Let $G$ be a connected (simple) graph with $n$ vertices and at least $n$ edges. Prove that it is possible to color the vertices of $G$ red and blue, so that the following conditions hold: i. There is at least one vertex of each color, ii. There is an even number of edges connecting a red vertex to a blue vertex, and iii. If all such edges are deleted, one is left with two connected graphs.

Finding all quads of integers $(a, b, c, p)$ where $p \ge 5$ is prime number such that the remainders of the numbers $am^3 + bm^2 + cm$, $m = 0, 1, . . . , p - 1$, upon division of $p$ are two by two different..

Mojtaba and Hooman are playing a game. Initially Mojtaba draws $2018$ vectors with zero sum. Then in each turn, starting with Mojtaba, the player takes a vector and puts it on the plane. After the first move, the players must put their vector next to the previous vector (the beginning of the vector must lie on the end of the previous vector). At last, there will be a closed polygon. If this polygon is not self-intersecting, Mojtaba wins. Otherwise Hooman. Who has the winning strategy? [i]Proposed by Mahyar Sefidgaran, Jafar Namdar [/i]

In a triangle $ABC$, points $P, Q$ and $ R$ distinct from the vertices of the triangle are chosen on sides $AB, BC$ and $CA$, respectively. The circumcircles of the triangles $APR$, $BPQ$, and $CQR$ are drawn. Prove that the centers of these circles are the vertices of a triangle similar to triangle $ABC$.

If the value of the infinite sum $$\frac{1}{2^2-1^2}+\frac{1}{4^2-2^2}+\frac{1}{8^2-4^2}+\frac{1}{16^2-8^2}+\dots.$$ can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a,b,$ evaluate $a+b.$ [i]Proposed by Alex Li[/i]

Find the largest prime that divides $1\cdot 2\cdot 3+2\cdot 3\cdot 4+\cdots +44\cdot 45\cdot 46$

Nine quadratics, $x^2+a_1x+b_1, x^2+a_2x+b_2,...,x^2+a_9x+b_9$ are written on the board. The sequences $a_1, a_2,...,a_9$ and $b_1, b_2,...,b_9$ are arithmetic. The sum of all nine quadratics has at least one real root. What is the the greatest possible number of original quadratics that can have no real roots?

Which pattern of identical squares could NOT be folded along the lines shown to form a cube? [asy] unitsize(12); draw((0,0)--(0,-1)--(1,-1)--(1,-2)--(2,-2)--(2,-3)--(4,-3)--(4,-2)--(3,-2)--(3,-1)--(2,-1)--(2,0)--cycle); draw((1,0)--(1,-1)--(2,-1)--(2,-2)--(3,-2)--(3,-3)); draw((7,0)--(8,0)--(8,-1)--(11,-1)--(11,-2)--(8,-2)--(8,-3)--(7,-3)--cycle); draw((7,-1)--(8,-1)--(8,-2)--(7,-2)); draw((9,-1)--(9,-2)); draw((10,-1)--(10,-2)); draw((14,-1)--(15,-1)--(15,0)--(16,0)--(16,-1)--(18,-1)--(18,-2)--(17,-2)--(17,-3)--(16,-3)--(16,-2)--(14,-2)--cycle); draw((15,-2)--(15,-1)--(16,-1)--(16,-2)--(17,-2)--(17,-1)); draw((21,-1)--(22,-1)--(22,0)--(23,0)--(23,-2)--(25,-2)--(25,-3)--(22,-3)--(22,-2)--(21,-2)--cycle); draw((23,-1)--(22,-1)--(22,-2)--(23,-2)--(23,-3)); draw((24,-2)--(24,-3)); draw((28,-1)--(31,-1)--(31,0)--(32,0)--(32,-2)--(31,-2)--(31,-3)--(30,-3)--(30,-2)--(28,-2)--cycle); draw((32,-1)--(31,-1)--(31,-2)--(30,-2)--(30,-1)); draw((29,-1)--(29,-2)); label("(A)",(0,-0.5),W); label("(B)",(7,-0.5),W); label("(C)",(14,-0.5),W); label("(D)",(21,-0.5),W); label("(E)",(28,-0.5),W); [/asy]

Let $M, \alpha, \beta \in \mathbb{R} $ with $M > 0$ and $\alpha, \beta \in (0,1)$. If $R>1$ is a real number, we say that a sequence of positive real numbers $\{ C_n \}_{n\geq 0}$ is $R$-[i]inoceronte[/i] if $ \sum_{i=1}^n R^{n-i}C_i \leq R^n \cdot M$ for all $n \geq 1$. Determine the smallest real $R>1$ for which exists a $R$-[i]inoceronte[/i] sequence $ \{ C_n \}_{n\geq 0}$ such that $\sum_{n=1}^{\infty} \beta ^n C_n^{\alpha}$ diverges.

A function $f$ is given by $f(x) = x^2 - 2x$ . Prove that there exists a number a which satisfies $f(f(a)) = a$ without satisfying $f(a) = a$ .

Let $ABC$ be a given acute triangle, in which $BC$ is the longest side. Let $H$ be the orthocenter of the triangle, and let $D$ and $E$ be the feet of the altitudes from $B$ and $C$, respectively. Let $F$ and $G$ be the midpoints of sides $AB$ and $AC$, respectively. $X$ is the point of intersection of lines $DF$ and $EG$. Let $O_1$ and $O_2$ be the circumcenters of triangles $EFX$ and $DGX$, respectively. Finally, $M$ is the midpoint of line segment $O_1O_2$. Prove that points $X, H$ and $M$ are collinear.

Let $ ABCD$ be a convex quadrilateral such that the sides $ AB, AD, BC$ satisfy $ AB \equal{} AD \plus{} BC.$ There exists a point $ P$ inside the quadrilateral at a distance $ h$ from the line $ CD$ such that $ AP \equal{} h \plus{} AD$ and $ BP \equal{} h \plus{} BC.$ Show that: \[ \frac {1}{\sqrt {h}} \geq \frac {1}{\sqrt {AD}} \plus{} \frac {1}{\sqrt {BC}} \]

Positive integer $m$ is such that the sum of decimal digits of $8^m$ equals 8. Can the last digit of $8^m$ be equal 6? (Author: V. Senderov) (compare with http://www.artofproblemsolving.com/Forum/viewtopic.php?f=57&t=431860)

An acute-angled triangle $ABC$ has altitudes $AD, BE$ and $CF$. Let $Q$ be an interior point of the segment $AD$, and let the circumcircles of the triangles $QDF$ and $QDE$ meet the line $BC$ again at points $X$ and $Y$ , respectively. Prove that $BX = CY$ .

Let $\{x\}$ be a sequence of positive reals $x_1, x_2, \ldots, x_n$, defined by: $x_1 = 1, x_2 = 9, x_3=9, x_4=1$. And for $n \geq 1$ we have: \[x_{n+4} = \sqrt[4]{x_{n} \cdot x_{n+1} \cdot x_{n+2} \cdot x_{n+3}}.\] Show that this sequence has a finite limit. Determine this limit.

Positive integers 1 to 9 are written in each square of a $ 3 \times 3 $ table. Let us define an operation as follows: Take an arbitrary row or column and replace these numbers $ a, b, c$ with either non-negative numbers $ a-x, b-x, c+x $ or $ a+x, b-x, c-x$, where $ x $ is a positive number and can vary in each operation. (1) Does there exist a series of operations such that all 9 numbers turn out to be equal from the following initial arrangement a)? b)? \[ a) \begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{array} )\] \[ b) \begin{array}{ccc} 2 & 8 & 5 \\ 9 & 3 & 4 \\ 6 & 7 & 1 \end{array} )\] (2) Determine the maximum value which all 9 numbers turn out to be equal to after some steps.

Points $M,\ N,\ K,\ L$ are taken on the sides $AB,\ BC,\ CD,\ DA$ of a rhombus $ABCD,$ respectively, in such a way that $MN\parallel LK$ and the distance between $MN$ and $KL$ is equal to the height of $ABCD.$ Show that the circumcircles of the triangles $ALM$ and $NCK$ intersect each other, while those of $LDK$ and $MBN$ do not.

Solve the equation $$\frac{10}{x+10}+\frac{10\cdot 9}{(x+10)(x+9)}+\frac{10\cdot 9\cdot 8}{(x+10)(x+9)(x+8)}+ ...+\frac{10\cdot 9\cdot ... \cdot 2 \cdot 1}{(x+10)(x+9)\cdot ... \cdot(x+1)}=11$$

\[\int_1^2 \frac{\sqrt{1-\frac 1x}}{x-1}\mathrm dx\] [i]Proposed by Connor Gordon[/i]

Let $A$, $B$, and $C$ be distinct points on a line with $AB=AC=1$. Square $ABDE$ and equilateral triangle $ACF$ are drawn on the same side of line $BC$. What is the degree measure of the acute angle formed by lines $EC$ and $BF$? [i]Ray Li[/i]

If $ g(x)\equal{}1\minus{}x^2$ and $ f(g(x)) \equal{} \frac{1\minus{}x^2}{x^2}$ when $ x\neq0$, then $ f(1/2)$ equals $ \textbf{(A)}\ 3/4 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ \sqrt2/2 \qquad \textbf{(E)}\ \sqrt2$

In convex polyhedron $ABCDE$ line segment $DE$ intersects the plane of triangle $ABC$ inside the triangle. Rotate the point $D$ outward into the plane of triangle $ABC$ around the lines $AB$, $BC$, $CA$; let the resulting points be $D_1$, $D_2$, and $D_3$. Similarly, rotate the point $E$ outward into the plane of triangle $ABC$ around the lines $AB$, $BC$, $CA$; let the resulting points be $E_1$, $E_2$, and $E_3$. Show that if the polyhedron has an inscribed sphere, then the circumcircles of $D_1D_2D_3$ and $E_1E_2E_3$ are concentric. [i]Proposed by: Géza Kós, Budapest[/i]

We say that a doubly infinite sequence $. . . , s_{−2}, s_{−1}, s_{0}, s_1, s_2, . . .$ is subaveraging if $s_n = (s_{n−1} + s_{n+1})/4$ for all integers n. (a) Find a subaveraging sequence in which all entries are different from each other. Prove that all entries are indeed distinct. (b) Show that if $(s_n)$ is a subaveraging sequence such that there exist distinct integers m, n such that $s_m = s_n$, then there are infinitely many pairs of distinct integers i, j with $s_i = s_j$ .

Let $S$ be the sum of all integers $b$ for which the polynomial $x^2+bx+2008b$ can be factored over the integers. Compute $|S|$.

Six points are placed on the plane such that each three of them are the vertices of a triangle with sides of different lengths. Prove that the shortest side of one of these triangles is also the longest side of another of them.