Find the coefficient of $x$ in the expansion of $(1 + x)(1 - 2x)(1 + 3x)(1 - 4x) ...(1 - 2008x)$.

In the acute triangle $\triangle ABC$ , $H$ is the orthocenter. $S$ is a point on $(AHC)$ st $\angle ASB = 90$. $P$ is on $AC$ and not on the extention of $AC$ from $A$ , st $\angle APS=\angle BAS$.Prove that $CS$ , the circle $(BPC)$ and the circle with diameter $AC$ are concurrent.

Let $T = \{n \in N|$n consists of $2$ digits $\}$ and $$P = \{x|x = n(n + 1)... (n + 7); n,n + 1,..., n + 7 \in T\}.$$ Determine the gcd of the elements of $P$.

Let $n$ be a positive integer not divisible by $3$. A triangular grid of length $n$ is obtained by dissecting a regular triangle with length $n$ into $n^2$ unit regular triangles. There is an orange at each vertex of the grid, which sums up to \[\frac{(n+1)(n+2)}{2}\] oranges. A triple of oranges $A,B,C$ is [i]good[/i] if each $AB,AC$ is some side of some unit regular triangles, and $\angle BAC = 120^{\circ}$. Each time, Yen can take away a good triple of oranges from the grid. Determine the maximum number of oranges Yen can take.

Find all real pairs $(a,b)$ that solve the system of equation \begin{align*} a^2+b^2 &= 25, \\ 3(a+b)-ab &= 15. \end{align*} [i](German MO 2016 - Problem 1)[/i]

Let $p$ be an odd prime and let $Z_{p}$ denote (the field of) integers modulo $p$. How many elements are in the set \[\{x^{2}: x \in Z_{p}\}\cap \{y^{2}+1: y \in Z_{p}\}?\]

Vertices $A, B, C$ of a equilateral triangle of side $1$ are in the surface of a sphere with radius $1$ and center $O$. Let $D$ be the orthogonal projection of $A$ on the plane $\alpha$ determined by points $B, C, O$. Let $N$ be one of the intersections of the line perpendicular to $\alpha$ passing through $O$ with the sphere. Find the angle $\angle DNO$.

Let $n$ be a positive integer. There is a pawn in one of the cells of an $n\times n$ table. The pawn moves from an arbitrary cell of the $k$th column, $k \in \{1,2, \cdots, n \}$, to an arbitrary cell in the $k$th row. Prove that there exists a sequence of $n^{2}$ moves such that the pawn goes through every cell of the table and finishes in the starting cell.

Anis, Banu, and Cholis are going to play a game. They are given an $n\times n$ board consisting of $n^2$ unit squares, where $n$ is an integer and $n > 5$. In the beginning of the game, the number $n$ is written on each unit square. Then Anis, Banu, and Cholis take turns playing the game, repeatedly in that order, according to the following procedure: On every turn, an arrangement of $n$ squares on the same row or column is chosen, and every number from the chosen squares is subtracted by $1$. The turn cannot be done if it results in a negative number, that is, no arrangement of $n$ unit squares on the same column or row in which all of its unit squares contain a positive number can be found. The last person to get a turn wins. Determine which player will win the game.

Let $a,b,c\in\mathbb{R^+}$ If $3=a+b+c\le 3abc$ , prove that $$\frac{1}{\sqrt{2a+1}}+ \frac{1}{\sqrt{2b+1}}+\frac{1}{\sqrt{2c+1}}\le \left( \frac32\right)^{3/2}$$ [i](Real Matrik)[/i]

Let \( S \) be a set consisting of \( 2024 \) points on a plane, such that no three points in \( S \) are collinear. A line \( \ell \) passing through two points in \( S \) is called a "weakly balanced line" if it satisfies the following condition: (Condition) The line \( \ell \) divides the plane into two regions, one containing exactly \( 1010 \) points of \( S \), and the other containing exactly \( 1012 \) points of \( S \) (where each region contains no points lying on \( \ell \)). Let \( \omega(S) \) denote the number of weakly balanced lines among the lines passing through two points in \( S \). Find the smallest possible value of \( \omega(S) \).

Prove that there exist at least $100!$ ways to write $100!$ as sum of elements of set {$1!,2!,3!...99!$} (each number in sum can be two or more times)

We are given a 5x5 square grid, divided to 1x1 tiles. Two tiles are called [b]linked[/b] if they lie in the same row or column, and the distance between their centers is 2 or 3. For example, in the picture the gray tiles are the ones linked to the red tile. [img]https://i.imgur.com/JVTQ9wB.png[/img] Sammy wants to mark as many tiles in the grid as possible, such that no two of them are linked. What is the maximal number of tiles he can mark?

A uniform disk ($ I = \dfrac {1}{2} MR^2 $) of mass 8.0 kg can rotate without friction on a fixed axis. A string is wrapped around its circumference and is attached to a 6.0 kg mass. The string does not slip. What is the tension in the cord while the mass is falling? [asy] size(250); pen p=linewidth(3), dg=gray(0.25), llg=gray(0.90), lg=gray(0.75),g=grey; void f(path P, pen p, pen q) { filldraw(P,p,q); } path P=CR((0,0),1); D((1,0)--(1,-2.5),p+lg); f(P,g,p); P=scale(0.4)*P; f(P,lg,p); path Q=arc((0,0),0.3,-40,130)--arc((-1,-1),0.5,130,320)--cycle; f(Q,llg,dg+p); P=scale(0.45)*P; f(P,llg,dg+p); P=shift((0.15,0.15))*((-1,-1)--(-1,-2)--(-1.1,-2)--(-1.1,-1.1)--(-2,-1.1)--(-2,-1)--cycle); f(P,llg,lg+p); P=shift((1.55,1.55))*scale(3)*P; f(P,llg,g+p); unfill((-1.23,-1.23)--(-1.23,-5)--(-5,-1.23)--cycle); clip((-3.8,-3.8)--(-3.8,3.8)--(3.8,3.8)--(3.8,-3.8)--cycle); P=(0.2,-2.5)--(1.8,-2.5)--(1.8,-4.1)--(0.2,-4.1)--cycle; f(P,llg,lg+p); MP("m",(1,-3.3),(0,0),fontsize(16)); MP("M",(0,-1),fontsize(16));[/asy] $ \textbf {(A) } \text {20.0 N} \qquad \textbf {(B) } \text {24.0 N} \qquad \textbf {(C) } \text {34.3 N} \qquad \textbf {(D) } \text {60.0 N} \qquad \textbf {(E) } \text {80.0 N} $

There are $10000$ cities in country, and roads between some cities. Every city has $<100$ roads. Every cycle route with odd number of road consists of $\geq 101 $ roads. Prove that we can divide all cities in $100$ groups with $100$ cities, such that every road leads from one group to other.

Solve the set of simultaneous equations \begin{align*} v^2+ w^2+ x^2+ y^2 &= 6 - 2u, \\ u^2+ w^2+ x^2+ y^2 &= 6 - 2v, \\ u^2+ v^2+ x^2+ y^2 &= 6- 2w, \\ u^2+ v^2+ w^2+ y^2 &= 6 - 2x, \\ u^2+ v^2+ w^2+ x^2 &= 6- 2y. \end{align*}

Let $ a,n$ be positive integers such that $ a\ge(n\minus{}1)!$. Prove that there exist $ n$ [i]distinct[/i] prime numbers $ p_1,\ldots,p_n$ so that $ p_i|a\plus{}i$, for all $ i\equal{}\overline{1,\ldots,n}$.

A line $r$ contains the points $A$, $B$, $C$, $D$ in that order. Let $P$ be a point not in $r$ such that $\angle{APB} = \angle{CPD}$. Prove that the angle bisector of $\angle{APD}$ intersects the line $r$ at a point $G$ such that: $\frac{1}{GA} + \frac{1}{GC} = \frac{1}{GB} + \frac{1}{GD}$

Let $\Delta$ denote the set of all triangles in a plane. Consider the function $f: \Delta\to(0,\infty)$ defined by $f(ABC) = \min \left( \dfrac ba, \dfrac cb \right)$, for any triangle $ABC$ with $BC=a\leq CA=b\leq AB = c$. Find the set of values of $f$.

Let $M$ be a set and $A,B,C$ given subsets of $M$. Find a necessary and sufficient condition for the existence of a set $X\subset M$ for which $(X\cup A)\backslash(X\cap B)=C$. Describe all such sets.

Find the greatest prime factor of $20!+20!+21!$.

Let $p$ be a sufficiently large prime. Show that the number of distinct residues taken by the set $$\{1 + \frac12 + ... + \frac{1}{n}: n = 1, 2,..., p - 1\}$$ modulo $p$ has at least $\sqrt[4]{p}$ elements. (Carlo Sanna)

The New York Public Library requires patrons to choose a 4-digit Personal Identification Number (PIN) to access its online system. (Leading zeros are allowed.) The PIN is not allowed to contain either of the following two forbidden patterns: * A digit that is repeated 3 or more times in a row. For example, 0001 and 5555 are not PINs, but 0010 is a PIN. * A pair of digits that is duplicated. For example, 1212 and 6363 are not PINs, but 1221 and 6633 are PINs. How many distinct possible PINs are there?

Let $\ell_A$ and $\ell_B$ be two distinct parallel lines. For positive integers $m$ and $n$, distinct points $A_1, A_2, \allowbreak A_3, \allowbreak \ldots, \allowbreak A_m$ lie on $\ell_A$, and distinct points $B_1, B_2, B_3, \ldots, B_n$ lie on $\ell_B$. Additionally, when segments $\overline{A_iB_j}$ are drawn for all $i=1,2,3,\ldots, m$ and $j=1,\allowbreak 2,\allowbreak 3, \ldots, \allowbreak n$, no point strictly between $\ell_A$ and $\ell_B$ lies on more than two of the segments. Find the number of bounded regions into which this figure divides the plane when $m=7$ and $n=5$. The figure shows that there are 8 regions when $m=3$ and $n=2$. [asy] import geometry; size(10cm); draw((-2,0)--(13,0)); draw((0,4)--(10,4)); label("$\ell_A$",(-2,0),W); label("$\ell_B$",(0,4),W); point A1=(0,0),A2=(5,0),A3=(11,0),B1=(2,4),B2=(8,4),I1=extension(B1,A2,A1,B2),I2=extension(B1,A3,A1,B2),I3=extension(B1,A3,A2,B2); draw(B1--A1--B2); draw(B1--A2--B2); draw(B1--A3--B2); label("$A_1$",A1,S); label("$A_2$",A2,S); label("$A_3$",A3,S); label("$B_1$",B1,N); label("$B_2$",B2,N); label("1",centroid(A1,B1,I1)); label("2",centroid(B1,I1,I3)); label("3",centroid(B1,B2,I3)); label("4",centroid(A1,A2,I1)); label("5",(A2+I1+I2+I3)/4); label("6",centroid(B2,I2,I3)); label("7",centroid(A2,A3,I2)); label("8",centroid(A3,B2,I2)); dot(A1); dot(A2); dot(A3); dot(B1); dot(B2); [/asy]

There are $n \ge 3$ distinct positive real numbers. Show that there are at most $n-2$ different integer power of three that can be written as the sum of three distinct elements from these $n$ numbers.