DE-MO 2015

Determine all pairs of real numbers $(x,y)$ satisfying \begin{align*} x^3+9x^2y&=10, y^3+xy^2 &=2. \end{align*}

A positive integer $n$ is called smooth if there exist integers $a_1,a_2,\dotsc,a_n$ satisfying $$a_1+a_2+\dotsc+a_n=a_1 \cdot a_2 \cdot \dotsc \cdot a_n=n.$$ Find all smooth numbers.

To prepare a stay abroad, students meet at a workshop including an excursion. To promote interaction between the students, they are to be distributed to two busses such that not too many of the students in the same bus know each other. Every student knows all those who know her. The number of such pairwise acquaintances is $k$ .

Prove that it is possible to distribute the students such that the number of pairwise acquaintances in each bus is at most $\frac{k}{3}$ .

Let $k$ be a positive integer. Define $n_k$ to be the number with decimal representation $70...01$ where there are exactly $k$ zeroes. Prove the following assertions:

a) None of the numbers $n_k$ is divisible by $13$ . b) Infinitely many of the numbers $n_k$ are divisible by $17$ .

Let $ABCD$ be a convex quadrilateral such that the circle with diameter $AB$ touches the line $CD$ . Prove that that the circle with diameter $CD$ touches the line $AB$ if and only if $BC$ and $AD$ are parallel.

Prove that for all $x,y,z>0$ , the inequality $$\frac{x+y+z}{3}+\frac{3}{\frac{1}{x}+\frac{1}{y}+\frac{1}{z}} \ge 5 \sqrt[3]{\frac{xyz}{16}}$$ holds. Determine if equality can hold and if so, in which cases it occurs.