Let $k$ be a natural number. For each function $f : \mathbb{N}\to \mathbb{N}$ define the sequence of functions $(f_{m})_{m\geq 1}$ by $f_{1}= f$ and $f_{m+1}= f \circ f_{m}$ for $m \geq 1$ . Function $f$ is called $k$-[i]nice[/i] if for each $n \in\mathbb{N}: f_{k}(n) = f (n)^{k}$. (a) For which $k$ does there exist an injective $k$-nice function $f$ ? (b) For which $k$ does there exist a surjective $k$-nice function $f$ ?

In a rectangular coordinate system we call a horizontal line parallel to the $x$ -axis triangular if it intersects the curve with equation \[y = x^4 + px^3 + qx^2 + rx + s\] in the points $A,B,C$ and $D$ (from left to right) such that the segments $AB, AC$ and $AD$ are the sides of a triangle. Prove that the lines parallel to the $x$ - axis intersecting the curve in four distinct points are all triangular or none of them is triangular.

Let the [i]square decomposition[/i] of a number be defined as the sequence of numbers given by the following algorithm. Given a positive integer $n$, add the largest possible perfect square that is less than or equal to $n$ to a sequence, and then subtract that number from $n$. Repeat as many times as necessary until your current $n$ is $0$. So for example, the square decomposition of $60$ would be $49, 9, 1, 1$. Define the size of a square decomposition to be the number of numbers in the sequence. Say that the maximal size of a square decomposition of a number in the range $[1, 2020]$ is $m$. Find the largest number in the range $[1, 2020]$ that has a square decomposition of size $m$. [i]Proposed by Timothy Qian[/i]

Let $n$ be a positive integer and let $P(x) = x^n + a_{n-1}x^{n-1} +... + a_1x + 1$ be a polynomial with positive real coefficients. Under the assumption that the roots of $P$ are all real, show that $P(x) \ge (x + 1)^n$ for all $x > 0$.

Let $f:\mathbb N\to \mathbb N$. Show that $f(m)+n\mid f(n)+m$ for all positive integers $m\le n$ if and only if $f(m)+n\mid f(n)+m$ for all positive integers $m\ge n$. [i]Proposed by Carl Schildkraut[/i]

Two parallel lines $\ell_1$ and $\ell_2$ lie on a plane, distance $d$ apart. On $\ell_1$ there are an infinite number of points $A_1, A_2, A_3, ...$ , in that order, with $A_nA_{n+1} = 2$ for all $n$. On $\ell_2$ there are an infinite number of points $B_1, B_2, B_3,...$ , in that order and in the same direction, satisfying $B_nB_{n+1} = 1$ for all $n$. Given that $A_1B_1$ is perpendicular to both $\ell_1$ and $\ell_2$, express the sum $\sum_{i=1}^{\infty} \angle A_iB_iA_{i+1}$ in terms of $d$. [img]https://cdn.artofproblemsolving.com/attachments/c/9/24b8000e19cffb401234be010af78a6eb67524.png[/img]

Determine the number of positive integers $a \le 250$ for which the set $\{a+1, a+2, \dots, a+1000\}$ contains $\bullet$ Exactly $333$ multiples of $3$, $\bullet$ Exactly $142$ multiples of $7$, and $\bullet$ Exactly $91$ multiples of $11$. [i]Based on a proposal by Rajiv Movva[/i]

Let $r$ be a given positive integer. Is is true that for every $r$-colouring of the natural numbers there exists a monochromatic solution of the equation $x+y=3z$? (proposed by B. Batbaysgalan, folklore)

What is the smallest positive integer $n$ such that $2016n$ is a perfect cube?

Let $f_n(x)=\underbrace{xx\cdots x}_{n\ \text{times}}$ that is, $f_n(x)$ is a $n$ digit number with all digits $x$, where $x\in \{1,2,\cdots,9\}$. Then which of the following is $\Big(f_n(3)\Big)^2+f_n(2)$? [list=1] [*] $f_n(5)$ [*] $f_{2n}(9)$ [*] $f_{2n}(1)$ [*] None of these [/list]

Prove that there don't exist positive integer numbers $k$ and $n$ which satisfy equation $n^n+(n+1)^{n+1}+(n+2)^{n+2} = 2023^k$. [i]Proposed by Mykhailo Shtandenko[/i]

Prove that there exists infinitely many primes $ p$ such that: \[ 13|p^3\plus{}1\]

In triangle $ABC$, $F$ is on segment $AB$ such that $CF$ bisects $\angle ACB$. Points $D$ and $E$ are on line $CF$ such that lines $AD,BE$ are perpendicular to $CF$. $M$ is the midpoint of $AB$. If $ME=13$, $AD=15$, and $BE=25$, find $AC+CB$. [i]Ray Li[/i]

Two circles $\gamma_1$ and $\gamma_2$ intersect at $A$ and $B$. A line $r$ through $B$ meets $\gamma_1$ at $C$ and $\gamma_2$ at $D$ so that $B$ is between $C$ and $D$. Let $s$ be the line parallel to $AD$ which is tangent to $\gamma_1$ at $E$, at the smaller distance from $AD$. Line $EA$ meets $\gamma_2$ in $F$. Let $t$ be the tangent to $\gamma_2$ at $F$. $(a)$ Prove that $t$ is parallel to $AC$. $(b)$ Prove that the lines $r,s,t$ are concurrent.

Let $s(n)$ denote the number of positive divisors of positive integer $n$. What is the largest prime divisor of the sum of numbers $(s(k))^3$ for all positive divisors $k$ of $2014^{2014}$? $ \textbf{(A)}\ 5 \qquad\textbf{(B)}\ 7 \qquad\textbf{(C)}\ 11 \qquad\textbf{(D)}\ 13 \qquad\textbf{(E)}\ \text{None of the preceding} $

Three squares $BCDE,CAFG$ and $ABHI$ are constructed outside the triangle $ABC$. Let $GCDQ$ and $EBHP$ be parallelograms. Prove that $APQ$ is an isosceles right triangle.

Is there a heptagon and a point $P$ inside it such that any vertex of the heptagon has its distance from $P$ equal to the length of the side opposite the vertex? [i]A side and a vertex are said to be opposite if the side is the fourth from the vertex page (in any direction).[/i]

Let $a$ and $b$ be the roots of the polynomial $x^2+2020x+c$. Given that $\frac{a}{b}+\frac{b}{a}=98$, compute $\sqrt c$.

Let $n$ be a positive integer. Determine the number of sequences $a_0, a_1, \ldots, a_n$ with terms in the set $\{0,1,2,3\}$ such that $$n=a_0+2a_1+2^2a_2+\ldots+2^na_n.$$

Let $n$ be a positive square free integer, $S$ is a subset of $[n]:=\{1,2,\ldots ,n\}$ such that $|S|\ge n/2.$ Prove that there exists three elements $a,b,c\in S$ (can be same), satisfy $ab\equiv c\pmod n.$ [i]Created by Zhenhua Qu[/i]

Prove that for any triangle each exradius is less than four times the circumradius.

$200$ natural numbers are written in a row. For any two adjacent numbers of the row, the right one is either $9$ times greater than the left one, $2$ times smaller than the left one. Can the sum of all these 200 numbers be equal to $24^{2022}$?

Let $O$ be the circumcenter of the triangle $ABC$. Points $D,E$ are on sides $AC,AB$ and points $P,Q,R,S$ are given in plane such that $P,C$ and $R,C$ are on different sides of $AB$ and pints $Q,B$ and $S,B$ are on different sides of $AC$ such that $R,S$ lie on circumcircle of $DAP,EAQ$ and $\triangle BCE \sim \triangle ADQ , \triangle CBD \sim \triangle AEP$(In that order), $\angle ARE=\angle ASD=\angle BAC$, If $RS\| PQ$ prove that $RE ,DS$ are concurrent on $AO$. [i]Proposed by Alireza Dadgarnia[/i]

If $ a$, $ b$, $ c$, $ d$, and $ e$ are constants such that every $ x > 0$ satisfies \[ \frac{5x^4 \minus{} 8x^3 \plus{} 2x^2 \plus{} 4x \plus{} 7}{(x \plus{} 2)^4} \equal{} a \plus{} \frac{b}{x \plus{} 2} \plus{} \frac{c}{(x \plus{} 2)^2} \plus{} \frac{d}{(x \plus{} 2)^3} \plus{} \frac{e}{(x \plus{} 2)^4} \, ,\] then what is the value of $ a \plus{} b \plus{} c \plus{} d \plus{} e$?

Square $ABCD$ has side length $1$. Point $P$, $Q$, $R$, and $S$ each lie on a side of $ABCD$ such that $APQCRS$ is an equilateral convex hexagon with side length $s$. What is $s$? $\textbf{(A) } \frac{\sqrt{2}}{3} \qquad \textbf{(B) } \frac{1}{2} \qquad \textbf{(C) } 2-\sqrt{2} \qquad \textbf{(D) } 1-\frac{\sqrt{2}}{4} \qquad \textbf{(E) } \frac{2}{3}$