## គណិតវិទ្យា​អូឡាំពិច​១៩៦១

ថ្ងៃ​ទី​មួយ
1. (HUN) ដោះ​ស្រាយ​ប្រព័ន្ធ​សមីការ

$x + y + z = a,$
$x^2 + y^2 + z^2 = b^2,$
$xy = z^2,$
where $a$ and $b$ are given real numbers. What conditions must hold on $a$ and $b$ for the solutions to be positive and distinct?
2. (POL) Let $a, b,$ and $c$ be the lengths of a triangle whose area is $S$. Prove that
$a^2 + b^2 + c^2 \geq 4S\sqrt{3}$ .
In what case does equality hold?
3. (BUL) ដោះ​ស្រាយ​សមីការ $\cos^n x-\sin^n x = 1$, ដែល​ $n$ ជា​ចំនួន​គត់​វិជ្ជមាន​ដែល​គេ​អោយ។

ថ្ងៃ​ទី​ពីរ

4. (GDR) In the interior of $P_1P_2P_3$ a point $P$ is given. Let $Q_1, Q_2,$ and $Q_3$ respectively be the intersections of $PP_1, PP_2,$ and $PP_3$ with the opposing edges of $P_1P_2P_3$. Prove that among the ratios $PP_1/PQ_1, PP_2/PQ_2,$ and $PP_3/PQ_3$ there exists at least one not larger than $2$ and at least one not smaller than $2$.

5. (CZS) Construct a triangle $ABC$ if the following elements are given: $AC = b, AB = c,$ and $\angle AMB = \omega (\omega < 90^{\circ})$, where $M$ is the midpointof $BC$. Prove that the construction has a solution if and only if $b \tan \frac{\omega}{2}\leq c .
In what case does equality hold?
6. (ROM) A plane is given and on one side of the plane three noncollinear points $A, B,$ and $C$ such that the plane determined by them is not parallel to $\epsilon$. Three arbitrary points $A', B',$ and $C'$ in are selected. Let $L, M,$ and $N$ be the midpoints of $AA', BB',$ and $CC',$ and $G$ the centroid of $\triangle LMN$. Find the locus of all points obtained for $G$ as$A', B',$ and $C'$ are varied (independently of each other) accros $\epsilon$.