A [i] magic square[/i] is a $3\times 3$ grid of numbers in which the sums of the numbers in each row, column, and long diagonal are all equal. How many magic squares exist where each of the integers from $11$ to $19$ inclusive is used exactly once and two of the numbers are already placed as shown below? $\begin{tabular}{|l|l|l|l|} \hline & & 18 \\ \hline & 15 & \\ \hline & & \\ \hline \end{tabular}$

Let $A=\{a_1,a_2,\cdots,a_n,b_1,b_2,\cdots,b_n\}$ be a set with $2n$ distinct elements, and $B_i\subseteq A$ for any $i=1,2,\cdots,m.$ If $\bigcup_{i=1}^m B_i=A,$ we say that the ordered $m-$tuple $(B_1,B_2,\cdots,B_m)$ is an ordered $m-$covering of $A.$ If $(B_1,B_2,\cdots,B_m)$ is an ordered $m-$covering of $A,$ and for any $i=1,2,\cdots,m$ and any $j=1,2,\cdots,n,$ $\{a_j,b_j\}$ is not a subset of $B_i,$ then we say that ordered $m-$tuple $(B_1,B_2,\cdots,B_m)$ is an ordered $m-$covering of $A$ without pairs. Define $a(m,n)$ as the number of the ordered $m-$coverings of $A,$ and $b(m,n)$ as the number of the ordered $m-$coverings of $A$ without pairs. $(1)$ Calculate $a(m,n)$ and $b(m,n).$ $(2)$ Let $m\geq2,$ and there is at least one positive integer $n,$ such that $\dfrac{a(m,n)}{b(m,n)}\leq2021,$ Determine the greatest possible values of $m.$

There is a square table with side $n$. Is it possible to enter the numbers $0$, $1$ or $2$ into the cells of this table so that all sums of numbers in rows and columns are different and take values from $1$ to $2n$, if: a) $n = 7$ ? b) $n = 8$ ?

Let $P_1P_2\ldots P_{24}$ be a regular $24$-sided polygon inscribed in a circle $\omega$ with circumference $24$. Determine the number of ways to choose sets of eight distinct vertices from these $24$ such that none of the arcs has length $3$ or $8$.

For a positive integer $n \ge 2$, define a sequence $a_1, a_2, \cdots ,a_n$ as the following. $$ a_1 = \frac{n(2n-1)(2n+1)}{3}$$ $$a_k = \frac{(n+k-1)(n-k+1)}{2(k-1)(2k+1)}a_{k-1}, \text{ } (k=2,3, \cdots n)$$ (a) Show that $a_1, a_2, \cdots a_n$ are all integers. (b) Prove that there are exactly one number out of $a_1, a_2, \cdots a_n$ which is not a multiple of $2n-1$ and exactly one number out of $a_1, a_2, \cdots a_n$ which is not a multiple of $2n+1$ if and only if $2n-1$ and $2n+1$ are all primes.

Alice is thinking of a positive real number $x$, and Bob is thinking of a positive real number $y$. Given that $x^{\sqrt{y}}=27$ and $(\sqrt{x})^y=9$, compute $xy$.

If $x, y$, and $z$ are real numbers with $\frac{x - y}{z}+\frac{y - z}{x}+\frac{z - x}{y}= 36$, find $2012 +\frac{x - y}{z}\cdot \frac{y - z}{x}\cdot\frac{z - x}{y}$ .

Three fair dice are tossed at random (i.e., all faces have the same probability of coming up). What is the probability that the three numbers turned up can be arranged to form an arithmetic progression with common difference one? $\textbf{(A) }\frac{1}{6}\qquad\textbf{(B) }\frac{1}{9}\qquad\textbf{(C) }\frac{1}{27}\qquad\textbf{(D) }\frac{1}{54}\qquad \textbf{(E) }\frac{7}{36}$

Show that if the point $ M $ is situated in the interior of a square $ ABCD, $ then, among the segments $ MA,MB,MC,MD, $ [b]a)[/b] at most one of them is greater with a factor of $ \sqrt 5/2 $ than the side of the square. [b]b)[/b] at most two of them are greater than the side of the square. [b]c)[/b] at most three of them are greater with a factor of $ \sqrt 2/2 $ than the side of the square.

We assume a truck as a $1 \times (k + 1)$ tile. Our parking is a $(2k + 1) \times (2k + 1)$ table and there are $t$ trucks parked in it. Some trucks are parked horizontally and some trucks are parked vertically in the parking. The vertical trucks can only move vertically (in their column) and the horizontal trucks can only move horizontally (in their row). Another truck is willing to enter the parking lot (it can only enter from somewhere on the boundary). For $3k + 1 < t < 4k$, prove that we can move other trucks forward or backward in such a way that the new truck would be able to enter the lot. Prove that the statement is not necessarily true for $t = 3k + 1$.

For nonnegative integers $m$ and $n$, define the sequence $a(m,n)$ of real numbers as follows. Set $a(0,0)=2$ and for every natural number $n$, set $a(0,n)=1$ and $a(n,0)=2$. Then for $m,n\geq1$, define \[ a(m,n)=a(m-1,n)+a(m,n-1). \] Prove that for every natural number $k$, all the roots of the polynomial $P_{k}(x)=\sum_{i=0}^{k}a(i,2k+1-2i)x^{i}$ are real.

Given natural number n. Suppose that $N$ is the maximum number of elephants that can be placed on a chessboard measuring $2 \times n$ so that no two elephants are mutually under attack. Determine the number of ways to put $N$ elephants on a chessboard sized $2 \times n$ so that no two elephants attack each other. Alternative Formulation: Determine the number of ways to put $2015$ elephants on a chessboard measuring $2 \times 2015$ so there are no two elephants attacking each othe PS. Elephant = Bishop

Define the modulo $2553$ distance $d(x, y)$ between two integers $x, y$ to be the smallest nonnegative integer $d$ equivalent to either $x - y$ or $y - x$ modulo $2553$. Show that, given a set S of integers such that $|S| \ge 70$, there must be $m, n \in S$ with $d(m, n) \le 36$.

Let $n$ be a positive integer. Prove that the number of lines which go through the origin and precisely one other point with integer coordinates $(x,y),0\le x,y\le n$, is at least $\frac{n^2}{4}$.

Evaluate $ \int_1^{e^2} \frac{(e^{\sqrt{x}}\minus{}e^{\minus{}\sqrt{x}})\cos \left(e^{\sqrt{x}}\plus{}e^{\minus{}\sqrt{x}}\plus{}\frac{\pi}{4}\right)\plus{}(e^{\sqrt{x}}\plus{}e^{\minus{}\sqrt{x}})\cos \left(e^{\sqrt{x}}\minus{}e^{\minus{}\sqrt{x}}\plus{}\frac{\pi}{4}\right)}{\sqrt{x}}\ dx.$

Let $\sum_{i=1}^n a_i x_i =0$, $a_i\in \mathbb{Z}$. It is known that however we color $\mathbb{N}$ with finite number of colors, then the upper equation has a solution $x_1,x_2,...,x_n$ in one color. Prove that there is some non-empty sum of its coefficients equal to 0.

Three points are given on the plane. With the help of compass and ruler construct a straight line in this plane, which will be equidistant from these three points. Explore how many solutions have this construction.

A rectangular piece of paper $ABCD$ has sides of lengths $AB = 1$, $BC = 2$. The rectangle is folded in half such that $AD$ coincides with $BC$ and $EF$ is the folding line. Then fold the paper along a line $BM$ such that the corner $A$ falls on line $EF$. How large, in degrees, is $\angle ABM$? [asy] size(180); pathpen = rgb(0,0,0.6)+linewidth(1); pointpen = black+linewidth(3); pointfontpen = fontsize(10); pen dd = rgb(0,0,0.6) + linewidth(0.7) + linetype("4 4"), dr = rgb(0.8,0,0), dg = rgb(0,0.6,0), db = rgb(0,0,0.6)+linewidth(1); pair A=(0,1), B=(0,0), C=(2,0), D=(2,1), E=A/2, F=(2,.5), M=(1/3^.5,1), N=reflect(B,M)*A; D(B--M--D("N",N,NE)--B--D("C",C,SE)--D("D",D,NE)--M); D(D("M",M,plain.N)--D("A",A,NW)--D("B",B,SW),dd); D(D("E",E,W)--D("F",F,plain.E),dd); [/asy]

Let the Nagel point of triangle $ABC$ be $N$. We draw lines from $B$ and $C$ to $N$ so that these lines intersect sides $AC$ and $AB$ in $D$ and $E$ respectively. $M$ and $T$ are midpoints of segments $BE$ and $CD$ respectively. $P$ is the second intersection point of circumcircles of triangles $BEN$ and $CDN$. $l_1$ and $l_2$ are perpendicular lines to $PM$ and $PT$ in points $M$ and $T$ respectively. Prove that lines $l_1$ and $l_2$ intersect on the circumcircle of triangle $ABC$. [i]Proposed by Nima Hamidi[/i]

Equilateral $\triangle ABC$ has side length $1$, and squares $ABDE$, $BCHI$, $CAFG$ lie outside the triangle. What is the area of hexagon $DEFGHI$? [asy] import graph; size(6cm); pen dps = linewidth(0.7) + fontsize(8); defaultpen(dps); pair B = (0,0); pair C = (1,0); pair A = rotate(60,B)*C; pair E = rotate(270,A)*B; pair D = rotate(270,E)*A; pair F = rotate(90,A)*C; pair G = rotate(90,F)*A; pair I = rotate(270,B)*C; pair H = rotate(270,I)*B; draw(A--B--C--cycle); draw(A--E--D--B); draw(A--F--G--C); draw(B--I--H--C); draw(E--F); draw(D--I); draw(I--H); draw(H--G); label("$A$",A,N); label("$B$",B,SW); label("$C$",C,SE); label("$D$",D,W); label("$E$",E,W); label("$F$",F,E); label("$G$",G,E); label("$H$",H,SE); label("$I$",I,SW); [/asy] $ \textbf{(A)}\ \dfrac{12+3\sqrt3}4\qquad\textbf{(B)}\ \dfrac92\qquad\textbf{(C)}\ 3+\sqrt3\qquad\textbf{(D)}\ \dfrac{6+3\sqrt3}2\qquad\textbf{(E)}\ 6 $

Let $ABC$ be a scalene triangle with circumcircle $\Gamma$. Let $M$ be the midpoint of $BC$. A variable point $P$ is selected in the line segment $AM$. The circumcircles of triangles $BPM$ and $CPM$ intersect $\Gamma$ again at points $D$ and $E$, respectively. The lines $DP$ and $EP$ intersect (a second time) the circumcircles to triangles $CPM$ and $BPM$ at $X$ and $Y$, respectively. Prove that as $P$ varies, the circumcircle of $\triangle AXY$ passes through a fixed point $T$ distinct from $A$.

The area and the perimeter of the triangle with sides $10,8,6$ are equal. Find all the triangles with integral sides whose area and perimeter are equal.

A [i]discrete hexagon with center \((a,b,c)\) \emph{(where \(a\), \(b\), \(c\) are integers)[/i] and radius \(r\) [i](a nonnegative integer)[/i]} is the set of lattice points \((x,y,z)\) such that \(x+y+z=a+b+c\) and \(\max(|x-a|,|y-b|,|z-c|)\le r\). Let \(n\) be a nonnegative integer and \(S\) be the set of triples \((x,y,z)\) of nonnegative integers such that \(x+y+z=n\). If \(S\) is partitioned into discrete hexagons, show that at least \(n+1\) hexagons are needed. [i]Proposed by Linus Tang[/i]

Into every lateral face of a quadrangular pyramid a circle is inscribed. The circles inscribed into adjacent faces are tangent (have one point in common). Prove that the points of contact of the circles with the base of the pyramid lie on a circle.

A cat and mouse live on a house mapped out by the points $(-1, 0)$, $(-1, 2)$, $(0, 3)$, $(1, 2)$, $(1, 0)$. The cat starts at the top of the house (point $(0, 3)$) and the mouse starts at the origin (0, 0). Both start running clockwise around the house at the same time. If the cat runs at $12$ units a minute and the mouse at 9 units a minute, how many laps around the house will the cat run before it catches the mouse?