For what positive integers $k$ there exists a function $f : N \to N$ such that for all $n \in N$ we have $\underbrace{\hbox{f(f(... f(n)....))}}_{\hbox{k times}} = f(n) + 2$ ?

Let $a,b,c$ be positive real numbers with sum $6$. Find the maximum value of \[S = \sqrt[3]{{{a^2} + 2bc}} + \sqrt[3]{{{b^2} + 2ca}} + \sqrt[3]{{{c^2} + 2ab}}.\]

Let $\mathcal{F}$ be the set of polynomials $f(x)$ with integer coefficients for which there exists an integer root of the equation $f(x)=1$. For all $k>1$, let $m_k$ be the smallest integer greater than one for which there exists $f(x)\in \mathcal{F}$ such that $f(x)=m_k$ has exactly $k$ distinct integer roots. If the value of $\sqrt{m_{2021}-m_{2020}}$ can be written as $m\sqrt{n}$ for positive integers $m,n$ where $n$ is squarefree, compute the largest integer value of $k$ such that $2^k$ divides $\frac{m}{n}$.

Coins of diameter $1$ have been placed on a square of side $11$, without overlapping or protruding from the square. Can there be $126$ coins? and $127$? and $128$?

Let $a, b, c$ be reals and consider three lines $y=ax+b, y=bx+c, y=cx+a$. Two of these lines meet at a point with $x$-coordinate $1$. Show that the third one passes through a point with two integer coordinates.

Define the numbers $a_0, a_1, \ldots, a_n$ in the following way: \[ a_0 = \frac{1}{2}, \quad a_{k+1} = a_k + \frac{a^2_k}{n} \quad (n > 1, k = 0,1, \ldots, n-1). \] Prove that \[ 1 - \frac{1}{n} < a_n < 1.\]

Let $f(x)$ be a polynomial of degree $n$ such that $f(x)^{p}$ is divisible by $f'(x)^{q}$ for some positive integers $p,q$. Prove that $f(x)$ is divisible by $f'(x)$ and that $f(x)$ has a single root of multiplicity $n$.

$(CZS 2)$ Let $p$ be a prime odd number. Is it possible to find $p-1$ natural numbers $n + 1, n + 2, . . . , n + p -1$ such that the sum of the squares of these numbers is divisible by the sum of these numbers?

Moor and nine friends are seated around a circular table. Moor starts out holding a bottle, and whoever holds the bottle passes it to the person on his left or right with equal probability until everyone has held the bottle. Compute the expected distance between Moor and the last person to receive the bottle, where distance is the fewest number of times the bottle needs to be passed in order to go back to Moor.

The target below is made up of concentric circles with diameters $4$, $8$, $12$, $16$, and $20$. The area of the dark region is $n\pi$. Find $n$. [asy] size(150); defaultpen(linewidth(0.8)); int i; for(i=5;i>=1;i=i-1) { if (floor(i/2)==i/2) { filldraw(circle(origin,4*i),white); } else { filldraw(circle(origin,4*i),red); } } [/asy]

The game of [b]Hive [/b]is played on a regular hexagonal grid (as shown in the figure) by 3 players. The grid consists of $k$ layers (where $k$ is a natural number) surrounding a regular hexagon, with each layer constructed around the previous layer. The figure below shows a grid with 2 layers. The players, [i]Ali[/i], [i]Shayan[/i], and [i]Sajad[/i], take turns playing the game. In each turn, a player places a tile, similar to the one shown in the figure, on the empty cells of the grid (rotation of the tile is also allowed). The first player who is unable to place a tile on the grid loses the game. Prove that two players can collaborate in such a way that the third player always loses. Proposed by [size=110]Pouria Mahmoudkhan Shirazi[/size].

Let $P,Q,R,S$ be non constant polynomials with real coefficients, such that $P(Q(x))=R(S(x)) $ and the degree of $P$ is multiple of the degree of $R. $ Prove that there exists a polynomial $T$ with real coefficients such that $$\displaystyle P(x)=R(T(x))$$

Marie repeatedly flips a fair coin and stops after she gets tails for the second time. What is the expected number of times Marie flips the coin?

We have $4$ circles in plane such that any two of them are tangent to each other. we connect the tangency point of two circles to the tangency point of two other circles. Prove that these three lines are concurrent. [i]proposed by Masoud Nourbakhsh[/i]

The closed curve in the figure is made up of $9$ congruent circular arcs each of length $\frac{2\pi}{3}$, where each of the centers of the corresponding circles is among the vertices of a regular hexagon of side $2$. What is the area enclosed by the curve? [asy] size(170); defaultpen(fontsize(6pt)); dotfactor=4; label("$\circ$",(0,1)); label("$\circ$",(0.865,0.5)); label("$\circ$",(-0.865,0.5)); label("$\circ$",(0.865,-0.5)); label("$\circ$",(-0.865,-0.5)); label("$\circ$",(0,-1)); dot((0,1.5)); dot((-0.4325,0.75)); dot((0.4325,0.75)); dot((-0.4325,-0.75)); dot((0.4325,-0.75)); dot((-0.865,0)); dot((0.865,0)); dot((-1.2975,-0.75)); dot((1.2975,-0.75)); draw(Arc((0,1),0.5,210,-30)); draw(Arc((0.865,0.5),0.5,150,270)); draw(Arc((0.865,-0.5),0.5,90,-150)); draw(Arc((0.865,-0.5),0.5,90,-150)); draw(Arc((0,-1),0.5,30,150)); draw(Arc((-0.865,-0.5),0.5,330,90)); draw(Arc((-0.865,0.5),0.5,-90,30)); [/asy] $ \textbf{(A)}\ 2\pi+6\qquad\textbf{(B)}\ 2\pi+4\sqrt3 \qquad\textbf{(C)}\ 3\pi+4 \qquad\textbf{(D)}\ 2\pi+3\sqrt3+2 \qquad\textbf{(E)}\ \pi+6\sqrt3 $

[b]a)[/b] Show that there exists exactly a sequence $ \left( x_n,y_n \right)_{n\ge 0} $ of pairs of nonnegative integers, that satisfy the property that $ \left( 1+\sqrt 33 \right)^n=x_n+y_n\sqrt 33, $ for all nonegative integers $ n. $ [b]b)[/b] Having in mind the sequence from [b]a),[/b] prove that, for any natural prime $ p, $ at least one of the numbers $ y_{p-1} ,y_p $ and $ y_{p+1} $ are divisible by $ p. $

Find all functions $f : \mathbb{R} \to \mathbb{R}$ that satisfy the condition \[f(f(x) + y) = 2x + f(f(y) - x)\quad \text{ for all } x, y \in\mathbb{R}.\]

Let $ABCD$ be a cyclic quadrilateral so that $\overline{AC} \perp \overline{BD}$. Let $E$ be the intersection of $\overline{AC}$ and $\overline{BD}$, and let $F$ be the foot of the altitude from $E$ to $\overline{AB}$. Let $\overline{EF}$ intersect $\overline{CD}$ at $G$, and let the foot of the perpendiculars from $G$ to $\overline{AC}$ and $\overline{BD}$ be $H, I$ respectively. If $\overline{AB} = \sqrt{5}, \overline{BC} = \sqrt{10}, \overline{CD} = 3\sqrt{5}, \overline{DA} = 2\sqrt{10}$, find the length of $\overline{HI}$. [i]Proposed by Timothy Qian[/i]

Numbers $a, b$ and $c$ are positive integers and $\frac{1}{a}+\frac{1}{b}+\frac{ 1}{c}< 1.$ Show that \[\frac{1}{a}+\frac{1}{b}+\frac{ 1}{c}\leq \frac{41}{42}.\]

Fill in each of the ten boxes with a 3-digit number so that the following conditions are satisfied. [list=1] [*]Every number has three distinct digits that sum to $15$. $0$ may not be a leading digit. One digit of each number has been given to you. [*]No two numbers in any pair of boxes use the same three digits. For example, it is not allowed for two different boxes to have the numbers $456$ and $645$. [*]Two boxes joined by an arrow must have two numbers that share an equal hundreds digit, tens digit, or ones digit. Also, the smaller number must point to the larger.[/list] You do not need to prove that your configuration is the only one possible; you merely need to find a configuration that satisfies the constraints above. (Note: In any other USAMTS problem, you need to provide a full proof. Only in this problem is an answer without justification acceptable.) [asy] size(200); defaultpen(linewidth(0.8)); path arrow; pair squares[]={(2,4),(6,4),(10,4),(0,0),(4,0),(8,0),(12,0),(2,-4),(6,-4),(10,-4)}; pair horizarrows[]={(4,4),(2,0),(6,0),(10,0),(4,-4),(8,-4)}; bool isLeft[]={false,false,true,false,false,false}; pair diagarrows[]={(1,2),(7,2),(9,2),(1,-2),(5,-2),(11,-2)}; bool isDown[]={true,false,true,false,false,true}; for(int i=0;i<=9;i=i+1) { draw(box(squares[i]-(1,1),squares[i]+(1,1))); label("$"+(string)i+"$",squares[i]); } for(int j=0;j<=5;j=j+1) { if(isLeft[j]) arrow=(horizarrows[j].x-1,horizarrows[j].y)--(horizarrows[j].x+1,horizarrows[j].y); else arrow=(horizarrows[j].x+1,horizarrows[j].y)--(horizarrows[j].x-1,horizarrows[j].y); draw(arrow,BeginArrow(size=7)); } for(int k=0;k<=5;k=k+1) { if(isDown[k]) arrow=(diagarrows[k].x-1/3,diagarrows[k].y-1)--(diagarrows[k].x+1/3,diagarrows[k].y+1); else arrow=(diagarrows[k].x-1/3,diagarrows[k].y+1)--(diagarrows[k].x+1/3,diagarrows[k].y-1); draw(arrow,BeginArrow(size=7)); } [/asy]

Given the sequence $(u_n)_{n=1}^{\infty}$, where $u_1 = 1, u_2 = 2$, and $u_{n + 2} = u_{n + 1} +u_ n+ \frac{(-1)^n-1}{2}$ for any positive integers $n$. Prove that every positive integers can be expressed as the sum of some distinguished numbers of the sequence of numbers $(u_n)_{n=1}^{\infty}$ Nguyen Duy Thai Son, The University of Danang, Da Nang.

In a convex quadrilateral $ABCD$, $\angle ABC = \angle ADC = 90^\circ$. A point $P$ is chosen from the diagonal $BD$ such that $\angle APB = 2\angle CPD$, points $X$, $Y$ is chosen from the segment $AP$ such that $\angle AXB = 2\angle ADB$, $\angle AYD = 2\angle ABD$. Prove that: $BD = 2XY$.

Find all nonnegative integers $x,y$ such that \[ 2 \cdot 3^{x} +1 = 7 \cdot 5^{y}. \]

Let $\{a_n\}$ be a sequence defined by $a_1=0$ and $$a_n=\frac{1}{n}+\frac{1}{\lceil \frac{n}{2} \rceil}\sum_{k=1}^{\lceil \frac{n}{2} \rceil}a_k$$ for any positive integer $n$. Find the maximal term of this sequence.

Let $P$ be a point chosen on the interior of side $\overline{BC}$ of triangle $\triangle ABC$ with side lengths $\overline{AB} = 10, \overline{BC} = 10, \overline{AC} = 12$. If $X$ and $Y$ are the feet of the perpendiculars from $P$ to the sides $AB$ and $AC$, then the minimum possible value of $PX^2 + PY^2$ can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [i]Proposed by Andrew Wen[/i]