A square is cut into several rectangles, none of which is a square, so that the sides of each rectangle are parallel to the sides of the square. For each rectangle with sides $a, b,a<b$, compute the ratio $a/b$. Prove that sum of these ratios is at least $1$.

A quadrilateral is inscribed in a circle. If an angle is inscribed into each of the four segments outside the quadrilateral, the sum of these four angles, expressed in degrees, is: $ \textbf{(A)}\ 1080\qquad \textbf{(B)}\ 900\qquad \textbf{(C)}\ 720\qquad \textbf{(D)}\ 540\qquad \textbf{(E)}\ 360$

What is the maximum of $|z^3 -z+2|$, where $z$ is a complex number with $|z|=1?$

Two functions are defined by equations: $f(a) = a^2 + 3a + 2$ and $g(b, c) = b^2 - b + 3c^2 + 3c$. Prove that for any positive integer $a$ there exist positive integers $b$ and $c$ such that $f(a) = g(b, c)$.

Let $\Gamma$ be a semicircle with center $O$ and diameter $AB$. $D$ is the midpoint of arc $AB$. On the ray $OD$, we take $E$ such that $OE = BD$. $BE$ intersects the semicircle at $F$ and $ P$ is the point on $AB$ such that $FP$ is perpendicular to $AB$. Prove that $BP=\frac13 AB$.

A cube with edge of length $n$ ($n\ge 3$) consists of $n^3$ unit cubes. Prove that it is possible to write different $n^3$ integers on all the unit cubes to provide the zero sum of all integers in the every row parallel to some edge.

$ A$ can do a piece of work in $ 9$ days. $ B$ is $ 50\%$ more efficient than $ A$. The number of days it takes $ B$ to do the same piece of work is: $ \textbf{(A)}\ 13\frac {1}{2} \qquad\textbf{(B)}\ 4\frac {1}{2} \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ \text{none of these answers}$

What is the number $x =\sqrt{4+\sqrt7}-\sqrt{4-\sqrt7}-\sqrt2$, positive, negative or zero?

Two circles $\omega_1$ and $\omega_2$ with centers $O_1$ and $O_2$ respectively intersect each other at points $A$ and $B$, and point $O_1$ lies on $\omega_2$. Let $P$ be an arbitrary point lying on $\omega_1$. Lines $BP, AP$ and $O_1O_2$ cut $\omega_2$ for the second time at points $X$, $Y$ and $C$, respectively. Prove that quadrilateral $XPYC$ is a parallelogram. [i]Proposed by Iman Maghsoudi[/i]

Let $N$ be a positive integer. Consider the sequence $a_1, a_2, ..., a_N$ of positive integers, none of which is a multiple of $2^{N+1}$. For $n \ge N +1$, the number $a_n$ is defined as follows: choose $k$ to be the number among $1, 2, ..., n - 1$ for which the remainder obtained when $a_k$ is divided by $2^n$ is the smallest, and define $a_n = 2a_k$ (if there are more than one such $k$, choose the largest such $k$). Prove that there exist $M$ for which $a_n = a_M$ holds for every $n \ge M$.

A $10\times10\times10$ cube has $1000$ unit cubes. $500$ of them are coloured black and $500$ of them are coloured white. Show that there are at least $100$ unit squares, being the common face of a white and a black unit cube.

Consider triangle $\vartriangle ABC$ with circumradius $R = 10$, inradius $r = 2$ and semi-perimeter $S = 18$. Let $I$ be the incenter, and we extend $AI$, $BI$ and $CI$ to intersect the circumcircle at $D, E$ and $F$ respectively. Compute the area of $\vartriangle DEF$.

[i]Version 1[/i]. Let $n$ be a positive integer, and set $N=2^{n}$. Determine the smallest real number $a_{n}$ such that, for all real $x$, \[ \sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant a_{n}(x-1)^{2}+x . \] [i]Version 2[/i]. For every positive integer $N$, determine the smallest real number $b_{N}$ such that, for all real $x$, \[ \sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant b_{N}(x-1)^{2}+x . \]

Joe is right at the middle of a train tunnel and he realizes that a train is coming. The train travels at a speed of $50$ miles per hour, and Joe can run at a speed of $10$ miles per hour. Joe hears the train whistle when the train is a half mile from the point where it will enter the tunnel. At that point in time, Joe can run toward the train and just exit the tunnel as the train meets him. Instead, Joe runs away from the train when he hears the whistle. How many seconds does he have to spare (before the train is upon him) when he gets to the tunnel entrance? $\textbf{(A) }7.2\hspace{14em}\textbf{(B) }14.4\hspace{14em}\textbf{(C) }36$ $\textbf{(D) }10\hspace{14.3em}\textbf{(E) }12\hspace{14.8em}\textbf{(F) }2.4$ $\textbf{(G) }25.2\hspace{13.5em}\textbf{(H) }123456789$

Let \(p\) a prime number, and \(N\) the number of matrices \(p \times p\) \[\begin{array}{cccc} a_{11} & a_{12} & \ldots & a_{1p}\\ a_{21} & a_{22} & \ldots & a_{2p}\\ \vdots & \vdots & \ddots & \vdots \\ a_{p1} & a_{p2} & \ldots & a_{pp} \end{array}\] such that \(a_{ij} \in \{0,1,2,\ldots,p\} \) and if \(i \leq i^\prime\) and \(j \leq j^\prime\), then \(a_{ij} \leq a_{i^\prime j^\prime}\). Find \(N \pmod{p}\).

Let $a_1$, $a_2, \dots, a_{2015}$ be a sequence of positive integers in $[1,100]$. Call a nonempty contiguous subsequence of this sequence [i]good[/i] if the product of the integers in it leaves a remainder of $1$ when divided by $101$. In other words, it is a pair of integers $(x, y)$ such that $1 \le x \le y \le 2015$ and \[a_xa_{x+1}\dots a_{y-1}a_y \equiv 1 \pmod{101}. \]Find the minimum possible number of good subsequences across all possible $(a_i)$. [i]Proposed by Yang Liu[/i]

Find the number of ordered pairs of integers $(a, b)$ such that $a, b$ are divisors of 720 but $ab$ is not.

Cyclic quadrilateral $ABCD$ satisfies $\angle ADC = 2 \cdot \angle BAD = 80^\circ$ and $\overline{BC} = \overline{CD}$. Let the angle bisector of $\angle BCD$ meet $AD$ at $P$. What is the measure, in degrees, of $\angle BP D$?

There are $n\ge 3$ students in a classroom. Every day, the teacher separates the students into at least two non-empty groups, and each pair of students from the same group will shake hands once. Suppose after $k$ days, each pair of students has shaken hands exactly once, and $k$ is as minimal as possible. Prove that $$\sqrt{n} \le k-1 \le 2\sqrt{n}$$ [i]Proposed by Wong Jer Ren[/i]

Let $\angle ABC=24^\circ$ and $\angle ABD =20^\circ$. What is the smallest possible degree measure for $\angle CBD$? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ 12 $

\[\int_0^{\frac \pi 2} \left(\frac{1}{1-\cos(x)}-\frac{2}{x^2}\right)\,\mathrm dx\] [i]Proposed by Vlad Oleksenko[/i]

There are $ 90$ contestants in a mathematics competition. Each contestant gets acquainted with at least $ 60$ other contestants. One of the contestants, Amin, state that at least four contestants have the same number of new friends. Prove or disprove his statement.

What is the maximum number of coins can be arranged in cells of the table $n \times n$ (each cell is not more of the one coin) so that any coin was not simultaneously below and to the right than any other?

Given is a triangle $ABC$.On the extensions of the side $AB$ we consider points $A_1,B_1$ such that $AB_1=BA_1$ (with $A_1$ lying closer to $B$).On the extensions of the side $BC$ we consider points $B_4,C_4$ such that $CB_4=BC_4$ (with $B_4$ lying closer to $C$).On the extensions of the side $AC$ we consider points $C_1,A_4$ such that $AC_1=CA_4$ (with $C_1$ lying closer to $A$).On the segment $A_1A_4$ we consider points $A_2,A_3$ such that $A_1A_2=A_3A_4=mA_1A_4$ where $0<m<\frac{1}{2}$.Points $B_2,B_3$ and $C_2,C_3$ are defined similarly,on the segments $B_1B_4,C_1C_4$ respectively.If $D\equiv BB_2\cap CC_2 \ , \ E\equiv AA_3\cap CC_2 \ , \ F\equiv AA_3\cap BB_3$, $\ G\equiv BB_3\cap CC_3 \ , \ H\equiv AA_2\cap CC_3$ and $I\equiv AA_2\cap BB_2$,prove that the diagonals $DG,EH,FI$ of the hexagon $DEFGHI$ are concurrent. [hide=Diagram][asy]import graph; size(12cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -7.984603447540051, xmax = 21.28710511372557, ymin = -6.555010307713199, ymax = 10.006614273002825; /* image dimensions */ pen aqaqaq = rgb(0.6274509803921569,0.6274509803921569,0.6274509803921569); pen uququq = rgb(0.25098039215686274,0.25098039215686274,0.25098039215686274); draw((1.1583842866003107,4.638449718549554)--(0.,0.)--(7.,0.)--cycle, aqaqaq); /* draw figures */ draw((1.1583842866003107,4.638449718549554)--(0.,0.), uququq); draw((0.,0.)--(7.,0.), uququq); draw((7.,0.)--(1.1583842866003107,4.638449718549554), uququq); draw((1.1583842866003107,4.638449718549554)--(1.623345080409327,6.500264738079558)); draw((0.,0.)--(-0.46496079380901606,-1.8618150195300045)); draw((-3.0803965232149757,0.)--(0.,0.)); draw((7.,0.)--(10.080396523214976,0.)); draw((1.1583842866003107,4.638449718549554)--(0.007284204967787214,5.552463941947242)); draw((7.,0.)--(8.151100081632526,-0.9140142233976905)); draw((-0.46496079380901606,-1.8618150195300045)--(8.151100081632526,-0.9140142233976905)); draw((-3.0803965232149757,0.)--(0.007284204967787214,5.552463941947242)); draw((10.080396523214976,0.)--(1.623345080409327,6.500264738079558)); draw((0.,0.)--(3.7376079411107392,4.8751985535596685)); draw((-0.7646359770779035,4.164347956460432)--(7.,0.)); draw((1.1583842866003107,4.638449718549554)--(5.997084862772141,-1.150964422430769)); draw((0.,0.)--(7.966133662513563,1.6250661845198895)); draw((-2.308476341169285,1.3881159854868106)--(7.,0.)); draw((1.1583842866003107,4.638449718549554)--(1.6890544250513695,-1.624864820496926)); draw((2.0395968109217,2.660375186246903)--(2.9561195753832448,0.6030390855677443), linetype("2 2")); draw((3.4388364046369224,1.909931693481981)--(1.4816619768719694,0.8229159040072803), linetype("2 2")); draw((1.3969966570225139,1.8221911417546572)--(4.301698851378541,0.8775330211014288), linetype("2 2")); /* dots and labels */ dot((1.1583842866003107,4.638449718549554),linewidth(3.pt) + dotstyle); label("$A$", (0.6263408942608304,4.2), NE * labelscalefactor); dot((0.,0.),linewidth(3.pt) + dotstyle); label("$B$", (-0.44658827292841696,0.04763072114368767), NE * labelscalefactor); dot((7.,0.),linewidth(3.pt) + dotstyle); label("$C$", (7.008893888822507,0.18518574257820614), NE * labelscalefactor); dot((1.623345080409327,6.500264738079558),linewidth(3.pt) + dotstyle); label("$B_1$", (1.7267810657369815,6.6777827542874775), NE * labelscalefactor); dot((-0.46496079380901606,-1.8618150195300045),linewidth(3.pt) + dotstyle); label("$A_1$", (-1.1068523758141076,-1.6305405403574376), NE * labelscalefactor); dot((10.080396523214976,0.),linewidth(3.pt) + dotstyle); label("$B_4$", (10.062615364668826,-0.612633381742001), NE * labelscalefactor); dot((-3.0803965232149757,0.),linewidth(3.pt) + dotstyle); label("$C_4$", (-3.3077327187664096,-0.612633381742001), NE * labelscalefactor); dot((0.007284204967787214,5.552463941947242),linewidth(3.pt) + dotstyle); label("$C_1$", (0.1036318128096586,5.714897604245849), NE * labelscalefactor); dot((8.151100081632526,-0.9140142233976905),linewidth(3.pt) + dotstyle); label("$A_4$", (8.521999124602214,-1.1903644717669786), NE * labelscalefactor); dot((-2.308476341169285,1.3881159854868106),linewidth(3.pt) + dotstyle); label("$C_3$", (-2.9776006673235647,1.7808239912186203), NE * labelscalefactor); dot((-0.7646359770779035,4.164347956460432),linewidth(3.pt) + dotstyle); label("$C_2$", (-1.1618743843879151,4.504413415622086), NE * labelscalefactor); dot((1.6890544250513695,-1.624864820496926),linewidth(3.pt) + dotstyle); label("$A_2$", (1.6167370485893664,-2.125738617521704), NE * labelscalefactor); dot((5.997084862772141,-1.150964422430769),linewidth(3.pt) + dotstyle); label("$A_3$", (6.211074764502297,-1.603029536070534), NE * labelscalefactor); dot((7.966133662513563,1.6250661845198895),linewidth(3.pt) + dotstyle); label("$B_3$", (8.081823056011753,1.7808239912186203), NE * labelscalefactor); dot((3.7376079411107392,4.8751985535596685),linewidth(3.pt) + dotstyle); label("$B_2$", (3.8451283958285725,5.027122497073257), NE * labelscalefactor); dot((2.0395968109217,2.660375186246903),linewidth(3.pt) + dotstyle); label("$D$", (1.7542920700238853,2.991308179842383), NE * labelscalefactor); dot((3.4388364046369224,1.909931693481981),linewidth(3.pt) + dotstyle); label("$E$", (3.542507348672631,2.083445038374561), NE * labelscalefactor); dot((4.301698851378541,0.8775330211014288),linewidth(3.pt) + dotstyle); label("$F$", (4.22,0.93), NE * labelscalefactor); dot((2.9561195753832448,0.6030390855677443),linewidth(3.pt) + dotstyle); label("$G$", (2.909754250073844,0.10265272971749505), NE * labelscalefactor); dot((1.4816619768719694,0.8229159040072803),linewidth(3.pt) + dotstyle); label("$H$", (0.9839839499905795,0.43278478116033936), NE * labelscalefactor); dot((1.3969966570225139,1.8221911417546572),linewidth(3.pt) + dotstyle); label("$I$", (0.9839839499905795,1.8908680083662353), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */[/asy][/hide]

The bisector of the angle $BAC$of the acute triangle $ABC$ ( $AC \ne AB$) intersects its circumscribed circle for the second time at the point $W$. Let $O$ be the center of the circumscribed circle $\Delta ABC$. The line $AW$ intersects for the second time the circumcribed circles of triangles $OWB$ and $OWC$ at the points $N$ and $M$, respectively. Prove that $BN + MC = AW$. (Mitrofanov V., Hilko D.)