Let $\ell$ be the perimeter of an acute-angled triangle $ABC$ which is not an equilateral triangle. Let $P$ be a variable points inside the triangle $ABC$, and let $D,E,F$ be the projections of $P$ on the sides $BC,CA,AB$ respectively. Prove that \[ 2(AF+BD+CE ) = \ell \] if and only if $P$ is collinear with the incenter and the circumcenter of the triangle $ABC$.

There are lamps in every field of $n\times n$ table. At start all the lamps are off. A move consists of chosing $m$ consecutive fields in a row or a column and changing the status of that $m$ lamps. Prove that you can reach a state in which all the lamps are on only if $m$ divides $n.$

We say a triangle in the coordinate plane is [i]integral[/i] if its three vertices have integer coordinates and if its three sides have integer lengths. (a) Find an integral triangle with perimeter of $42$. (b) Is there an integral triangle with perimeter of $43$?

Each grid point of a cartesian plane is colored with one of three colors, whereby all three colors are used. Show that one can always find a right-angled triangle, whose three vertices have pairwise different colors.

Let $E$ be a point outside the square $ABCD$ such that $m(\widehat{BEC})=90^{\circ}$, $F\in [CE]$, $[AF]\perp [CE]$, $|AB|=25$, and $|BE|=7$. What is $|AF|$? $ \textbf{(A)}\ 29 \qquad\textbf{(B)}\ 30 \qquad\textbf{(C)}\ 31 \qquad\textbf{(D)}\ 32 \qquad\textbf{(E)}\ 33 $

In a $ 19\times 19$ board, a piece called [i]dragon[/i] moves as follows: It travels by four squares (either horizontally or vertically) and then it moves one square more in a direction perpendicular to its previous direction. It is known that, moving so, a dragon can reach every square of the board. The [i]draconian distance[/i] between two squares is defined as the least number of moves a dragon needs to move from one square to the other. Let $ C$ be a corner square, and $ V$ the square neighbor of $ C$ that has only a point in common with $ C$. Show that there exists a square $ X$ of the board, such that the draconian distance between $ C$ and $ X$ is greater than the draconian distance between $ C$ and $ V$.

Let us consider $a,b$ two integers. Prove that there exists and it is unique a pair of integers $(x,y)$ such that: \[(x+2y-a)^{2}+(2x-y-b)^{2}\leq 1.\]

On rectangular coordinates, point $A = (1,2)$, $B = (3,4)$. $P = (a, 0)$ is on $x$-axis. Given that $P$ is chosen such that $AP + PB$ is minimized, compute $60a$.

On the graph of a polynomial with integral coefficients are two points with integral coordinates. Prove that if the distance between these two points is integral, then the segment connecting them is parallel to the $x$-axis. [i](4 points)[/i]

Two points $A$ and $B$ are given in the plane. A point $X$ is called their [i]preposterous midpoint[/i] if there is a Cartesian coordinate system in the plane such that the coordinates of $A$ and $B$ in this system are non-negative, the abscissa of $X$ is the geometric mean of the abscissae of $A$ and $B$, and the ordinate of $X$ is the geometric mean of the ordinates of $A$ and $B$. Find the locus of all the [i]preposterous midpoints[/i] of $A$ and $B$. (K. Tyschu)

The equation $x^3+px+q=0$ has three distinct real roots. Show that $p<0$

For a function $f(x)\ (x\geq 1)$ satisfying $f(x)=(\log_e x)^2-\int_1^e \frac{f(t)}{t}dt$, answer the questions as below. (a) Find $f(x)$ and the $y$-coordinate of the inflection point of the curve $y=f(x)$. (b) Find the area of the figure bounded by the tangent line of $y=f(x)$ at the point $(e,\ f(e))$, the curve $y=f(x)$ and the line $x=1$.

Let $\gamma$ be the incircle of $\triangle ABC$ (i.e. the circle inscribed in $\triangle ABC$) for which $AB+AC=3BC$. Let the point where $AC$ is tangent to $\gamma$ be $D$. Let the incenter of $I$. Let the intersection of the circumcircle of $\triangle BCI$ with $\gamma$ that is closer to $B$ be $P$. Show that $PID$ is collinear.

Let $\omega$ be the unit circle centered at the origin of $R^2$. Determine the largest possible value for the radius of the circle inscribed to the triangle $OAP$ where $ P$ lies the circle and $A$ is the projection of $P$ on the axis $OX$.

here is the most difficult and the most beautiful problem occurs in 21th iranian (2003) olympiad assume that P is n-gon ,lying on the plane ,we name its edge 1,2,..,n. if S=s1,s2,s3,.... be a finite or infinite sequence such that for each i, si is in {1,2,...,n}, we move P on the plane according to the S in this form: at first we reflect P through the s1 ( s1 means the edge which iys number is s1)then through s2 and so on like the figure below. a)show that there exist the infinite sequence S sucth that if we move P according to S we cover all the plane b)prove that the sequence in a) isn't periodic. c)assume that P is regular pentagon ,which the radius of its circumcircle is 1,and D is circle ,with radius 1.00001 ,arbitrarily in the plane .does exist a sequence S such that we move P according to S then P reside in D completely?

Square $A$ is adjacent to square $B$ which is adjacent to square $C$. The three squares all have their bottom sides along a common horizontal line. The upper left vertices of the three squares are collinear. If square $A$ has area $24$, and square $B$ has area $36$, find the area of square $C$. [asy] import graph; size(8cm); real labelscalefactor = 0.5; pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); real xmin = -4.89, xmax = 13.61, ymin = -1.39, ymax = 9; draw((0,0)--(2,0)--(2,2)--(0,2)--cycle, linewidth(1.2)); draw((2,0)--(5,0)--(5,3)--(2,3)--cycle, linewidth(1.2)); draw((5,4.5)--(5,0)--(9.5,0)--(9.5,4.5)--cycle, linewidth(1.2)); draw((2,0)--(2,2), linewidth(1.2)); draw((2,2)--(0,2), linewidth(1.2)); draw((0,2)--(0,0), linewidth(1.2)); draw((2,0)--(5,0), linewidth(1.2)); draw((5,0)--(5,3), linewidth(1.2)); draw((5,3)--(2,3), linewidth(1.2)); draw((2,3)--(2,0), linewidth(1.2)); draw((5,4.5)--(5,0), linewidth(1.2)); draw((5,0)--(9.5,0), linewidth(1.2)); draw((9.5,0)--(9.5,4.5), linewidth(1.2)); draw((9.5,4.5)--(5,4.5), linewidth(1.2)); label("A",(0.6,1.4),SE*labelscalefactor); label("B",(3.1,1.76),SE*labelscalefactor); label("C",(6.9,2.5),SE*labelscalefactor); draw((13.13,8.56)--(-3.98,0), linewidth(1.2)); draw((-3.98,0)--(15.97,0), linewidth(1.2));[/asy]

Square $PQRS$ lies in the first quadrant. Points $(3,0), (5,0), (7,0),$ and $(13,0)$ lie on lines $SP, RQ, PQ$, and $SR$, respectively. What is the sum of the coordinates of the center of the square $PQRS$? $ \textbf{(A)}\ 6\qquad\textbf{(B)}\ 6.2\qquad\textbf{(C)}\ 6.4\qquad\textbf{(D)}\ 6.6\qquad\textbf{(E)}\ 6.8 $

The numbers $S_1=2^2, S_2=2^4,\ldots, S_n=2^{2n}$ are given. A rectangle $OABC$ is constructed on the Cartesian plane according to these numbers. For this, starting from the point $O$ the points $A_1,A_2,\ldots,A_n$ are consistently marked along the axis $Ox$, and the points $C_1,C_2,\ldots,C_n$ are consistently marked along the axis $Oy$ in such a way that for all $k$ from $1$ to $n$ the lengths of the segments $A_{k-1}A_k=x_k$ and $C_{k-1}C_k=y_k$ are positive integers (let $A_0=C_0=O$, $A_n=A$, and $C_n=C$) and $x_k\cdot y_k=S_k$. [b]a)[/b] Find the maximal possible value of the area of the rctangle $OABC$ and all pairs of sets $(x_1,x_2,\ldots,x_k)$ and $(y_1,y_2,\ldots,y_k)$ at which this maximal area is achieved. [b]b)[/b] Find the minimal possible value of the area of the rctangle $OABC$ and all pairs of sets $(x_1,x_2,\ldots,x_k)$ and $(y_1,y_2,\ldots,y_k)$ at which this minimal area is achieved. [i](E. Manzhulina, B. Rublyov)[/i]

The circle $x^2+ y^2 = r^2$ meets the coordinate axis at $A = (r,0), B = (-r,0), C = (0,r)$ and $D = (0,-r).$ Let $P = (u,v)$ and $Q = (-u,v)$ be two points on the circumference of the circle. Let $N$ be the point of intersection of $PQ$ and the $y$-axis, and $M$ be the foot of the perpendicular drawn from $P$ to the $x$-axis. If $r^2$ is odd, $u = p^m > q^n = v,$ where $p$ and $q$ are prime numbers and $m$ and $n$ are natural numbers, show that \[ |AM| = 1, |BM| = 9, |DN| = 8, |PQ| = 8. \]

a) The incircle of a triangle $ABC$ touches $AC$ and $AB$ at points $B_0$ and $C_0$ respectively. The bisectors of angles $B$ and $C$ meet the perpendicular bisector to the bisector $AL$ in points $Q$ and $P$ respectively. Prove that the lines $PC_0, QB_0$ and $BC$ concur. b) Let $AL$ be the bisector of a triangle $ABC$. Points $O_1$ and $O_2$ are the circumcenters of triangles $ABL$ and $ACL$ respectively. Points $B_1$ and $C_1$ are the projections of $C$ and $B$ to the bisectors of angles $B$ and $C$ respectively. Prove that the lines $O_1C_1, O_2B_1,$ and $BC$ concur. c) Prove that the two points obtained in pp. a) and b) coincide.

A grasshopper is jumping about in a grid. From the point with coordinates $(a, b)$ it can jump to either $(a + 1, b),(a + 2, b),(a + 1, b + 1),(a, b + 2)$ or $(a, b + 1)$. In how many ways can it reach the line $x + y = 2014?$ Where the grasshopper starts in $(0, 0)$.

Let $ABC$ be a triangle with vertices at lattice points. Suppose one of its sides in $\sqrt{n}$, where $n$ is square-free. Prove that $\frac{R}{r}$ is irraational . The symbols have usual meanings.

Two lines parallel to the $x$-axis cut the graph of $y=ax^3+bx^2+cx+d$ in points $A,C,E$ and $B,D,F$ respectively, in that order from left to right. Prove that the length of the projection of the segment $CD$ onto the $x$-axis equals the sum of the lengths of the projections of $AB$ and $EF$.

The pentagon $ABCDE$ is inscirbed in a circle $w$. Suppose that $w_a,w_b,w_c,w_d,w_e$ are reflections of $w$ with respect to sides $AB,BC,CD,DE,EA$ respectively. Let $A'$ be the second intersection point of $w_a,w_e$ and define $B',C',D',E'$ similarly. Prove that \[2\le \frac{S_{A'B'C'D'E'}}{S_{ABCDE}}\le 3,\] where $S_X$ denotes the surface of figure $X$. [i]Proposed by Morteza Saghafian, Ali khezeli[/i]

Let $n \geq 2$ be a natural number, and let $\left( a_{1};\;a_{2};\;...;\;a_{n}\right)$ be a permutation of $\left(1;\;2;\;...;\;n\right)$. For any integer $k$ with $1 \leq k \leq n$, we place $a_k$ raisins on the position $k$ of the real number axis. [The real number axis is the $x$-axis of a Cartesian coordinate system.] Now, we place three children A, B, C on the positions $x_A$, $x_B$, $x_C$, each of the numbers $x_A$, $x_B$, $x_C$ being an element of $\left\{1;\;2;\;...;\;n\right\}$. [It is not forbidden to place different children on the same place!] For any $k$, the $a_k$ raisins placed on the position $k$ are equally handed out to those children whose positions are next to $k$. [So, if there is only one child lying next to $k$, then he gets the raisin. If there are two children lying next to $k$ (either both on the same position or symmetric with respect to $k$), then each of them gets one half of the raisin. Etc..] After all raisins are distributed, a child is unhappy if he could have received more raisins than he actually has received if he had moved to another place (while the other children would rest on their places). For which $n$ does there exist a configuration $\left( a_{1};\;a_{2};\;...;\;a_{n}\right)$ and numbers $x_A$, $x_B$, $x_C$, such that all three children are happy?