Suppose $f:\mathbb{N}\rightarrow \mathbb{N}$ is a function such that $$f^n(n) = 2n$$ for all $n\in \mathbb{N}$. Must $f(n) = n+1$ for all $n$?

Find all composite numbers $n$ such that for each decomposition of $n=xy$, $x+y$ be a power of $2$.

Given acute triangle $\vartriangle ABC$ with orthocenter $H$ and circumcenter $O$ ($O \ne H$) . Let $\Gamma$ be the circumcircle of $\vartriangle BOC$ . Segment $OH$ untersects $\Gamma$ at point $P$. Extension of $AO$ intersects $\Gamma$ at point $K$. If $AP \perp OH$, prove that $PK$ bisects $BC$. [img]https://cdn.artofproblemsolving.com/attachments/a/b/267053569c41692f47d8f4faf2a31ebb4f4efd.png[/img]

Let $n$ be a fixed positive integer. Find all the positive integers $m$ such that $$\frac{m^2+4m}{a_1}+\frac{m^2+8m}{a_1+a_2}+\frac{m^2+12m}{a_1+a_2+a_3}+...+\frac{m^2+4nm}{a_1+a_2+...+a_n}<2500 \left(\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_n}\right)$$ for any positive numbers $a_1,a_2,...,a_n$. Justify your answer.

Three prime numbers $p,q,r$ and a positive integer $n$ are given such that the numbers \[ \frac{p+n}{qr}, \frac{q+n}{rp}, \frac{r+n}{pq} \] are integers. Prove that $p=q=r $. [i]Nazar Agakhanov[/i]

[u]Round 1[/u] [b]p1. [/b] What is the sum of the digits in the binary representation of $2023$? [b]p2.[/b] Jack is buying fruits at the EMCCmart. Three apples and two bananas cost $\$11.00$. Five apples and four bananas cost $\$19.00$. In cents, how much more does an apple cost than a banana? [b]p3.[/b] Define $a \sim b$ as $a! - ab$. What is $(4 \sim 5) \sim (5 \sim (3 \sim 1))$? [u] Round 2[/u] [b]p4.[/b] Alan has $24$ socks in his drawer. Of these socks, $4$ are red, $8$ are blue, and $12$ are green. Alan takes out socks one at a time from his drawer at random. What is the minimum number of socks he must pull out to guarantee that the number of green socks is at least twice the number of red socks? [b]p5.[/b] What is the remainder when the square of the $24$th smallest prime number is divided by $24$? [b]p6.[/b] A cube and a sphere have the same volume. If $k$ is the ratio of the length of the longest diagonal of the cube to the diameter of the sphere, find $k^6$. [u]Round 3[/u] [b]p7.[/b] Equilateral triangle $ABC$ has side length $3\sqrt3$. Point $D$ is drawn such that $BD$ is tangent to the circumcircle of triangle $ABC$ and $BD = 4$. Find the distance from the circumcenter of triangle $ABC$ to $D$. [b]p8.[/b] If $\frac{2023!}{2^k}$ is an odd integer for an integer $k$, what is the value of $k$? [b]p9.[/b] Let $S$ be a set of 6 distinct positive integers. If the sum of the three smallest elements of $S$ is $8$, and the sum of the three largest elements of $S$ is $19$, find the product of the elements in $S$. [u]Round 4[/u] [b]p10.[/b] For some integers $b$, the number $1 + 2b + 3b^2 + 4b^3 + 5b^4$ is divisible by $b + 1$. Find the largest possible value of $b$. [b]p11.[/b] Let $a, b, c$ be the roots of cubic equation $x^3 + 7x^2 + 8x + 1$. Find $a^2 + b^2 + c^2 + \frac{1}{a} + \frac{1}{b} + \frac{1}{c}$ [b]p12.[/b] Let $C$ be the set of real numbers $c$ such that there are exactly two integers n satisfying $2c < n < 3c$. Find the expected value of a number chosen uniformly at random from $C$. PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h3131590p28370327]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In triangle $ABC$, $AB=AC$, $|BC|$ and $d(A,BC)$ are both integers. Then, $\sin A$ and $\cos A$ $\text{(A)}$ one is a rational number while the other is not $\text{(B)}$ both are rational numbers $\text{(C)}$ neither is rational number $\text{(D)}$ not sure

Find the least value of $k$ such that for all $a,b,c,d \in \mathbb{R}$ the inequality \[ \begin{array} c \sqrt{(a^2+1)(b^2+1)(c^2+1)} +\sqrt{(b^2+1)(c^2+1)(d^2+1)} +\sqrt{(c^2+1)(d^2+1)(a^2+1)} +\sqrt{(d^2+1)(a^2+1)(b^2+1)} \\ \ \\ \ge 2( ab+bc+cd+da+ac+bd)-k \end{array}\] holds.

$(4^{-1} - 3^{-1})^{-1} =$ $ \textbf{(A)}\ -12\qquad\textbf{(B)}\ -1\qquad\textbf{(C)}\ \frac{1}{12}\qquad\textbf{(D)}\ 1\qquad\textbf{(E)}\ 12 $

A clock chimes once at $ 30$ minutes past each hour and chimes on the hour according to the hour. For example, at 1 PM there is one chime and at noon and midnight there are twelve chimes. Starting at 11:15 AM on February $ 26$, $ 2003$, on what date will the $ 2003^{\text{rd}}$ chime occur? $ \textbf{(A)}\ \text{March 8} \qquad \textbf{(B)}\ \text{March 9} \qquad \textbf{(C)}\ \text{March 10} \qquad \textbf{(D)}\ \text{March 20} \qquad \textbf{(E)}\ \text{March 21}$

Let $M$ be a set of $n \ge 4$ points in the plane, no three of which are collinear. Initially these points are connected with $n$ segments so that each point in $M$ is the endpoint of exactly two segments. Then, at each step, one may choose two segments $AB$ and $CD$ sharing a common interior point and replace them by the segments $AC$ and $BD$ if none of them is present at this moment. Prove that it is impossible to perform $n^3 /4$ or more such moves. [i]Proposed by Vladislav Volkov, Russia[/i]

You want to color a flag like the one shown in the following image, for which four different colors are available. Two regions of the flag that share a side (or a segment of a side) must have different colors. The flag cannot be flipped, rotated, or reflected. How many different flags can be colored with these conditions? [img]https://cdn.artofproblemsolving.com/attachments/4/9/879d1e144acdbc63ee2ffe34cf13a920d5d836.png[/img]

Let $p$ be a prime number and the decimal notation of $\frac{1}{p}$ is periodical with a length of the period $4k$, $\frac{1}{p}=0,a_1 a_2…a_{4k} a_1 a_2…a_{4k}…$ .Prove that $a_1+a_3+...+a_{4k-1}=a_2+a_4+...+a_{4k}$.

Let $A$ be the product of all positive integers less than $1000$ whose ones or hundreds digit is $7$. Compute the remainder when $A/101$ is divided by $101$.

Let $ABC$ be a triangle such that $CA \neq CB$, the points $A_{1}$ and $B_{1}$ are tangency points for the ex-circles relative to sides $CB$ and $CA$, respectively, and $I$ the incircle. The line $CI$ intersects the cincumcircle of the triangle $ABC$ in the point $P$. The line that trough $P$ that is perpendicular to $CP$, intersects the line $AB$ in $Q$. Prove that the lines $QI$ and $A_{1}B_{1}$ are parallels.

Let $ABC$ be a triangle with $\angle ABC$ obtuse. The [i]$A$-excircle[/i] is a circle in the exterior of $\triangle ABC$ that is tangent to side $BC$ of the triangle and tangent to the extensions of the other two sides. Let $E$, $F$ be the feet of the altitudes from $B$ and $C$ to lines $AC$ and $AB$, respectively. Can line $EF$ be tangent to the $A$-excircle? [i]Proposed by Ankan Bhattacharya, Zack Chroman, and Anant Mudgal[/i]

Kelvin the Frog lives in the 2-D plane. Each day, he picks a uniformly random direction (i.e. a uniformly random bearing $\theta\in [0,2\pi]$) and jumps a mile in that direction. Let $D$ be the number of miles Kelvin is away from his starting point after ten days. Determine the expected value of $D^4$.

The opposite sides of the reentrant hexagon $AFBDCE$ intersect at the points $K,L,M$ (as shown in the figure). It is given that $AL = AM = a, BM = BK = b$, $CK = CL = c, LD = DM = d, ME = EK = e, FK = FL = f$. [img]http://imgur.com/LUFUh.png[/img] $(a)$ Given length $a$ and the three angles $\alpha, \beta$ and $\gamma$ at the vertices $A, B,$ and $C,$ respectively, satisfying the condition $\alpha+\beta+\gamma<180^{\circ}$, show that all the angles and sides of the hexagon are thereby uniquely determined. $(b)$ Prove that \[\frac{1}{a}+\frac{1}{c}=\frac{1}{b}+\frac{1}{d}\] Easier version of $(b)$. Prove that \[(a + f)(b + d)(c + e)= (a + e)(b + f)(c + d)\]

The [i]cross[/i] of a convex $n$-gon is the quadratic mean of the lengths between the possible pairs of vertices. For example, the cross of a $3 \times 4$ rectangle is $\sqrt{ \dfrac{3^2 + 3^2 + 4^2 + 4^2 + 5^2 + 5^2}{6} } = \dfrac{5}{3} \sqrt{6}$. Suppose $S$ is a dodecagon ($12$-gon) inscribed in a unit circle. Find the greatest possible cross of $S$.

Let $\omega,\omega_1,\omega_2$ be three mutually tangent circles such that $\omega_1,\omega_2$ are externally tangent at $P$, $\omega_1,\omega$ are internally tangent at $A$, and $\omega,\omega_2$ are internally tangent at $B$. Let $O,O_1,O_2$ be the centers of $\omega,\omega_1,\omega_2$, respectively. Given that $X$ is the foot of the perpendicular from $P$ to $AB$, prove that $\angle{O_1XP}=\angle{O_2XP}$. [i]David Yang.[/i]

Prove that : \[\int_1^{\sqrt{e}} (\ln x)^n dx=(-1)^{n-1}n!+\sqrt{e}\sum_{m=0}^{n} (-1)^{n-m}\frac{n!}{m!}\left(\frac 12\right)^m\ (n=1,\ 2,\ \cdots)\] [i]2010 Miyazaki University entrance exam/Medicine[/i]

Let $ABC$ be an acute triangle with orthocenter $H$ and circumcenter $O$. The perpendicular bisector of segment $CH$ intersects the sides $AC$ and $BC$ in points $X$ and $Y$ , respectively. The lines $XO$ and $YO$ intersect the side $AB$ in points $P$ and $Q$, respectively. Prove that if $XP + Y Q = AB + XY$ then $\angle OHC = 90^o$.

Two disjoint circles are inscribed in an angle with vertex $A$, whose measure is equal to $a$. The distance between their centers is $d$. A straight line tangent to both circles and not passing through $A$ intersects the sides of the angle at points $B$ and $C$. Find the radius of the circle circumscribed about triangle $ABC$.

Let $p(x, y)$ and $q(x, y)$ be polynomials in two variables such that for $x \ge 0, y \ge 0$ the following conditions hold: $(i) p(x, y)$ and $q(x, y)$ are increasing functions of $x$ for every fixed $y$. $(ii) p(x, y)$ is an increasing and $q(x)$ is a decreasing function of $y$ for every fixed $x$. $(iii) p(x, 0) = q(x, 0)$ for every $x$ and $p(0, 0) = 0$. Show that the simultaneous equations $p(x, y) = a, q(x, y) = b$ have a unique solution in the set $x \ge 0, y \ge 0$ for all $a, b$ satisfying $0 \le b \le a$ but lack a solution in the same set if $a < b$.

Sergei skated around an ice rink, gliding along different paths. The gray lines in the figures below show four of the paths labeled $P$, $Q$, $R$, and $S$. What is the sorted order of the four paths from shortest to longest? [center][img]https://wiki-images.artofproblemsolving.com/9/94/2024_AMC_8_Problem_6.png[/img][/center] $\textbf{(A) }\text{P, Q, R, S}\qquad\textbf{(B) }\text{P, R, S, Q}\qquad\textbf{(C) }\text{Q, S, P, R}\qquad\textbf{(D) }\text{R, P, S, Q}\qquad\textbf{(E) }\text{R, S, P, Q}$