The incircle $(I)$ of a triangle $ABC$ touches $BC,CA,AB$ at $D,E, F$. Let $ID, IE, IF$ intersect $EF, FD,DE$ at $X,Y,Z$, respectively. The lines $\ell_a, \ell_b, \ell_c$ through $A,B,C$ respectively and are perpendicular to $YZ,ZX,XY$ . Prove that $\ell_a, \ell_b, \ell_c$ are concurrent at a point that is on the line segment joining $I$ and the centroid of triangle $ABC$ . Nguyễn Minh Hà

Positive numbers $x, y, z$ satisfy $x + y + z \le 1$. Prove that $\big( \frac{1}{x}-1\big) \big( \frac{1}{y}-1\big)\big( \frac{1}{z}-1\big) \ge 8$.

We consider a point $P$ in a plane $p$ and a point $Q \not\in p$. Determine all the points $R$ from $p$ for which \[ \frac{QP+PR}{QR} \] is maximum.

Squares $ABCD$, $EFGH$, and $GHIJ$ are equal in area. Points $C$ and $D$ are the midpoints of sides $IH$ ad $HE$, respectively. What is the ratio of the area of the shaded pentagon $AJICB$ to the sum of the areas of the three squares? [asy] pair A,B,C,D,E,F,G,H,I,J; A = (0.5,2); B = (1.5,2); C = (1.5,1); D = (0.5,1); E = (0,1); F = (0,0); G = (1,0); H = (1,1); I = (2,1); J = (2,0); draw(A--B); draw(C--B); draw(D--A); draw(F--E); draw(I--J); draw(J--F); draw(G--H); draw(A--J); filldraw(A--B--C--I--J--cycle,grey); draw(E--I); dot("$A$", A, NW); dot("$B$", B, NE); dot("$C$", C, NE); dot("$D$", D, NW); dot("$E$", E, NW); dot("$F$", F, SW); dot("$G$", G, S); dot("$H$", H, N); dot("$I$", I, NE); dot("$J$", J, SE);[/asy] $\textbf{(A)}\ \frac14 \qquad \textbf{(B)}\ \frac7{24} \qquad \textbf{(C)}\ \frac13 \qquad \textbf{(D)}\ \frac38 \qquad \textbf{(E)}\ \frac5{12}$

Let $ABC$ be a acute-angled triangle with centroid $G$, the angle bisector of $\angle ABC$ intersects $AC$ in $D$. Let $P$ and $Q$ be points in $BD$ where $\angle PBA = \angle PAB$ and $\angle QBC = \angle QCB$. Let $M$ be the midpoint of $QP$, let $N$ be a point in the line $GM$ such that $GN = 2GM$(where $G$ is the segment $MN$), prove that: $\angle ANC + \angle ABC = 180$

The points $P = (a, b)$ and $Q = (c, d)$ are in the first quadrant of the $xy$ plane, and $a, b, c$ and $d$ are integers satisfying $a < b, a < c, b < d$ and $c < d$. A route from point $P$ to point $Q$ is a broken line consisting of unit steps in the directions of the positive coordinate axes. An allowed route is a route not touching the line $x = y$. Tetermine the number of allowed routes.

Real numbers $a,b,c,d$ are given. Solve the system of equations (unknowns $x,y,z,u)$\[ x^{2}-yz-zu-yu=a\] \[ y^{2}-zu-ux-xz=b\] \[ z^{2}-ux-xy-yu=c\] \[ u^{2}-xy-yz-zx=d\]

Triangle $\triangle ABC$ has $AB= 3$, $BC = 4$, and $AC = 5$. Let $M$ and $N$ be the midpoints of $AC$ and $BC$, respectively. If line $AN$ intersects the circumcircle of triangle $\triangle BMC$ at points $X$ and $Y$, then $XY^2 = \frac{m}{n}$ for some relatively prime positive integers $m,n$. Find $m+n$. [i]Proposed by [b]Th3Numb3rThr33[/b][/i]

(V.Yasinsky, Ukraine) Suppose $ X$ and $ Y$ are the common points of two circles $ \omega_1$ and $ \omega_2$. The third circle $ \omega$ is internally tangent to $ \omega_1$ and $ \omega_2$ in $ P$ and $ Q$ respectively. Segment $ XY$ intersects $ \omega$ in points $ M$ and $ N$. Rays $ PM$ and $ PN$ intersect $ \omega_1$ in points $ A$ and $ D$; rays $ QM$ and $ QN$ intersect $ \omega_2$ in points $ B$ and $ C$ respectively. Prove that $ AB \equal{} CD$.

Let $f(t) = \displaystyle\sum_{j=1}^{N} a_j \sin (2\pi jt)$, where each $a_j$ is areal and $a_N$ is not equal to $0$. Let $N_k$ denote the number of zeroes (including multiplicites) of $\dfrac{d^k f}{dt^k}$. Prove that \[ N_0 \le N_1 \le N_2 \le \cdots \text { and } \lim_{k \rightarrow \infty} N_k = 2N. \] [color=green][Only zeroes in [0, 1) should be counted.][/color]

Determine the natural numbers that cannot be written as $\lfloor n + \sqrt{n} + \frac{1}{2} \rfloor$ for any $n \in \mathbb{N}$.

Triangle $ABC$ has $AB=25$, $AC=29$, and $BC=36$. Additionally, $\Omega$ and $\omega$ are the circumcircle and incircle of $\triangle ABC$. Point $D$ is situated on $\Omega$ such that $AD$ is a diameter of $\Omega$, and line $AD$ intersects $\omega$ in two distinct points $X$ and $Y$. Compute $XY^2$. [i]Proposed by David Altizio[/i]

In a triangle $ABC$, $\angle A$ is $60^\circ$. On sides $AB$ and $AC$ we make two equilateral triangles (outside the triangle $ABC$) $ABK$ and $ACL$. $CK$ and $AB$ intersect at $S$ , $AC$ and $BL$ intersect at $R$ , $BL$ and $CK$ intersect at $T$. Prove the radical centre of circumcircle of triangles $BSK, CLR$ and $BTC$ is on the median of vertex $A$ in triangle $ABC$. [i]Proposed by Ali Zamani[/i]

Suppose that each of the 5 persons knows a piece of information, each piece is different, about a certain event. Each time person $A$ calls person $B$, $A$ gives $B$ all the information that $A$ knows at that moment about the event, while $B$ does not say to $A$ anything that he knew. (a) What is the minimum number of calls are necessary so that everyone knows about the event? (b) How many calls are necessary if there were $n$ persons?

Let $x=-2016$. What is the value of $\left| \ \bigl \lvert { \ \lvert x\rvert -x }\bigr\rvert -|x|{\frac{}{}}^{}_{}\right|-x$? $\textbf{(A)}\ -2016\qquad\textbf{(B)}\ 0\qquad\textbf{(C)}\ 2016\qquad\textbf{(D)}\ 4032\qquad\textbf{(E)}\ 6048$

A cylindrical log has diameter $ 12$ inches. A wedge is cut from the log by making two planar cuts that go entirely through the log. The first is perpendicular to the axis of the cylinder, and the plane of the second cut forms a $ 45^\circ$ angle with the plane of the first cut. The intersection of these two planes has exactly one point in common with the log. The number of cubic inches in the wedge can be expressed as $ n\pi,$ where $ n$ is a positive integer. Find $ n.$

Given a parallelogram $ABCD$ with $AB<BC$, show that the circumcircles of the triangles $APQ$ share a second common point (apart from $A$) as $P,Q$ move on the sides $BC,CD$ respectively s.t. $CP=CQ$.

Let $D$ be the interior of the circle $C$ and let $A \in C$. Show that the function $f : D \to \mathbb R, f(M)=\frac{|MA|}{|MM'|}$ where $M' = AM \cap C$, is strictly convex; i.e., $f(P) <\frac{f(M_1)+f(M_2)}{2}, \forall M_1,M_2 \in D, M_1 \neq M_2$ where $P$ is the midpoint of the segment $M_1M_2.$

Let $a,b$ be positive real numbers and $n\geq 2$ a positive integer. Prove that if $x^n \leq ax+b$ holds for a positive real number $x$, then it also satisfies the inequality $x < \sqrt[n-1]{2a} + \sqrt[n]{2b}.$

Calculate the sum $$\frac{1}{1+2^{-2006}}+...+ \frac{1}{1+2^{-1}}+ \frac{1}{1+2^{0}}+ \frac{1}{1+2^{1}}+...+ \frac{1}{1+2^{2006}}$$

Find all positive integers $n$ such that for all positive integers $m$, $1<m<n$, relatively prime to $n$, $m$ must be a prime number.

The polynomial $1976(x+x^2+ \cdots +x^n)$ is decomposed into a sum of polynomials of the form $a_1x + a_2x^2 + \cdots + a_nx^n$, where $a_1, a_2, \ldots , a_n$ are distinct positive integers not greater than $n$. Find all values of $n$ for which such a decomposition is possible.

Find all positive integers $x$ and $y$ such that $x+y^2+z^3 = xyz$, where $z$ is the greatest common divisor of $x$ and $y$

Define a $\textit{tasty residue}$ of $n$ to be an integer $1<a<n$ such that there exists an integer $m>1$ satisfying \[a^m\equiv a\pmod n.\] Find the number of tasty residues of $2016$.

The cheering team of Ubon Ratchathani University sits on the amphitheater that has $441$ seats arranged into a $21\times 21$ grid. Every seat is occupied by exactly one person, and each person has a blue sign and a yellow sign. Count the number of ways for each person to raise one sign so that each row and column has an odd number of people raising a blue sign.