DE-MO 2019

Determine all real solutions $(x,y)$ of the following system of equations: \begin{align*} x&=3x^2y-y^3, y &= x^3-3xy^2 \end{align*}

Let $a$ and $b$ be two circles, intersecting in two distinct points $Y$ and $Z$ . A circle $k$ touches the circles $a$ and $b$ externally in the points $A$ and $B$ .

Show that the angular bisectors of the angles $\angle ZAY$ and $\angle YBZ$ intersect on the line $YZ$ .

In the cartesian plane consider rectangles with sides parallel to the coordinate axes. We say that one rectangle is below another rectangle if there is a line $g$ parallel to the $x$ -axis such that the first rectangle is below $g$ , the second one above $g$ and both rectangles do not touch $g$ .

Similarly, we say that one rectangle is to the right of another rectangle if there is a line $h$ parallel to the $y$ -axis such that the first rectangle is to the right of $h$ , the second one to the left of $h$ and both rectangles do not touch $h$ .

Show that any finite set of $n$ pairwise disjoint rectangles with sides parallel to the coordinate axes can be enumerated as a sequence $(R_1,\dots,R_n)$ so that for all indices $i,j$ with $1 \le i<j \le n$ the rectangle $R_i$ is to the right of or below the rectangle $R_j$

Show that for each non-negative integer $n$ there are unique non-negative integers $x$ and $y$ such that we have $$n=\frac{(x+y)^2+3x+y}{2}.$$

We are given two positive integers $p$ and $q$ .

Step by step, a rope of length $1$ is cut into smaller pieces as follows: In each step all the currently longest pieces are cut into two pieces with the ratio $p:q$ at the same time. After an unknown number of such operations, the currently longest pieces have the length $x$ .

Determine in terms of $x$ the number $a(x)$ of different lengths of pieces of rope existing at that time.

Suppose that real numbers $x,y$ and $z$ satisfy the following equations:

\begin{align*} x+\frac{y}{z} &=2, y+\frac{z}{x} &=2, z+\frac{x}{y} &=2. \end{align*}

Show that $s=x+y+z$ must be equal to $3$ or $7$ .

Note: It is not required to show the existence of such numbers $x,y,z$ .