Let $BCDK$ be a convex quadrilateral such that $BC=BK$ and $DC=DK$. $A$ and $E$ are points such that $ABCDE$ is a convex pentagon such that $AB=BC$ and $DE=DC$ and $K$ lies in the interior of the pentagon $ABCDE$. If $\angle ABC=120^{\circ}$ and $\angle CDE=60^{\circ}$ and $BD=2$ then determine area of the pentagon $ABCDE$.

Find the smallest positive integer $c$ for which the following statement holds: Let $k$ and $n$ be positive integers. Suppose there exist pairwise distinct subsets $S_1$, $S_2$, $\dots$, $S_{2k}$ of $\{1, 2, \dots, n\}$, such that $S_i \cap S_j \neq \varnothing$ and $S_i \cap S_{j+k} \neq \varnothing$ for all $1 \le i,j \le k$. Then $1000k \le c \cdot 2^n$. [i]Proposed by Yang Liu[/i]

On the board all the 20-digit numbers which have 10 ones and 10 twos in their decimal form are written. It is allowed to choose two different digits in any number and to reverse the order of digits in the interval between them. What is the maximal quantity of equal numbers which is possible to get on the board using such operations?

Let $\{a_n\}$ be a nonnegative real sequence. Define $$X_k = \sum_{i=1}^{2^k}a_i, Y_k = \sum_{i=1}^{2^k}\left\lfloor \frac{2^k}{i}\right\rfloor a_i, k=0,1,2,...$$ Prove that $X_n\le Y_n - \sum_{i=0}^{n-1} Y_i \le \sum_{i=0}^n X_i$ for all positive integer $n$. Here $\lfloor\alpha\rfloor$ denotes the largest integer that does not exceed $\alpha$.

For a nonnegative integer $n$ and a strictly increasing sequence of real numbers $t_0, t_1, \ldots, t_n$, let $f(t)$ be the corresponding real-valued function defined for $t \geq t_0$ by the following properties: (a) $f(t)$ is continuous for $t \geq t_0$, and is twice differentiable for all $t>t_0$ other than $t_1, \ldots, t_n$; (b) $f\left(t_0\right)=1 / 2$; (c) $\lim _{t \rightarrow t_k^{+}} f^{\prime}(t)=0$ for $0 \leq k \leq n$; (d) For $0 \leq k \leq n-1$, we have $f^{\prime \prime}(t)=k+1$ when $t_k<t<t_{k+1}$, and $f^{\prime \prime}(t)=n+1$ when $t>t_n$. Considering all choices of $n$ and $t_0, t_1, \ldots, t_n$ such that $t_k \geq t_{k-1}+1$ for $1 \leq k \leq n$, what is the least possible value of $T$ for which $f\left(t_0+T\right)=2023$?

For a positive integer $n$, let $\omega(n)$ denote the number of positive prime divisors of $n$. Find the smallest positive tinteger $k$ such that $2^{\omega(n)}\leq k\sqrt[4]{n}\forall n\in\mathbb{N}$.

The circle touches the square and goes through its two vertices as shown in the figure. Find the area of the square. (Distance in the picture is measured horizontally from the midpoint of the side of the square.) [img]https://cdn.artofproblemsolving.com/attachments/7/5/ab4b5f3f4fb4b70013e6226ce5189f3dc2e5be.png[/img]

The circle $\omega$ lies inside the circle $\Omega$ and touches it internally at $T.$ Let $XY{}$ be a variable chord of the circle $\Omega$ touching $\omega.$ Denote by $X'$ and $Y'$ the midpoints of the arcs $TY{}$ and $TX{}$ which do not contain $X{}$ and $Y{}$ respectively. Prove that all possible lines $X'Y'$ pass through a fixed point. [i]Proposed by F. Petrov[/i]

Let $f(x)$ be a continuous function defined on the closed interval $0 \leq x \leq 1$. Let $G(f)$ denote the graph of $f(x): G(f) = \{(x, y) \in \mathbb R^2 | 0 \leq$$ x \leq 1, y = f(x) \}$. Let $G_a(f)$ denote the graph of the translated function $f(x - a)$ (translated over a distance $a$), defined by $G_a(f) = \{(x, y) \in \mathbb R^2 | a \leq x \leq a + 1, y = f(x - a) \}$. Is it possible to find for every $a, \ 0 < a < 1$, a continuous function $f(x)$, defined on $0 \leq x \leq 1$, such that $f(0) = f(1) = 0$ and $G(f)$ and $G_a(f)$ are disjoint point sets ?

Let a triangle $\triangle ABC$ and the real numbers $x$, $y$, $z>0$. Prove that $$x^n\cos\frac{A}{2}+y^n\cos\frac{B}{2}+z^n\cos\frac{C}{2}\geq (yz)^{\frac{n}{2}}\sin A +(zx)^{\frac{n}{2}}\sin B +(xy)^{\frac{n}{2}}\sin C$$

Given two positive integers $r > s$, and let $F$ be an infinite family of sets, each of size $r$, no two of which share fewer than $s$ elements. Prove that there exists a set of size $r -1$ that shares at least $s$ elements with each set in $F$.

Let $A$ be a set of positive integers having the following property: for each positive integer $n$ exactly one of the three numbers $n, 2n$ and $3n$ is an element of $A$. Furthermore, it is given that $2 \in A$. Prove that $13824 \notin A$.

Construct a geometric gure in a sequence of steps. In step $1$, begin with a $4\times 4$ square. In step $2$, attach a $1\times 1$ square onto the each side of the original square so that the new squares are on the outside of the original square, have a side along the side of the original square, and the midpoints of the sides of the original square and the attached square coincide. In step $3$, attach a $\frac14\times  \frac14$ square onto the centers of each of the $3$ exposed sides of each of the $4$ squares attached in step $2$. For each positive integer $n$, in step $n + 1$, attach squares whose sides are $\frac14$ as long as the sides of the squares attached in step n placing them at the centers of the $3$ exposed sides of the squares attached in step $n$. The diagram shows the gure after step $4$. If this is continued for all positive integers $n$, the area covered by all the squares attached in all the steps is $\frac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. Find $p + q$. [img]https://cdn.artofproblemsolving.com/attachments/2/1/d963460373b56906e93c4be73bc6a15e15d0d6.png[/img]

$BC$ is a diameter of a circle and the points $X,Y$ are on the circle such that $XY\perp BC$. The points $P,M$ are on $XY,CY$ (or their stretches), respectively, such that $CY||PB$ and $CX||PM$. Let $K$ be the meet point of the lines $XC,BP$. Prove that $PB\perp MK$.

Let $N$ be the set of natural numbers, and let $f: N \to N$ be a function satisfying $f(x) + f(x + 2) < 2 f(x + 1)$ for any $x \in N$. Prove that there exists a straight line in the $xy$-plane which contains infinitely many points with coordinates $(n,f(n))$.

For positive integers $n$ and $k$, let $f(n,k)$ be the remainder when $n$ is divided by $k$, and for $n>1$ let $F(n) = \displaystyle\max_{1 \le k \le \frac{n}{2}} f(n,k)$. Find the remainder when $\displaystyle\sum_{n=20}^{100} F(n)$ is divided by $1000$.