In how many ways can $1, 2,\ldots, 2n$ be arranged in a $2 \times n$ rectangular array $\left(\begin{array}{cccc}a_1& a_2 & \cdots & a_n\\b_1& b_2 & \cdots & b_n\end{array}\right)$ for which: [b](i)[/b] $a_1 < a_2 < \cdots < a_n,$ [b](ii) [/b] $b_1 < b_2 <\cdots < b_n,$ [b](iii) [/b]$a_1 < b_1, a_2 < b_2, \ldots, a_n < b_n \ ?$

Given is the equation $x^2+y^2-axy+2=0$ where $a$ is a positive integral parameter. $i.$Show that,for $a\neq 4$ there exist no pairs $(x,y)$ of positive integers satisfying the equation. $ii.$ Show that,for $a=4$ there exist infinite pairs $(x,y)$ of positive integers satisfying the equation,and determine those pairs.

Find the set of all possible values of the expression $\lfloor m^2+\sqrt{2} n \rfloor$, where $m$ and $n$ are positive integers. Note: The symbol $\lfloor x\rfloor$ denotes the largest integer less than or equal to $x$.

In convex quadrilateral $KALE$, angles $\angle KAL$, $\angle AKL$, and $\angle ELK$ measure $110^\circ$, $50^\circ$, and $10^\circ$, respectively. Given that $KA = LE$ and that $\overline{KL}$ and $\overline{AE}$ intersect at point $X$, compute the value of $\tfrac{KX^2}{AL\cdot EX}$.

Prove that if all faces of a parallelepiped are equal parallelograms, they are rhombuses.

$ABC$ is a right triangle with $\angle C = 90^o$. Let $N$ be the middle of arc $BAC$ of the circumcircle and $K$ be the intersection point of $CN$ and $AB$. Assume $T$ is a point on a line $AK$ such that $TK=KA$. Prove that the circle with center $T$ and radius $TK$ is tangent to $BC$. (Mykhailo Sydorenko)

Let $f:[-\pi/2,\pi/2]\to\mathbb{R}$ be a twice differentiable function which satisfies \[\left(f''(x)-f(x)\right)\cdot\tan(x)+2f'(x)\geqslant 1,\]for all $x\in(-\pi/2,\pi/2)$. Prove that \[\int_{-\pi/2}^{\pi/2}f(x)\cdot \sin(x) \ dx\geqslant \pi-2.\]

$\omega_A, \omega_B, \omega_C$ are three concentric circles with radii $2,3,$ and $7,$ respectively. We say that a point $P$ in the plane is [i]nice[/i] if there are points $A, B,$ and $C$ on $\omega_A, \omega_B,$ and $\omega_C,$ respectively, such that $P$ is the centroid of $\triangle{ABC}.$ If the area of the smallest region of the plane containing all nice points can be expressed as $\tfrac{a\pi}{b},$ where $a$ and $b$ are relatively prime positive integers , what is $a+b$?

Let $\mathbb{R}$ denote the set of the reals. Find all $f : \mathbb{R} \to \mathbb{R}$ such that $$ f(x)f(y) = xf(f(y-x)) + xf(2x) + f(x^2) $$ for all real $x, y$.

What is the maximal number of regions a circle can be divided in by segments joining $n$ points on the boundary of the circle ? [i]Posted already on the board I think...[/i]

Determine all triples $(m , n , p)$ satisfying : \[n^{2p}=m^2+n^2+p+1\] where $m$ and $n$ are integers and $p$ is a prime number.

[i]Mediovagio[/i] is a computer game that consists in a $3 \times 3$ table in which each of the nine cells has a integer number from $1$ to $n$. When one clicks a cell, the numbers in the clicked cell and in the cells that share an edge with it are increased by $1$ and the sum is evaluated${}\bmod n$. Determine the values of $n$ for which it's possible, with a finite number of clicks, obtain any combination of numbers from an given initial combination. EDIT: I corrected the statement.

A train car held $6000$ pounds of mud which was $88$ percent water. Then the train car sat in the sun, and some of the water evaporated so that now the mud is only $82$ percent water. How many pounds does the mud weigh now?

Let $a,b,c$ be sides of a triangle and $p$ its semiperimeter. Show that $a\sqrt{\frac{(p-b)(p-c)}{bc}}+b \sqrt{\frac{(p-c)(p-a)}{ac}}+c\sqrt{\frac{(p-a)(p-b)}{ab}}\geq p$

As shown in the figure, it is known that the quadrilateral $ABCD$ satisfies $\angle ADB = \angle ACB = 90^o$. Suppose $AC$ and $BD$ intersect at point $P$, point $R$ lies on $CD$ and $RP \perp AB$. $M$ and $N$ are the midpoints of $AB$ and $CD$ respectively. Point $K$ is a point on the extension line of $NM$, the circumscribed circles of $\vartriangle DKC$ and $\vartriangle AKB$ intersect at point $S$. Prove that $KS \perp SR$. [img]https://cdn.artofproblemsolving.com/attachments/5/d/fc0a391f8ebcdee792e9b226cbf55a058251a1.png[/img]

Given is triangle $ABC$ with an inscribed circle with center $M$ and radius $r$. The tangent to this circle parallel to $BC$ intersects $AC$ in $D$ and $AB$ in $E$. The tangent to this circle parallel to $AC$ intersects $AB$ in $F$ and $BC$ in $G$. The tangent to this circle parallel to $AB$ intersects $BC$ in $H$ and $AC$ in $K$. Name the centers of the inscribed circles of triangle $AED$, triangle $FBG$ and triangle $KHC$ successively $M_A, M_B, M_C$ and the rays successively $r_A, r_B$ and $r_C$. Prove that $r_A + r_B + r_C = r$.

How many integers greater than $10$ and less than $100$, written in base-$10$ notation, are increased by $9$ when their digits are reversed? $\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 8 \qquad \textbf{(D)}\ 9 \qquad \textbf{(E)}\ 10$

In an acute-angled triangle $ABC$, the altitudes from $A,B,C$ meet the sides of $ABC$ at $A_1$, $B_1$, $C_1$, and meet the circumcircle of $ABC$ at $A_2$, $B_2$, $C_2$, respectively. Line $A_1 C_1$ intersects the circumcircles of triangles $AC_1 C_2$ and $CA_1 A_2$ at points $P$ and $Q$ ($Q\neq A_1$, $P\neq C_1$). Prove that the circle $PQB_1$ touches the line $AC$.

Let $G$ be a finite simple graph and let $k$ be the largest number of vertices of any clique in $G$. Suppose that we label each vertex of $G$ with a non-negative real number, so that the sum of all such labels is $1$. Define the [i]value of an edge[/i] to be the product of the labels of the two vertices at its ends. Define the [i]value of a labelling[/i] to be the sum of values of the edges. Prove that the maximum possible value of a labelling of $G$ is $\frac{k-1}{2k}$. (A [i]finite simple graph[/i] is a graph with finitely many vertices, in which each edge connects two distinct vertices and no two edges connect the same two vertices. A [i]clique[/i] in a graph is a set of vertices in which any two are connected by an edge.)

Find all function $ f: \mathbb{N}\rightarrow\mathbb{N}$ satisfy $ f(mn)\plus{}f(m\plus{}n)\equal{}f(m)f(n)\plus{}1$ for all natural number $ n$

Suppose that a rectangle with sides $ a$ and $ b$ is arbitrarily cut into $ n$ squares with sides $ x_{1},\ldots,x_{n}$. Show that $ \frac{x_{i}}{a}\in\mathbb{Q}$ and $ \frac{x_{i}}{b}\in\mathbb{Q}$ for all $ i\in\{1,\cdots, n\}$.

Lucy starts by writing $s$ integer-valued $2022$-tuples on a blackboard. After doing that, she can take any two (not necessarily distinct) tuples $\mathbf{v}=(v_1,\ldots,v_{2022})$ and $\mathbf{w}=(w_1,\ldots,w_{2022})$ that she has already written, and apply one of the following operations to obtain a new tuple: \begin{align*} \mathbf{v}+\mathbf{w}&=(v_1+w_1,\ldots,v_{2022}+w_{2022}) \\ \mathbf{v} \lor \mathbf{w}&=(\max(v_1,w_1),\ldots,\max(v_{2022},w_{2022})) \end{align*} and then write this tuple on the blackboard. It turns out that, in this way, Lucy can write any integer-valued $2022$-tuple on the blackboard after finitely many steps. What is the smallest possible number $s$ of tuples that she initially wrote?

There are $ 2010 $ people sitting around a round table. First, we give one person $ x $ a candy. Next, we give candies to $1$ st person, $1+2$ th person, $ 1+2+3$ th person, $\cdots$ , and $1+2+\cdots + 2009 $ th person clockwise from $ x $. Find the number of people who get at least one candy.

Let $ ABC$ be an acute triangle. Points $ D$, $ E$, and $ F$ lie on segments $ BC$, $ CA$, and $ AB$, respectively, and each of the three segments $ AD$, $ BE$, and $ CF$ contains the circumcenter of $ ABC$. Prove that if any two of the ratios $ \frac{BD}{DC}$, $ \frac{CE}{EA}$, $ \frac{AF}{FB}$, $ \frac{BF}{FA}$, $ \frac{AE}{EC}$, $ \frac{CD}{DB}$ are integers, then triangle $ ABC$ is isosceles.

Let $\Delta ABC$ be a triangle. The internal bisectors of angles $\angle CAB$ and $\angle ABC$ intersect segments $BC$, respectively $AC$ in $D$, respectively $E$. Prove that \[DE\leq (3-2\sqrt{2})(AB+BC+CA).\]