Draw a $2004 \times 2004$ array of points. What is the largest integer $n$ for which it is possible to draw a convex $n$-gon whose vertices are chosen from the points in the array?

Let $ABC$ be an acute,non-isosceles triangle with $AB<AC<BC$.Let $D,E,Z$ be the midpoints of $BC,AC,AB$ respectively and segments $BK,CL$ are altitudes.In the extension of $DZ$ we take a point $M$ such that the parallel from $M$ to $KL$ crosses the extensions of $CA,BA,DE$ at $S,T,N$ respectively (we extend $CA$ to $A$-side and $BA$ to $A$-side and $DE$ to $E$-side).If the circumcirle $(c_{1})$ of $\triangle{MBD}$ crosses the line $DN$ at $R$ and the circumcirle $(c_{2})$ of $\triangle{NCD}$ crosses the line $DM$ at $P$ prove that $ST\parallel PR$.

On the side $AC$ of the triangle $ABC$, choose any point $D$ different from the vertices $A$ and C. Let $O_1$ and $O_2$ be circumcenters the triangles $ABD$ and $CBD$, respectively. Prove that the triangles $O_1DO_2$ and $ABC$ are similar.

A planar section of a parallelepiped is a regular hexagon. Show that this parallelepiped is a cube.

Find the number of finite sequences $ \{a_1,a_2,\ldots,a_{2n\plus{}1}\}$, formed with nonnegative integers, for which $ a_1\equal{}a_{2n\plus{}1}\equal{}0$ and $ |a_k \minus{}a_{k\plus{}1}|\equal{}1$, for all $ k\in\{1,2,\ldots,2n\}$.

You are given a convex quadrilateral $ABCD$. It is known that $\angle CAD = \angle DBA = 40^o$, $\angle CAB = 60^o$, $\angle CBD = 20^o$. Find the angle $\angle CDB $.

In a convex quadrilateral $ABCD$, diagonals $AC$ and $BD$ are equal, and they intersect at $E$. Perpendicular bisectors of $AB$ and $CD$ intersect at point $P$ lying inside triangle $AED$, and perpendicular bisectors of $BC$ and $DA$ intersect at point $Q$ lying inside triangle $CED$. Prove that $\angle PEQ = 90^\circ$.

There are two circles of perimeter $1979$. Let $1979$ points be marked on the first circle, and several arcs with the total length of $1$ on the second. Show that it is possible to place the second circle onto the first in such a way that none of the marked points is covered by a marked arc.

Let $ABC$ be a triangle such that $BC=AC+\frac{1}{2}AB$. Let $P$ be a point of $AB$ such that $AP=3PB$. Show that $\widehat{PAC} = 2 \widehat{CPA}.$

[b]p4.[/b] For how many three-digit multiples of $11$ in the form $\underline{abc}$ does the quadratic $ax^2 + bx + c$ have real roots? [b]p5.[/b] William draws a triangle $\vartriangle ABC$ with $AB =\sqrt3$, $BC = 1$, and $AC = 2$ on a piece of paper and cuts out $\vartriangle ABC$. Let the angle bisector of $\angle ABC$ meet $AC$ at point $D$. He folds $\vartriangle ABD$ over $BD$. Denote the new location of point $A$ as $A'$. After William folds $\vartriangle A'CD$ over $CD$, what area of the resulting figure is covered by three layers of paper? [b]p6.[/b] Compute $(1)(2)(3) + (2)(3)(4) + ... + (18)(19)(20)$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $E$ be a point on the side $AD$ of the square $ABCD$. Find such points $M$ and $K$ on the sides $AB$ and $BC$ respectively, such that the segments $MK$ and $EC$ are parallel, and the quadrilateral $MKCE$ has the largest area.

Given an acute-angled triangle $ABC$. Let $CB$ be the altitude and $E$ a random point on the line $CD$. Finally, let $P, Q, R$ and $S$ are the projections of $D$ on the straight lines $AC, AE, BE$ and $BC$. Prove that the points $P, Q, R$ and $S$ lie either on a circle or on one straight line.

We call a set $ S$ on the real line $ \mathbb{R}$ [i]superinvariant[/i] if for any stretching $ A$ of the set by the transformation taking $ x$ to $ A(x) \equal{} x_0 \plus{} a(x \minus{} x_0), a > 0$ there exists a translation $ B,$ $ B(x) \equal{} x\plus{}b,$ such that the images of $ S$ under $ A$ and $ B$ agree; i.e., for any $ x \in S$ there is a $ y \in S$ such that $ A(x) \equal{} B(y)$ and for any $ t \in S$ there is a $ u \in S$ such that $ B(t) \equal{} A(u).$ Determine all [i]superinvariant[/i] sets.

Let $ABC$ be a triangle. For any point $X$ on $BC$, let $AX$ meet the circumcircle of $ABC$ in $X'$. Prove or disprove: $XX'$ has maximum length if and only if $AX$ lies between the median and the internal angle bisector from $A$.

Find the area of the part, in the $x$-$y$ plane, enclosed by the curve $|ye^{2x}-6e^{x}-8|=-(e^{x}-2)(e^{x}-4).$ [i]2010 Tokyo University of Agriculture and Technology entrance exam[/i]

Find the area of the region in the $xy$-plane satisfying $x^6-x^2+y^2 \le 0$.

The perimeter of a rectangle is $100$ and its diagonal has length $x$. What is the area of this rectangle? $\textbf{(A) }625-x^2\qquad\textbf{(B) }625-\dfrac{x^2}2\qquad\textbf{(C) }1250-x^2\qquad\textbf{(D) }1250-\dfrac{x^2}2\qquad\textbf{(E) }2500-\dfrac{x^2}2$

Let $CH$ be the altitude of a right-angled triangle $ABC$ ($\angle C = 90^{\circ}$) with $BC = 2AC$. Let $O_1$, $O_2$ and $O$ be the incenters of triangles $ACH$, $BCH$ and $ABC$ respectively, and $H_1$, $H_2$, $H_0$ be the projections of $O_1$, $O_2$, $O$ respectively to $AB$. Prove that $H_1H = HH_0 = H_0H_2$.

The median, bisector, and height, all originate at the same vertex of a triangle. Given the intersection points of the median, bisector, and height with the circumscribed circle, construct the triangle.

Circle is divided into $n$ arcs by $n$ marked points on the circle. After that circle rotate an angle $ 2\pi k/n $ (for some positive integer $ k $), marked points moved to $n$ [i] new points [/i], dividing the circle into $ n $ [i] new arcs[/i]. Prove that there is a new arc that lies entirely in the one of the old arсs. (It is believed that the endpoints of arcs belong to it.) [i]I. Mitrophanov[/i]

Let ABCD be a square and K be a circle with diameter AB. For an arbitrary point P on side CD, segments AP and BP meet K again at points M and N, respectively, and lines DM and CN meet at point Q. Prove that Q lies on the circle K and that AQ : QB = DP : PC.

[color=darkred]Find all functions $f:\mathbb{R}\to\mathbb{R}$ with the following property: for any open bounded interval $I$, the set $f(I)$ is an open interval having the same length with $I$ .[/color]

Circle $k$ and its diameter $AB$ are given. Find the locus of the centers of circles inscribed in the triangles having one vertex on $AB$ and two other vertices on $k.$

Two squares of side length $2$ are glued together along their boundary so that the four vertices of the first square are glued to the midpoints of the four sides of the other square, and vice versa. This gluing results in a convex polyhedron. If the square of the volume of this polyhedron is written in simplest form as $\tfrac{a+b\sqrt c}d$, what is the value of $a+b+c+d$?

Prove that for any integer $n$ greater than $5$, a square can be divided into $n$ squares.