In how many ways rooks can be placed on a $8$ by $8$ chess board such that every row and every column has at least one rook? (Any number of rooks are available,each square can have at most one rook and there is no relation of attacking between them)

Let $ABCDEF$ be a regular hexagon of side lengths $1$ and $O$ its centre, Join $O$ cto each of the six vertices , thus getting $12$ unit line segments in all. Find the number of closed paths from (i) $O$ to $O$ (ii) $A$ to $A$ each of length $2004$

Let $ n$ be a positive integer. Given an integer coefficient polynomial $ f(x)$, define its [i]signature modulo $ n$[/i] to be the (ordered) sequence $ f(1), \ldots , f(n)$ modulo $ n$. Of the $ n^n$ such $ n$-term sequences of integers modulo $ n$, how many are the signature of some polynomial $ f(x)$ if a) $ n$ is a positive integer not divisible by the square of a prime. b) $ n$ is a positive integer not divisible by the cube of a prime.

The four points $A(-1,2), B(3,-4), C(5,-6),$ and $D(-2,8)$ lie in the coordinate plane. Compute the minimum possible value of $PA + PB + PC + PD$ over all points P .

Find all pairs of integers, $n$ and $k$, $2 < k < n$, such that the binomial coefficients \[\binom{n}{k-1}, \binom{n}{k}, \binom{n}{k+1}\] form an increasing arithmetic series.

Let $A,B,C,D$ be the vertices of a rhombus, let $k_1$ be the circle through $B,C$ and $D$, let $k_2$ be the circle through $A,C$ and $D$, let $k_3$ be the circle through $A,B$ and $D$, let $k_4$ be the circle through $A,B$ and $C$. Prove that the tangents to $k_1$ and $k_3$ at $B$ form the same angle as the tangents to $k_2$ and $k_4$ at $A$.

There $100$ people seated at a round table $50$ women and $50$ men. Show that there are two people of opposite gender that stay between two people of opposite gender. (WWMM, MMWW, WMWM, MWMW)

An equilateral triangle with integer side length $n$ is subdivided into smaller equilateral triangles of side length $1$ by drawing lines parallel to its sides, as shown in the figure for $n = 4$. [asy] size(5cm); // Function to draw an equilateral triangle with subdivisions and mark vertices void drawTriangleWithDots(pair A, pair B, pair C, int n) { real step = 1.0 / n; // Draw horizontal lines for (int i = 0; i <= n; ++i) { pair start = A + i * step * (C - A); pair end = start + i * step * (B - C); draw(start -- end, gray(0.5)); } // Draw left-leaning diagonal lines for (int i = 0; i <= n; ++i) { pair start = A + i * step * (B - A); pair end = start + (n - i) * step * (C - A); draw(start -- end, gray(0.5)); } // Draw right-leaning diagonal lines for (int i = 0; i <= n; ++i) { pair start = B + i * step * (C - B); pair end = start + (n - i) * step * (A - B); draw(start -- end, gray(0.5)); } // Mark dots at all vertices for (int i = 0; i <= n; ++i) { for (int j = 0; j <= i; ++j) { pair vertex = A + i * step * (C - A) + j * step * (B - C); dot(vertex, black); } } // Draw the outer triangle draw(A -- B -- C -- cycle, black+linewidth(1)); } // Main triangle vertices pair A = (0, 0); pair B = (4, 0); pair C = (2, 3.464); // Height = sqrt(3)/2 * side length // Subdivisions int n = 4; // Draw the subdivided equilateral triangle with dots drawTriangleWithDots(A, B, C, n); [/asy] Consider the set $A$ consisting of all points that are vertices of any of these smaller triangles. A [i]subtriangle[/i] is defined as any equilateral triangle whose three vertices belong to the set $A$ and whose three sides lie along the lines of the initial subdivision. We wish to color all points in $A$ either red or blue such that no subtriangle has all three vertices of the same color. Let $C(n)$ denote the number of such valid colorings for each positive integer $n$. Calculate, in terms of $n$, the value of $C(n)$.

The point $P$ lies inside $\vartriangle ABC$ so that $\vartriangle BPC$ is isosceles, and angle $P$ is a right angle. Furthermore both $\vartriangle BAN$ and $\vartriangle CAM$ are isosceles with a right angle at $A$, and both are outside $\vartriangle ABC$. Show that $\vartriangle MNP$ is isosceles and right-angled. [img]https://1.bp.blogspot.com/-i9twOChu774/XzcBLP-RIXI/AAAAAAAAMXA/n5TJCOJypeMVW28-9GDG4st5C47yhvTCgCLcBGAsYHQ/s0/2005%2BMohr%2Bp3.png[/img]

How many positive three-digit integers have a remainder of 2 when divided by 6, a remainder of 5 when divided by 9, and a remainder of 7 when divided by 11? $\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad \textbf{(E) }5$

Let $d_n$ be the last nonzero digit of the decimal representation of $n!$. Prove that $d_n$ is aperiodic; that is, there do not exist $T$ and $n_0$ such that for all $n \geq n_0, d_{n+T} = d_n.$

Segment $AB$ on the surface of the cube is the shortest polyline on the surface that connects $A$ and $B$. Triangle $ABC$ consisted of such segments $AB, BC,CA$. What may be the sum of angles of such triangle if none of the vertex is on the edge of the cube ?

As shown in the figure, $BQ$ is a diameter of the circumcircle of $ABC$, and $D$ is the midpoint of arc $BC$ (excluding point $A$) . The bisector of the exterior angle of $\angle BAC$ intersects and the extension of $BC$ at point $E$. The ray $EQ$ intersects $\odot (ABC)$ at point $P$. Point $S$ lies on $PQ$ so that $SA = SP$. Point $T$ lies on $BC$ such that $TB = TD$. Prove that $TS \perp SE$. [img]https://cdn.artofproblemsolving.com/attachments/c/4/01460565e70b32b29cddb65d92e041bea40b25.png[/img]

Determine the largest positive integer that is a divisor of all three of $A=2^{2010}\times3^{2010}, B=3^{2010}\times5^{2010},$ and $C=5^{2010}\times2^{2010}.$

Peter made a paper rectangle, put it on an identical rectangle and pasted both rectangles along their perimeters. Then he cut the upper rectangle along one of its diagonals and along the perpendiculars to this diagonal from two remaining vertices. After this he turned back the obtained triangles in such a way that they, along with the lower rectangle form a new rectangle. Let this new rectangle be given. Restore the original rectangle using compass and ruler.

Find a function $f: \mathbb N \to \mathbb N$ such that for all positive integers $n$, \[ f(f(n))\equal{}n^2.\]

$ \lim_{n\to\infty } \frac{1}{n}\sum_{i,j=1}^n \frac{i+j}{i^2+j^2} $

Let $n$ be a positive integer. Initially a few positive integers are written on the blackboard. On one move Igor chooses two numbers $a, b$ of the same parity on the blackboard and writes $\frac{a+b} {2}$. After a few moves the numbers on the blackboard were exactly $1, 2, \ldots, n$. Find the smallest possible number of positive integers that were initially written on the blackboard.

Find all pairs of integers $(m, n)$ such that $$m+n = 3(mn+10).$$

Let $ABC$ be an acute triangle with circumcenter $O$ and orthocenter $H$. Let $M_B$ and $M_C$ be the midpoints of $AC$ and $AB$, respectively. Place points $X$ and $Y$ on line $BC$ such that $\angle HM_BX = \angle HM_CY = 90^{\circ}$. Prove that triangles $OXY$ and $HBC$ are similar.

In the lake, there are $23$ stones arranged along a circle. There are $22$ frogs numbered $1, 2, \cdots, 22$ (each number appears once). Initially, each frog randomly sits on a stone (several frogs might sit on the same stone). Every minute, all frogs jump at the same time as follows: the frog number $i$ jumps $i$ stones forward in the clockwise direction. (In particular, the frog number $22$ jumps $1$ stone in the counter-clockwise direction.) Prove that at some point, at least $6$ stones will be empty.

Determine if there exist functions $f, g : R \to R$ satisfying for every $x \in R$ the following equations $f(g(x)) = x^3$ and $g(f(x)) = x^2$.

Prove there are no such integers $ k, m $ which satisfy $ k \ge 0, m \ge 0 $ and $ k!+48=48(k+1)^m $.

Given a cyclic quadrilateral $ABCD$ with circumcenter $O$, let the circle $(AOD)$ intersect the segments $AB$, $AC$, $DB$, $DC$ at $P$, $Q$, $R$, $S$ respectively. Suppose $X$ is the reflection of $D$ about $PQ$ and $Y$ is the reflection of $A$ about $RS$. Prove that the circles $(AOD)$, $(BPX)$, $(CSY)$ meet at a common point. [i]Proposed by Leia Mayssa & Ivan Chan Kai Chin[/i]

Find all $l,m,n \in\mathbb{N}$ that satisfies the equation $5^l43^m+1=n^3$