Let $SABC$ be a pyramid with right angles at the vertex $S$. Points $A', B', C'$ lie on the edges $SA, SB, SC$ respectively in such a way that the triangles $ABC$ and $A'B'C'$ are similar. Does this yield that the planes $ABC$ and $A'B'C'$ are parallel?

The teacher wrote $5$ distinct real numbers on the board. After this, Petryk calculated the sums of each pair of these numbers and wrote them on the left part of the board, and Vasyl calculated the sums of each triple of these numbers and wrote them on the left part of the board (each of them wrote $10$ numbers). Could the multisets of numbers written by Petryk and Vasyl be identical?

Let $m$ be a nonnegative integer. Show that the number of tilings of a $(2m + 2) \times (2m + 2)$ grid of squares by $1 \times 2$ or $2 \times 1$ rectangles is at least $$2 \cdot 2^{\frac{5}{2}m} \cdot 5120^{\frac{1}{8}m^2}.$$ [i]Proposed by Milan Haiman[/i]

Given a quadrilateral $ABCD$ simultaneously inscribed and circumscribed, assume that none of its diagonals or sides is a diameter of the circumscribed circle. Let $P$ be the intersection point of the external bisectors of the angles near $A$ and $B$. Similarly, let $Q$ be the intersection point of the external bisectors of the angles $C$ and $D$. If $J$ and $O$ respectively are the incenter and circumcenter of $ABCD$ prove that $OJ\perp PQ$.

Andrew, Benji, and Carlson want to split a pile of $5$ indistinguishable left shoes and $7$ indistinguishable right shoes. Andrew is practical and wants the same number of left and right shoes. Benji is greedy and wants the most shoes out of the three of them. Carlson is a trickster and wants Benji to have a different number of left and right shoes. How many ways are there to split up the shoes in a way that suits everyone’s desires?

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]