Moldova National Mathematical Olympiad 2008.

Posted on March 2, 2008 by aimingbeyond

Yesterday was the first day and today the second day. Here are the problems for 12th grade.

Check here for the problems in pdf format. Next week I have holiday, so I will try to upload problems for other grades, eventually with solutions.

Day 1

Problem 1

Consider the equation $x^4-4x^3+4x^2+ax+b=0$ , where $a,b\in\mathbb{R}$ . Determine the largest value $a+b$ can take, so that the given equation has two distinct positive roots $x_1,x_2$ so that $x_1+x_2=2x_1x_2$ .

Problem 2

Evaluate

$\displaystyle E=\int_0^{\frac\pi2}\cos^{1003}x\text{d}x\cdot\int_0^{\frac\pi2}\cos^{1004}x\text{d}x$ .

Problem 3

In the usual coordinate system $xOy$ , line $d$ intersects circles $C_1:(x+1)^2+y^2=1$ and $C_2:(x-2)^2+y^2=4$ in the points $A,B,C,D$ (in this order), all having positive $Oy$ coordinates. Given that $A\left(-\frac32,\frac{\sqrt3}2\right)$ and $m(\angle{BOC})=60^\circ$ find the slope of $d$ .

Problem 4

Define the sequence $(a_p)_{pge0}$ as follows:

$a_p=\displaystyle\frac{\binom p0}{2\cdot 4}-\frac{\binom p1}{3\cdot5}+\frac{\binom p2}{4\cdot6}-\ldots+(-1)^p\cdot\frac{\binom pp}{(p+2)(p+4)}$

Compute $\displaystyle\lim_{n\to\infty}(a_0+a_1+\ldots+a_n)$ .

Day 2

Problem 5

Find the least positive integer $n$ so that the polynomial $P(X)=\sqrt3X^{n+1}-X^n-1$ has at least one root of modulus $1$ .

Problem 6

For $n\ge1$ , let

$\displaystyle a_n=\frac1{\sqrt{n^2+8n-1}}+\frac1{\sqrt{n^2+16n-1}}+\frac1{\sqrt{n^2+24n-1}}+\ldots+\frac1{\sqrt{9n^2-1}}$

Find $\displaystyle\lim_{n\to\infty}a_n$ .

Problem 7

Vertices $B,C$ of triangle are fixed and $BC=2$ , while $A$ is variable. Denote by $H$ and $G$ the orthocenter and centroid respectively of triangle $ABC$ . Let $F\in (HG)$ so that $HF/FG=3$ . Find the locus of the point $A$ so that $F\in BC$ .