Moldova, Second Team Selection Test for IMO-BMO 2008

Posted on March 29, 2008 by aimingbeyond

Here is the pdf file. Time allowed – 4 hours, 30 minutes. Each problem is worth 7 points.

Moldova, Second Team Selection Test for IMO-BMO 2008, March 29

Problem 1

Find all solutions $(x,y)\in\mathbb{R}\times\mathbb{R}$ of the system:

$\begin{cases} x^3+3xy^2=49,\\ x^2+8xy+y^2=8y+17x.\end{cases}$

Problem 2

Let $a_1,a_2,\ldots,a_n$ be positive reals so that $a_1+a_2+\ldots+a_n\le\dfrac n2$ . Find the minimal value of

$\displaystyle\sqrt{a_1^2+\frac1{a_2^2}}+\sqrt{a_2^2+\frac1{a_3^2}}+\ldots+\sqrt{a_n^2+\frac1{a_1^2}}$

Problem 3

Let $\omega$ be the circumcircle of $ABC$ and let $D$ be a fixed point on $(BC)$ . $X$ is a variable point on $(BC)$ , $X\neq D$ . Denote by $Y$ the second intersection of $AX$ and $\omega$ . Prove that the circumcircle of triangle $XYD$ passes through a fixed point.