On square $ABCD,$ points $E,F,G,$ and $H$ lie on sides $\overline{AB},\overline{BC},\overline{CD},$ and $\overline{DA},$ respectively, so that $\overline{EG} \perp \overline{FH}$ and $EG=FH = 34.$ Segments $\overline{EG}$ and $\overline{FH}$ intersect at a point $P,$ and the areas of the quadrilaterals $AEPH, BFPE, CGPF,$ and $DHPG$ are in the ratio $269:275:405:411.$ Find the area of square $ABCD$. [asy] size(200); defaultpen(linewidth(0.8)+fontsize(10.6)); pair A = (0,sqrt(850)); pair B = (0,0); pair C = (sqrt(850),0); pair D = (sqrt(850),sqrt(850)); draw(A--B--C--D--cycle); dotfactor = 3; dot("$A$",A,dir(135)); dot("$B$",B,dir(215)); dot("$C$",C,dir(305)); dot("$D$",D,dir(45)); pair H = ((2sqrt(850)-sqrt(120))/6,sqrt(850)); pair F = ((2sqrt(850)+sqrt(306)+7)/6,0); dot("$H$",H,dir(90)); dot("$F$",F,dir(270)); draw(H--F); pair E = (0,(sqrt(850)-6)/2); pair G = (sqrt(850),(sqrt(850)+sqrt(100))/2); dot("$E$",E,dir(180)); dot("$G$",G,dir(0)); draw(E--G); pair P = extension(H,F,E,G); dot("$P$",P,dir(60)); label("$w$", (H+E)/2,fontsize(15)); label("$x$", (E+F)/2,fontsize(15)); label("$y$", (G+F)/2,fontsize(15)); label("$z$", (H+G)/2,fontsize(15)); label("$w:x:y:z=269:275:405:411$",(sqrt(850)/2,-4.5),fontsize(11)); [/asy]

On the plane are three straight lines $\ell_1, \ell_2,\ell_3$, forming a triangle, and the point $O$ is marked, the center of the circumscribed circle of this triangle. For an arbitrary point X of the plane, we denote by $X_i$ the point symmetric to the point X with respect to the line $\ell_i, i = 1,2,3$. a) Prove that for an arbitrary point $M$ the straight lines connecting the midpoints of the segments $O_1O_2$ and $M_1M_2, O_2O_3$ and $M_2M_3, O_3O_1$ and $M_3M_1$ intersect at one point, b) where can this intersection point lie?

Consider an arc of a planar curve; let the radius of curvature at any point of the arc be a differentiable function of the arc length and its derivative be everywhere different from zero; moreover, let the total curvature be less than $ \frac{\pi}{2}$. Let $ P_1,P_2,P_3,P_4,P_5$ and $ P_6$ be any points on this arc, subject to the only condition that the radius of curvature at $ P_k$ is greater than at $ P_j$ if $ j<k$. Prove that the radius of the circle passing through the points $ P_1,P_3$ and $ P_5$ is less than the radius of the circle through $ P_2,P_4$ and $ P_6$

Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$, such that $$f(xy+f(x^2))=xf(x+y)$$ for all reals $x, y$.

Points $A', B', C'$ lie on sides $BC, CA, AB$ of triangle $ABC.$ for a point $X$ one has $\angle AXB =\angle A'C'B' + \angle ACB$ and $\angle BXC = \angle B'A'C' +\angle BAC.$ Prove that the quadrilateral $XA'BC'$ is cyclic.

In a building there are 6018 desks in 2007 rooms, and in every room there is at least one desk. Every room can be cleared dividing the desks in the oher rooms such that in every room is the same number of desks. Find out what methods can be used for dividing the desks initially.

Which of the following triangles cannot exist? $\textbf{(A)}\ \text{An acute isosceles triangle}$ $\textbf{(B)}\ \text{An isosceles right triangle}$ $\textbf{(C)}\ \text{An obtuse right triangle}$ $\textbf{(D)}\ \text{A scalene right triangle}$ $\textbf{(E)}\ \text{A scalene obtuse triangle}$

Prove that there doesn't exist any positive integer $n$ such that $2n^2+1,3n^2+1$ and $6n^2+1$ are perfect squares.

Let $a$ be a real number greater than $1$ such that $\frac{20a}{a^2+1} = \sqrt{2}$. Find $\frac{14a}{a^2 - 1}$.

For $n$ an odd positive integer, the unit squares of an $n\times n$ chessboard are coloured alternately black and white, with the four corners coloured black. A [i]tromino[/i] is an $L$-shape formed by three connected unit squares. $(a)$ For which values of $n$ is it possible to cover all the black squares with non-overlapping trominoes lying entirely on the chessboard? $(b)$ When it is possible, find the minimum number of trominoes needed.

Let $a_1>0$ and for $n \ge 1$ define \[a_{n+1}=a_n+\frac{1}{a_1+a_2+\dots+a_n}.\] Prove that \[\lim_{n \to \infty} \frac{a_n^2}{\ln n}=2.\]

An equilateral triangle is drawn with a side of length $ a$. A new equilateral triangle is formed by joining the midpoints of the sides of the first one. Then a third equilateral triangle is formed by joining the midpoints of the sides of the second; and so on forever. The limit of the sum of the perimeters of all the triangles thus drawn is: $ \textbf{(A)}\ \text{Infinite} \qquad\textbf{(B)}\ 5\frac {1}{4}a \qquad\textbf{(C)}\ 2a \qquad\textbf{(D)}\ 6a \qquad\textbf{(E)}\ 4\frac {1}{2}a$

Given an isosceles triangle $ABC$ with a vertex at the point $B$. Based on $AC$, an arbitrary point $D $ is selected, different from the vertices $A$ and $C $. On the line $AC $ select the point $E $ outside the segment $AC$, for which $AE = CD$. Prove that the perimeter $\Delta BDE$ is larger than the perimeter $\Delta ABC$.

Prove that there exists a positive integer $k$ such that $k\cdot2^n+1$ is composite for every integer $n$.

Natural number $N$ is given. Let $p_N$ - biggest prime, that $ \leq N$. On every move we replace $N$ by $N-p_N$. We repeat this until we get $0$ or $1$. If we get $1$ then $N$ is called as good, else is bad. For example, $95$ is good because we get $95 \to 6 \to 1$. Prove that among numbers from $1$ to $1000000$ there are between one quarter and half good numbers

Let $n\not\equiv 2\pmod{3}$. Is $\sqrt{\lfloor n+\tfrac {2n}{3}\rfloor+7},\forall n \in \mathbb {N}$, a natural number?

Numbers $0, 1$ and $2$ are placed in a table $2005 \times 2006$ so that total sums of the numbers in each row and in each column are factors of $3$. Find the maximal possible number of $1$'s that can be placed in the table. [i](6 points)[/i]

We say that a positive integer $n$ is [i]nice[/i] if we can split the numbers $1,2,\ldots,n$ into three sets, so that the sum of the numbers in each set is the same. For example, the number $12$ is nice because we can divide the numbers $1,2,\ldots,12$ into the sets $\left\{1,2,4,5,6,8\right\}$, $\left\{7,9,10\right\}$, and $\left\{3,11,12\right\}$, and the sum of the numbers in each set is $26$. Find all nice positive integers.

Let $x_1$ and $x_2$ be the roots of the equation $x^2 + 3x + 1 = 0$. Compute \[\left(\frac{x_1}{x_2 + 1}\right)^2 + \left(\frac{x_2}{x_1 + 1}\right)^2\]

There are real numbers $a$ and $b$ such that for every positive number $x$, we have the identity \[ \tan^{-1} \bigl( \frac{1}{x} - \frac{x}{8} \bigr) + \tan^{-1}(ax) + \tan^{-1}(bx) = \frac{\pi}{2} \, . \] (Throughout this equation, $\tan^{-1}$ means the inverse tangent function, sometimes written $\arctan$.) What is the value of $a^2 + b^2$?

Let $ABCDE$ be a convex pentagon such that $\angle ABC = \angle AED = 90^\circ$. Suppose that the midpoint of $CD$ is the circumcenter of triangle $ABE$. Let $O$ be the circumcenter of triangle $ACD$. Prove that line $AO$ passes through the midpoint of segment $BE$.

Let $ABC$ be a triangle and $H$ and $D$ be the feet of the height and bisector relative to $A$ in $BC$, respectively. Let $E$ be the intersection of the tangent to the circumcircle of $ABC$ by $A$ with $BC$ and $M$ be the midpoint of $AD$. Finally, let $r$ be the line perpendicular to $BC$ that passes through $M$. Show that $r$ is tangent to the circumcircle of $AHE$.

Calculate: [b]a)[/b] $ \int \frac{1-x^2-x^6+x^8}{1+x^{10}} dx $ [b]b)[/b] $ \int\frac{x^{n-1}+x^{5n-1}}{1+x^{6n}} dx $ [i]Dumitru Acu[/i]

The inscribed circle inside triangle $ABC$ intersects side $AB$ in $D$. The inscribed circle inside triangle $ADC$ intersects sides $AD$ in $P$ and $AC$ in $Q$.The inscribed circle inside triangle $BDC$ intersects sides $BC$ in $M$ and $BD$ in $N$. Prove that $P , Q, M, N$ are cyclic.

For positive integers $n,d$, define $n \% d$ to be the unique value of the positive integer $r < d$ such that $n = qd + r$, for some positive integer $q$. What is the smallest value of $n$ not divisible by $5,7,11,13$ for which $n^2 \% 5 < n^2 \% 7 < n^2 \% 11 < n^2 \% 13$?