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 \[ (x+y)(1+xy) = 2^{n} \]

Investigate the behaviour at $t\to\infty$ of the solution of the problem \[u_t+(u\sin x)_x=\epsilon u_{xx},\;u|_{t=0}=1,\;\epsilon\ll1\]

Let $x, y$ be distinct positive integers. Show that the number $$\frac{(x + y)^2}{x^3 + xy^2- x^2y -y^3}$$ is not an integer.

How many ways are there to color a tetrahedron’s faces, edges, and vertices in red, green, and blue so that no face shares a color with any of its edges, and no edge shares a color with any of its endpoints? (Rotations and reflections are considered distinct.) [i]Tiebreaker #2[/i]

Find two positive integers $a,b$ such that $a | b^2, b^2 | a^3, a^3 | b^4, b^4 | a^5$, but $a^5$ does not divide $b^6$

Given right hexagon. Each side is divided onto $1000$ equal segments. All the points of division are connected with the segments, parallel to sides. Let us paint in turn the triples of unpainted nodes of obtained net, if they are vertices of the unilateral triangle, doesn't matter of what size an orientation. Suppose, we have managed to paint all the vertices except one. Prove that the unpainted node is not a hexagon vertex.

Consider the graph in $3$-space of \[0 = xyz(x + y)(y + z)(z + x)(x - y)(y - z)(z - x).\] This graph divides $3$-space into $N$ connected regions. What is $N$?

The point $P$ lies inside $\vartriangle ABC$ so that $\vartriangle BPC$ is isosceles, and angle $P$ is a right angle. Furthermore both $\vartriangle BAN$ and $\vartriangle CAM$ are isosceles with a right angle at $A$, and both are outside $\vartriangle ABC$. Show that $\vartriangle MNP$ is isosceles and right-angled. [img]https://1.bp.blogspot.com/-i9twOChu774/XzcBLP-RIXI/AAAAAAAAMXA/n5TJCOJypeMVW28-9GDG4st5C47yhvTCgCLcBGAsYHQ/s0/2005%2BMohr%2Bp3.png[/img]