ $~$ is given. A line $~$ $l$ $~$ parallel to side $~$ $AB$ $~$ passing through vertex $~$ $C$ $~$ is drawn. Let the angle bisectors of $~$ $\angle{BAC}$ $~$ and $~$ $\angle{ABC}$ $~$ intersect the sides $~$ $BC$ and $~$ $AC$ at points $~$ $D$ $~$ and $~$ $F$ , and line $~$ $l$ $~$ at points $~$ $E$ $~$ and $~$ $G$ $~$ respectively. Prove that if $~$ $\overline{DE}=\overline{GF}$ $~$ then $~$ $\overline{AC}=\overline{BC} .$

Find all natural numbers $n$ for which each natural number written with $~$ $n-1$ $~$ 'ones' and one 'seven' is prime.

Find all functions $~$ $f: \mathbb{R} \to \mathbb{R}$ $~$ which satisfy the equation

$$f(x+yf(x)) = f(f(x)) + xf(y) .$$

A table of the type $~$ $(n_1, n_2, ... , n_m) ,\ n_1 \ge n_2 \ge ... \ge n_m$ $~$ is defined in the following way: $~$ $n_1$ $~$ squares are ordered horizontally one next to another, then $~$ $n_2$ $~$ squares are ordered horizontally beneath the already ordered $~$ $n_1$ $~$ squares. The procedure continues until a net composed of $~$ $n_1$ $~$ squares in the first row, $~$ $n_2$ $~$ in the second, $~$ $n_i$ $~$ in the $~$ $i$ -th row is obtained, such that there are totally $~$ $n=n_1+n_2+...+n_m$ $~$ squares in the net. The ordered rows form a straight line on the left, as shown in the example. The obtained table is filled with the numbers from $~$ $1$ $~$ till $~$ $n$ $~$ in a way that the numbers in each row and column become greater from left to right and from top to bottom, respectively. An example of a table of the type $~$ $(5,4,2,1)$ $~$ and one possible way of filling it is attached to the post. Find the number of ways the table of type $~$ $(4,3,2)$ $~$ can be filled.

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