Let $P$ and $Q$ be distinct points in the plane of a triangle $ABC$ such that $AP : AQ = BP : BQ = CP : CQ$. Prove that the line $PQ$ passes through the circumcenter of the triangle.

On the sides $ AB,BC,CD,DA $ of the parallelogram $ ABCD, $ consider the points $ M,N,P, $ respectively, $ Q, $ such that $ \overrightarrow{MN} +\overrightarrow{QP} =\overrightarrow{AC} . $ Show that $ \overrightarrow{PN} +\overrightarrow{QM} = \overrightarrow{DB} . $

Let $ABC$ be a triangle. The angle bisectors of $\angle ABC$ and $\angle ACB$ intersect at $D$. If $\angle BAC =80^o$ , what are all possible values for $\angle BDC$ ?

Let $$F(x)=\frac{x^4}{\exp(x^3)}\int^x_0\int^{x-u}_0\exp(u^3+v^3)dvdu.$$Find $\lim_{x\to\infty}F(x)$ or prove that it does not exist.

Prove that $\overline{a0... 09}$ (in which $a > 0$ is a digit and there is at least one zero) is not a perfect square. (VA Senderov)

Solve the following system of equations in real numbers: $\begin{cases} a^2 = \cfrac{\sqrt{bc}\sqrt[3]{bcd}}{(b+c)(b+c+d)} \\ b^2 =\cfrac{\sqrt{cd}\sqrt[3]{cda}}{(c+d)(c+d+a)} \\ c^2 =\cfrac{\sqrt{da}\sqrt[3]{dab}}{(d+a)(d+a+b)} \\ d^2 =\cfrac{\sqrt{ab}\sqrt[3]{abc}}{(a+b)(a+b+c)} \end{cases}$

Given an $n\times n$ grid of dots, let $f(n)$ be the largest number of segments between adjacent dots which can be drawn such that (i) at most one segment is drawn between each pair of dots, and (ii) each dot has $1$ or $3$ segments coming from it. (For example, $f(4)=16$.) Compute $f(2000)$. [i]Proposed by David Stoner[/i]

Find the number of positive integer divisors of $20^{17}$ that are either perfect squares or perfect cubes.

There are no integers $ a,b,c $ that satisfy $ \left( a+b\sqrt{-3}\right)^{17}=c+\sqrt{-3} . $ [i]Dorin Andrica, Mihai Piticari[/i]

Two sequences of integers, $ a_1, a_2, a_3, \ldots$ and $ b_1, b_2, b_3, \ldots$, satisfy the equation \[ (a_n \minus{} a_{n \minus{} 1})(a_n \minus{} a_{n \minus{} 2}) \plus{} (b_n \minus{} b_{n \minus{} 1})(b_n \minus{} b_{n \minus{} 2}) \equal{} 0 \] for each integer $ n$ greater than $ 2$. Prove that there is a positive integer $ k$ such that $ a_k \equal{} a_{k \plus{} 2008}$.

Let $A\in \mathcal{M}_n(\mathbb{R}^*)$. If $A\cdot\ ^t A=I_n$, prove that: a)$|\text{Tr}(A)|\le n$; b)If $n$ is odd, then $\det(A^2-I_n)=0$.

Let $A, B, C, D, E$ points in circle of radius r, in that order, such that $AC = BD = CE = r$. The points $H_1, H_2, H_3$ are the orthocenters of the triangles $ACD$, $BCD$ and $BCE$, respectively. Prove that $H_1H_2H_3$ is a right triangle .

The side length of the largest square below is $8\sqrt{2}$, as shown. Find the area of the shaded region. [asy] size(10cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-18.99425911800572,xmax=23.81538435842469,ymin=-15.51769962526155,ymax=6.464951807764648; pen zzttqq=rgb(0.6,0.2,0.); pair A=(0.,1.), B=(0.,0.), C=(1.,0.), D=(1.,1.), F=(1.,2.), G=(2.,3.), H=(0.,3.), I=(0.,5.), J=(-2.,3.), K=(-4.,5.), L=(-4.,1.), M=(-8.,1.), O=(-8.,-7.), P=(0.,-7.); draw(B--A--D--C--cycle); draw(A--C--(2.,1.)--F--cycle); draw(A--(2.,1.)--G--H--cycle); draw(A--G--I--J--cycle); draw(A--I--K--L--cycle); draw(A--K--M--(-4.,-3.)--cycle); draw(A--M--O--P--cycle); draw(A--O--(0.,-15.)--(8.,-7.)--cycle); filldraw(A--B--C--D--cycle,opacity(0.2)+black); filldraw(A--(2.,1.)--F--cycle,opacity(0.2)+black); filldraw(A--G--H--cycle,opacity(0.2)+black); filldraw(A--I--J--cycle,opacity(0.2)+black); filldraw(A--K--L--cycle,opacity(0.2)+black); filldraw(A--M--(-4.,-3.)--cycle, opacity(0.2)+black); filldraw(A--O--P--cycle,opacity(0.2)+black); draw(B--A); draw(A--D); draw(D--C); draw(C--B); draw(A--C); draw(C--(2.,1.)); draw((2.,1.)--F); draw(F--A); draw(A--(2.,1.)); draw((2.,1.)--G); draw(G--H); draw(H--A); draw(A--G); draw(G--I); draw(I--J); draw(J--A); draw(A--I); draw(I--K); draw(K--L); draw(L--A); draw(A--K); draw(K--M); draw(M--(-4.,-3.)); draw((-4.,-3.)--A); draw(A--M); draw(M--O); draw(O--P); draw(P--A); draw(A--O); draw(O--(0.,-15.)); draw((0.,-15.)--(8.,-7.)); draw((8.,-7.)--A); draw(A--B,black); draw(B--C,black); draw(C--D,black); draw(D--A,black); draw(A--(2.,1.),black); draw((2.,1.)--F,black); draw(F--A,black); draw(A--G,black); draw(G--H,black); draw(H--A,black); draw(A--I,black); draw(I--J,black); draw(J--A,black); draw(A--K,black); draw(K--L,black); draw(L--A,black); draw(A--M,black); draw(M--(-4.,-3.),black); draw((-4.,-3.)--A,black); draw(A--O,black); draw(O--P,black); draw(P--A,black); label("$8\sqrt{2}$",(-8,-7)--(0,-15)); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); [/asy] [i]Lightning 2.4[/i]

It is given $n$ a natural number and a circle with circumference $n$. On the circle, in clockwise direction, numbers $0,1,2,\dots n-1$ are written, in this order and in the same distance to each other. Every number is colored red or blue, and there exists a non-zero number of numbers of each color. It is known that there exists a set $S\subsetneq \{0,1,2,\dots n-1\}, |S|\geq 2$, for wich it holds: if $(x,y), x<y$ is a circle sector whose endpoints are of distinct colors, whose distance $y-x$ is in $S$, then $y$ is in $S$. Prove that there is a divisor $d$ of $n$ different from $1$ and $n$ for wich holds: if $(x,y),x<y$ are different points of distinct colors, such that their distance is divisible by $d$, then both $x,y$ are divisible by $d$.

Find the root that the following three polynomials have in common: \begin{align*} & x^3+41x^2-49x-2009 \\ & x^3 + 5x^2-49x-245 \\ & x^3 + 39x^2 - 117x - 1435\end{align*}

The $8$ eyelets for the lace of a sneaker all lie on a rectangle, four equally spaced on each of the longer sides. The rectangle has a width of $50$ mm and a length of $80$ mm. There is one eyelet at each vertex of the rectangle. The lace itself must pass between the vertex eyelets along a width side of the rectangle and then crisscross between successive eyelets until it reaches the two eyelets at the other width side of the rectrangle as shown. After passing through these final eyelets, each of the ends of the lace must extend at least $200$ mm farther to allow a knot to be tied. Find the minimum length of the lace in millimeters. [asy] size(200); defaultpen(linewidth(0.7)); path laceL=(-20,-30)..tension 0.75 ..(-90,-135)..(-102,-147)..(-152,-150)..tension 2 ..(-155,-140)..(-135,-40)..(-50,-4)..tension 0.8 ..origin; path laceR=reflect((75,0),(75,-240))*laceL; draw(origin--(0,-240)--(150,-240)--(150,0)--cycle,gray); for(int i=0;i<=3;i=i+1) { path circ1=circle((0,-80*i),5),circ2=circle((150,-80*i),5); unfill(circ1); draw(circ1); unfill(circ2); draw(circ2); } draw(laceL--(150,-80)--(0,-160)--(150,-240)--(0,-240)--(150,-160)--(0,-80)--(150,0)^^laceR,linewidth(1));[/asy]