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 $s$? $\textbf{(A) } \frac{\sqrt{2}}{3} \qquad \textbf{(B) } \frac{1}{2} \qquad \textbf{(C) } 2-\sqrt{2} \qquad \textbf{(D) } 1-\frac{\sqrt{2}}{4} \qquad \textbf{(E) } \frac{2}{3}$

a) A castle is equilateral triangle with the side of $100$ metres. It is divided onto $100$ triangle rooms. Each wall between the rooms is $10$ metres long and contain one door. You are inside and are allowed to pass through every door not more than once. Prove that you can visit not more than $91$ room (not exiting the castle). b) Every side of the triangle is divided onto $k$ parts by the lines parallel to the sides. And the triangle is divided onto $k^2$ small triangles. Let us call the "chain" such a sequence of triangles, that every triangle in it is included only once, and the consecutive triangles have the common side. What is the greatest possible number of the triangles in the chain?