Let $x,y$, and $z$ be positive real numbers such that $xy+yz+zx=3xyz$. Prove that $$x^2y+y^2z+z^2x \geq 2(x+y+z)-3.$$ In which cases do we have equality?

Given a triangle $ABC$, let the bisector of $\angle BAC$ meets the side $BC$ and circumcircle of triangle $ABC$ at $D$ and $E$, respectively. Let $M$ and $N$ be the midpoints of $BD$ and $CE$, respectively. Circumcircle of triangle $ABD$ meets $AN$ at $Q$. Circle passing through $A$ that is tangent to $BC$ at $D$ meets line $AM$ and side $AC$ respectively at $P$ and $R$. Show that the four points $B,P,Q,R$ lie on the same line. [i]Proposer: Fajar Yuliawan[/i]

Let $D_n$ be the determinant of order $n$ of which the element in the $i$-th row and the $j$-th column is $|i-j|.$ Show that $D_n$ is equal to $$(-1)^{n-1}(n-1)2^{n-2}.$$

Ten teams of five runners each compete in a cross-country race. A runner finishing in [i]n[/i]th place contributes [i]n[/i] points to his team, and there are no ties. The team with the lowest score wins. Assuming the first place team does not have the same score as any other team, how many winning scores are possible?

Let $S$ be a set of size $11$. A random $12$-tuple $(s_1, s_2, . . . , s_{12})$ of elements of $S$ is chosen uniformly at random. Moreover, let $\pi : S \to S$ be a permutation of $S$ chosen uniformly at random. The probability that $s_{i+1}\ne \pi (s_i)$ for all $1 \le i \le 12$ (where $s_{13} = s_1$) can be written as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Compute $a$.

Let $a$ and $b$ be real, positive and distinct numbers. We consider the set: \[M=\{ ax+by\mid x,y\in\mathbb{R},\ x>0,\ y>0,\ x+y=1\} \] Prove that: (i) $\frac{2ab}{a+b}\in M;$ (ii) $\sqrt{ab}\in M.$

In triangle $ ABC$, $ AB \equal{} AC \equal{} 100$, and $ BC \equal{} 56$. Circle $ P$ has radius $ 16$ and is tangent to $ \overline{AC}$ and $ \overline{BC}$. Circle $ Q$ is externally tangent to $ P$ and is tangent to $ \overline{AB}$ and $ \overline{BC}$. No point of circle $ Q$ lies outside of $ \triangle ABC$. The radius of circle $ Q$ can be expressed in the form $ m \minus{} n\sqrt {k}$, where $ m$, $ n$, and $ k$ are positive integers and $ k$ is the product of distinct primes. Find $ m \plus{} nk$.

[b]8.[/b] An Oblique lattice-cubs is a lattice-cube of the three-dimensional fundamental lattice no edge of which is perpendicular to any coordinate axis. Prove that for any integer $h= 8n-1$ ($n= 1, 2, \dots $) there existis an oblique lattice-cube with edges of length $h$. Propose a method for finding such a cube. [b](N. 20)[/b]

A quadrilateral $ABCD$ is inscribed in a circle $k$ where $AB$ $>$ $CD$,and $AB$ is not paralel to $CD$.Point $M$ is the intersection of diagonals $AC$ and $BD$, and the perpendicular from $M$ to $AB$ intersects the segment $AB$ at a point $E$.If $EM$ bisects the angle $CED$ prove that $AB$ is diameter of $k$. Proposed by Emil Stoyanov,Bulgaria

Let $ A$ be one of the two points of intersection of two circles with centers $ X, Y$ respectively.The tangents at $ A$ to the two circles meet the circles again at $ B, C$. Let a point $ P$ be located so that $ PXAY$ is a parallelogram. Show that $ P$ is also the circumcenter of triangle $ ABC$.

Prove that a positive integer $n$ is composite if and only if there exist positive integers $a,b,x,y$ such that $a+b = n$ and $\frac{x}{a}+\frac{y}{b}= 1$.

Let $a,b$ be integers with $0<a<b$. A set $\{x,y,z\}$ of non-negative integers is [i]olympic[/i] if $x<y<z$ and if $\{z-y,y-x\}=\{a,b\}$. Show that the set of all non-negative integers is the union of pairwise disjoint olympic sets.

There are $n{}$ stones in a heap. Two players play the game by alternatively taking either 1 stone from the heap or a prime number of stones which divides the current number of stones in the heap. The player who takes the last stone wins. For which $n{}$ does the first player have a strategy so that he wins no matter how the other player plays? [i]Fedor Ivlev[/i]

Given a non-isosceles triangle $ABC$, let $D,E$, and $F$ denote the midpoints of the sides $BC,CA$, and $AB$ respectively. The circle $BCF$ and the line $BE$ meet again at $P$, and the circle $ABE$ and the line $AD$ meet again at $Q$. Finally, the lines $DP$ and $FQ$ meet at $R$. Prove that the centroid $G$ of the triangle $ABC$ lies on the circle $PQR$. [i](United Kingdom) David Monk[/i]

For any two coprime positive integers $p, q$, define $f(i)$ to be the remainder of $p\cdot i$ divided by $q$ for $i = 1, 2,\ldots,q -1$. The number $i$ is called a[b] large [/b]number (resp. [b]small[/b] number) when $f(i)$ is the maximum (resp. the minimum) among the numbers $f(1), f(2),\ldots,f(i)$. Note that $1$ is both large and small. Let $a, b$ be two fixed positive integers. Given that there are exactly $a$ large numbers and $b$ small numbers among $1, 2,\ldots , q - 1$, find the least possible number for $q$. [i] Proposed by usjl[/i]

On the sides $BC$, $CA$, $AB$ of an acute-angled triangle $ABC$, we erect (outwardly) the squares $BB_aC_aC$, $CC_bA_bA$, $AA_cB_cB$, respectively. On the sides $B_cB_a$ and $C_aC_b$ of the triangles $BB_cB_a$ and $CC_aC_b$, we erect (outwardly) the squares $B_cB_vB_uB_a$ and $C_aC_uC_vC_b$. Prove that $B_uC_u\parallel BC$. [i]Comment.[/i] This problem originates in the 68th Moscow MO 2005, and a solution was posted in http://www.mathlinks.ro/Forum/viewtopic.php?t=30184 . However ingenious this solution is, there is a different one which shows a bit more: $B_uC_u=4\cdot BC$. Darij

The circle $\Gamma$ through $A$ of triangle $ABC$ meets sides $AB,AC$ at $E$,$F$ respectively, and circumcircle of $ABC$ at $P$. Prove: Reflection of $P$ across $EF$ is on $BC$ if and only if $\Gamma$ passes through $O$ (the circumcentre of $ABC$).

Masaru randomly paints $50\%$ of the area of a square. What is the probability that at least $60\%$ of the left side of the square is painted? [asy] size(2cm); defaultpen(fontsize(7)); draw((0,0)--(4,0)--(4,4)--(0,4)--cycle,linewidth(1.5)); fill(circle((0.3,0.3),0.3),paleblue); fill(circle((1,1),0.5),palered); fill(circle((0.8,2),0.8),purple); fill(circle((3,3),0.6),orange); fill(circle((3.4,1.5),0.6),mediumgreen); fill(circle((1.4,1.6),0.4),yellow); fill(circle((2.5,2.8),0.4),cyan); fill(circle((1,3.5),0.5),red); fill((3,0)--(4,0)--(4,0.4)--(3.5,0.8)--cycle,magenta); fill((2,4)--(3,4)--(3.1,3.8)--(2.7,3.5)--(2.4,3.1)--cycle,olive); draw((2,0)--(2,4),dashed); [/asy] $\textbf{(A) } 25\%\qquad\textbf{(B) } 30\%\qquad\textbf{(C) } 35\%\qquad\textbf{(D) } 40\%\qquad\textbf{(E) } 45\%$

In triangle ABC, consider the sizes $\tan \angle A, \tan \angle B$, and $\tan \angle C$ into another such as the numbers $1, 2$ and $3$. Find the ratio of the sidelenghts $AC$ and $AB$ of the triangle.

Prove that there is no function $f : Z \to Z$ such that $f(f(x)) = x+1$ for all $x$.

Given an integer $n\geq 2$, determine all non-constant polynomials $f$ with complex coefficients satisfying the condition \[1+f(X^n+1)=f(X)^n.\]

For $a, b > 0$, denote by $t(a,b)$ the positive root of the equation $$(a+b)x^2-2(ab-1)x-(a+b) = 0.$$ Let $M = \{ (a.b) | \, a \ne b \,\,\, and \,\,\,t(a,b) \le \sqrt{ab} \}$ Determine, for $(a, b)\in M$, the mmimum value of $t(a,b)$.

Prove that the polynomial $z^{2n} + z^n + 1\ (n \in \mathbb{N})$ is divisible by the polynomial $z^2 + z + 1$ if and only if $n$ is not a multiple of $3$.

Let $f(x)$ be a real-valued function defined for $0<x<1.$ If $$ \lim_{x \to 0} f(x) =0 \;\; \text{and} \;\; f(x) - f \left( \frac{x}{2} \right) =o(x),$$ prove that $f(x) =o(x),$ where we use the O-notation.

In a certain forest there are at least three crossroads, and for any three crossroads of roads A, B and C there is a road from A to B without passing through C. A deer and a hunter are standing at crossroads of different paths. Is it possible that they can exchange positions without their paths crossing at other points, that are not their initial positions?