Let $ABC$ be a triangle and let $O$ be its circumcentre. The internal and external bisectrices of the angle $BAC$ meet the line $BC$ at points $D$ and $E$, respectively. Let further $M$ and $L$ respectively denote the midpoints of the segments $BC$ and $DE$. The circles $ABC$ and $ALO$ meet again at point $N$. Show that the angles $BAN$ and $CAM$ are equal.

Find all solutions of the equation $y\partial u/\partial x-\sin x\partial u/\partial y=u^2$ in a neighbourhood of the point $0,0$.

Given positive integers $a,b,c$ such that $a^3$ is divisible by $b$, $b^3$ is divisible by $c$, $c^3$ is divisible by $a$. Prove that $(a+b+c)^{13}$ is divisible by $abc$.

Let the roots $a,b,c$ of $$f(x)=x^3 +p x^2 + qx+r$$ be real, and let $a\leq b\leq c$. Prove that $f'(x)$ has a root in the interval $\left[\frac{b+c}{2}, \frac{b+2c}{3}\right]$. What will be the form of $f(x)$ if the root in question falls at either end of the interval?

A [u]positive sequence[/u] is a finite sequence of positive integers. [u]Sum of a sequence[/u] is the sum of all the elements in the sequence. We say that a sequence $A$ can be [u]embedded[/u] into another sequence $B$, if there exists a strictly increasing function $$\phi : \{1,2, \ldots, |A|\} \rightarrow \{1,2, \ldots, |B|\},$$ such that $\forall i \in \{1, 2, \ldots ,|A|\}$, $$A[i] \leq B[\phi(i)],$$ where $|S|$ denotes the length of a sequence $S$. For example, $(1,1,2)$ can be embedded in $(1,2,3)$, but $(3,2,1)$ can not be in $(1,2,3)$\\ Given a positive integer $n$, construct a positive sequence $U$ with sum $O(n \, \log \, n)$, such that all the positive sequences with sum $n$, can be embedded into $U$.\\

We call a positive integer $n{}$ [i]peculiar[/i] if, for any positive divisor $d{}$ of $n{}$ the integer $d(d + 1)$ divides $n(n + 1).$ Prove that for any four different peculiar positive integers $A, B, C$ and $D{}$ the following holds: \[\gcd(A, B, C, D) = 1.\]

Let $x=\frac{\sqrt{6+2\sqrt5}+\sqrt{6-2\sqrt5}}{\sqrt{20}}$. The value of $$H=(1+x^5-x^7)^{{2012}^{3^{11}}}$$ is (A) $1$ (B) $11$ (C) $21$ (D) $101$ (E) None of the above

Let be two complex numbers $ a,b $ chosen such that $ |a+b|\ge 2 $ and $ |a+b|\ge 1+|ab|. $ Prove that $$ \left| a^{n+1} +b^{n+1} \right|\ge \left| a^{n} +b^{n} \right| , $$ for any natural number $ n. $ [i]Alin Pop[/i]

Let $ABC$ be an acute triangle with incenter $I$; ray $AI$ meets the circumcircle $\Omega$ of $ABC$ at $M \neq A$. Suppose $T$ lies on line $BC$ such that $\angle MIT=90^{\circ}$. Let $K$ be the foot of the altitude from $I$ to $\overline{TM}$. Given that $\sin B = \frac{55}{73}$ and $\sin C = \frac{77}{85}$, and $\frac{BK}{CK} = \frac mn$ in lowest terms, compute $m+n$. [i]Proposed by Evan Chen[/i]

Let $x,y$ be positive real numbers .Find the minimum of $x+y+\frac{|x-1|}{y}+\frac{|y-1|}{x}$.

The graph of $ y \equal{} 2x^2 \plus{} 4x \plus{} 3$ has its: $ \textbf{(A)}\ \text{lowest point at } {(\minus{}1,9)}\qquad \textbf{(B)}\ \text{lowest point at } {(1,1)}\qquad \\ \textbf{(C)}\ \text{lowest point at } {(\minus{}1,1)}\qquad \textbf{(D)}\ \text{highest point at } {(\minus{}1,9)}\qquad \\ \textbf{(E)}\ \text{highest point at } {(\minus{}1,1)}$

Adam has a secret natural number $x$ which Eve is trying to discover. At each stage Eve may only ask questions of the form "is $x+n$ a prime number?" for some natural number $n$ of her choice. Prove that Eve may discover $x$ using finitely many questions.

Let $f(x)$ be a continuous function defined on $0\leq x\leq \pi$ and satisfies $f(0)=1$ and \[\left\{\int_0^{\pi} (\sin x+\cos x)f(x)dx\right\}^2=\pi \int_0^{\pi}\{f(x)\}^2dx.\] Evaluate $\int_0^{\pi} \{f(x)\}^3dx.$

Let the side $AC$ of triangle $ABC$ touch the incircle and the corresponding excircle at points $K$ and $L$ respectively. Let $P$ be the projection of the incenter onto the perpendicular bisector of $AC$. It is known that the tangents to the circumcircle of triangle $BKL$ at $K$ and $L$ meet on the circumcircle of $ABC$. Prove that the lines $AB$ and $BC$ touch the circumcircle of triangle $PKL$.

Points $A, B,$ and $C$ are on a circle such that $\triangle ABC$ is an acute triangle. $X, Y ,$ and $Z$ are on the circle such that $AX$ is perpendicular to $BC$ at $D$, $BY$ is perpendicular to $AC$ at $E$, and $CZ$ is perpendicular to $AB$ at $F$. Find the value of \[ \frac{AX}{AD}+\frac{BY}{BE}+\frac{CZ}{CF}, \] and prove that this value is the same for all possible $A, B, C$ on the circle such that $\triangle ABC$ is acute. [asy] pathpen = linewidth(0.7); pair B = (0,0), C = (10,0), A = (2.5,8); path cir = circumcircle(A,B,C); pair D = foot(A,B,C), E = foot(B,A,C), F = foot(C,A,B), X = IP(D--2*D-A,cir), Y = IP(E--2*E-B,cir), Z = IP(F--2*F-C,cir); D(MP("A",A,N)--MP("B",B,SW)--MP("C",C,SE)--cycle); D(cir); D(A--MP("X",X)); D(B--MP("Y",Y,NE)); D(C--MP("Z",Z,NW)); D(rightanglemark(B,F,C,12)); D(rightanglemark(A,D,B,12)); D(rightanglemark(B,E,C,12));[/asy]

There are $ k$ rooks on a $ 10 \times 10$ chessboard. We mark all the squares that at least one rook can capture (we consider the square where the rook stands as captured by the rook). What is the maximum value of $ k$ so that the following holds for some arrangement of $ k$ rooks: after removing any rook from the chessboard, there is at least one marked square not captured by any of the remaining rooks.

Integer numbers $x, y, z$ satisfy the equation \[ x^3 + y^3 = z^3.\] Prove that at least one of them is divisible by $3$.

Let be given four positive real numbers $ a$, $ b$, $ A$, $ B$. Consider a sequence of real numbers $ x_1$, $ x_2$, $ x_3$, $ \ldots$ is given by $ x_1 \equal{} a$, $ x_2 \equal{} b$ and $ x_{n \plus{} 1} \equal{} A\sqrt [3]{x_n^2} \plus{} B\sqrt [3]{x_{n \minus{} 1}^2}$ ($ n \equal{} 2, 3, 4, \ldots$). Prove that there exist limit $ \lim_{n\to \plus{} \propto}x_n$ and find this limit.

Given an inscribed pentagon $ABCDE$ with circumcircle $\Gamma$. Line $\ell$ passes through vertex $A$ and is tangent to $\Gamma$. Points $X,Y$ lie on $\ell$ so that $A$ lies between $X$ and $Y$. Circumcircle of triangle $XED$ intersects segment $AD$ at $Q$ and circumcircle of triangle $YBC$ intersects segment $AC$ at $P$. Lines $XE,YB$ intersects each other at $S$ and lines $XQ, Y P$ at $Z$. Prove that circumcircle of triangles $XY Z$ and $BES$ are tangent.

Given acute triangle $\vartriangle ABC$ with orthocenter $H$ and circumcenter $O$ ($O \ne H$) . Let $\Gamma$ be the circumcircle of $\vartriangle BOC$ . Segment $OH$ untersects $\Gamma$ at point $P$. Extension of $AO$ intersects $\Gamma$ at point $K$. If $AP \perp OH$, prove that $PK$ bisects $BC$. [img]https://cdn.artofproblemsolving.com/attachments/a/b/267053569c41692f47d8f4faf2a31ebb4f4efd.png[/img]

A mass $m$ is resting at equilibrium suspended from a vertical spring of natural length $L$ and spring constant $k$ inside a box as shown: [asy] //The Spring import graph; size(10cm); guide coil(path g, real width=0.1, real margin = 1*width) { real L = arclength(g); real r = width / 2; pair startpoint = arcpoint(g, margin); real[][] isectiontimes = intersections(g, circle(c=startpoint,r=r)); real initialcirclecentertime = (isectiontimes.length == 1 ? isectiontimes[0][0] : isectiontimes[1][0]); pair startdir = dir(startpoint - point(g,initialcirclecentertime)); real startangle = atan2(startdir.y, startdir.x); real startarctime = arclength(subpath(g, 0, initialcirclecentertime)); write(startarctime); pair endpoint = arcpoint(g, L - margin); real finalcirclecentertime = intersections(g, circle(c=endpoint,r=r))[0][0]; pair enddir = dir(endpoint - point(g,finalcirclecentertime)); real endangle = atan2(enddir.y, enddir.x); real endarctime = arclength(subpath(g, 0, finalcirclecentertime)); write(endarctime); real coillength = 2r; real lengthalongcoils = L - 2*margin; int numcoils = ceil(lengthalongcoils / coillength); real anglesubtended = 2pi * numcoils - startangle + endangle; real angleat(real arctime) { return (arctime - startarctime) * (anglesubtended / (endarctime - startarctime)) + startangle; } pair f(real t) { return arcpoint(g,t) + r * expi(angleat(t)); } return subpath(g, 0, arctime(g, margin)) & graph(f, startarctime, endarctime, n=max(length(g), 20*numcoils+2), operator..) & subpath(g, arctime(g, L-margin), length(g)); } draw(coil((0,0.25)--(0,1))); //Outer Box draw((-1,1)--(1,1),linewidth(2)); draw((-1,1)--(-1,-1.2),linewidth(2)); draw((-1,-1.2)--(1,-1.2),linewidth(2)); draw((1,1)--(1,-1.2),linewidth(2)); //Inner Box draw((-0.2,0.25)--(0.2,0.25),linewidth(2)); path arc1=arc((-0.2,0.15),(-0.2,0.25),(-0.3,0.15)); path arc2=arc((0.2,0.15),(0.3,0.15),(0.2,0.25)); draw(arc1,linewidth(2)); draw(arc2,linewidth(2)); draw((-0.3,0.15)--(-0.3,-0.3),linewidth(2)); draw((0.3,0.15)--(0.3,-0.3),linewidth(2)); path arc3=arc((-0.2,-0.3),(-0.3,-0.3),(-0.2,-0.4)); draw(arc3,linewidth(2)); path arc4=arc((0.2,-0.3),(0.2,-0.4),(0.3,-0.3)); draw((-0.2,-0.4)--(0.2,-0.4),linewidth(2)); draw(arc4,linewidth(2)); [/asy] The box begins accelerating upward with acceleration $a$. How much closer does the equilibrium position of the mass move to the bottom of the box? (a) $(a/g)L$ (b) $(g/a)L$ (c) $m(g + a)/k$ (d) $m(g - a)/k$ (e) $ma/k$

Prove that there is a natural number $n > 1$ such that the product of some $n$ consecutive natural numbers is equal to the product of some $n + 100$ consecutive natural numbers.

Find the function $f(x)$ which satisfies the following integral equation. \[f(x)=\int_0^x t(\sin t-\cos t)dt+\int_0^{\frac{\pi}{2}} e^t f(t)dt\]

Prove that in the set consisting of $\binom{2n}{n}$ people we can find a group of $n+1$ people in which everyone knows everyone or noone knows noone.

Find all triples $(x,y,n)$ of positive integers such that \[ (x+y)(1+xy) = 2^{n} \]