$32$ stones, with pairwise different weights, and lever scales without weights are given. How to determine by $35$ scaling, which stone is the heaviest and which is the second by weight?

A positive integer $b$ and a sequence $a_0,a_1,a_2,\dots$ of integers $0\le a_i<b$ is given. It is known that $a_0\neq 0$ and the sequence $\{a_i\}$ is eventually periodic but has infinitely many nonzero terms. Let $S$ be the set of positive integers $n$ so that $n\mid (a_0a_1\dots a_n)_b$. Given that $S$ is infinite, show that there are infinitely many primes that divide at least one element of $S$. [i]Proposed by Carl Schildkraut and Holden Mui[/i]

Fiona has a deck of cards labelled $1$ to $n$, laid out in a row on the table in order from $1$ to $n$ from left to right. Her goal is to arrange them in a single pile, through a series of steps of the following form: [list] [*]If at some stage the cards are in $m$ piles, she chooses $1\leq k<m$ and arranges the cards into $k$ piles by picking up pile $k+1$ and putting it on pile $1$; picking up pile $k+2$ and putting it on pile $2$; and so on, working from left to right and cycling back through as necessary. [/list] She repeats the process until the cards are in a single pile, and then stops. So for example, if $n=7$ and she chooses $k=3$ at the first step she would have the following three piles: $ \begin{matrix} 7 & \ &\ \\ 4 & 5 & 6 \\ 1 &2 & 3 \\ \hline \end{matrix} $ If she then chooses $k=1$ at the second stop, she finishes with the cards in a single pile with cards ordered $6352741$ from top to bottom. How many different final piles can Fiona end up with?

$a,b,c$ are positive real numbers such that $a+b+c\ge 3$ and $a^2+b^2+c^2=2abc+1$. Prove that $$a+b+c\le 2\sqrt{abc}+1$$

On side $AB$ of triangle $ABC$, point $M$ is selected. A straight line passing through $M$ intersects the segment $AC$ at point $N$ and the ray $CB$ at point $K$. The circumscribed circle of the triangle $AMN$ intersects $\omega$, the circumscribed circle of the triangle $ABC$, at points $A$ and $S$. Straight lines $SM$ and $SK$ intersect with $\omega$ for the second time at points $P$ and $Q$, respectively. Prove that $AC = PQ$.

Let $ f(x)\equal{}\left(1\plus{}\frac{1}{x}\right)^{x}\ (x>0)$. Find $ \lim_{n\to\infty}\left\{f\left(\frac{1}{n}\right)f\left(\frac{2}{n}\right)f\left(\frac{3}{n}\right)\cdots\cdots f\left(\frac{n}{n}\right)\right\}^{\frac{1}{n}}$.

Simon expands factored polynomials with his favorite AI, ChatSFFT. However, he has not paid for a premium ChatSFFT account, so when he goes to expand $(m - a)(n - b),$ where $a, b, m, n$ are integers, ChatSFFT returns the sum of the two factors instead of the product. However, when Simon plugs in certain pairs of integer values for $m$ and $n,$ he realizes that the value of ChatSFFT’s result is the same as the real result in terms of $a$ and $b$. How many such pairs are there?

The incircle of the acute-angled triangle $ABC$ is tangent to the sides $AB, BC, CA$ at points $C_1, A_1, B_1,$ respectively, and $I$ is the incenter. Let the midpoint of side $BC$ be $M.$ Let $J$ be the foot of the altitude drawn from $M$ to $C_1B_1.$ The tangent drawn from $B$ to the circumcircle of $\triangle BIC$ intersects $IJ$ at $X.$ If the circumcircle of $\triangle AXI$ intersects $AB$ at $Y,$ prove that $BY = BM.$

Can a set of six numbers $\left\{a, b,c, \frac{a^2}{b} , \frac{b^2}{c} , \frac{c^2}{a} \right\}$ , where $a, b, c$ positive numbers, turn out to be exactly exactly three different numbers?

Given real numbers $x_i,y_i (i=1,2,\ldots ,n)$, let $A$ be the $n\times n$ matrix given by $a_{ij}=1$ if $x_i\ge y_j$ and $a_{ij}=0$ otherwise. Suppose $B$ is a $n\times n$ matrix whose entries are $0$ and $1$ such that the sum of entries in any row or column of $B$ equals the sum of entries in the corresponding row or column of $A$. Prove that $B=A$.

Let $ABC$ be a triangle with $AB=13$, $BC=14$, and $AC=15$. Denote by $\omega$ its incircle. A line $\ell$ tangent to $\omega$ intersects $\overline{AB}$ and $\overline{AC}$ at $X$ and $Y$ respectively. Suppose $XY=5$. Compute the positive difference between the lengths of $\overline{AX}$ and $\overline{AY}$.

Let $ABC$ be a triangle such that $AC$ is its shortest side. A point $P$ is inside it and satisfies that $BP = AC$. Let $R$ be the midpoint of $BC$ and let $M$ be the midpoint of $AP$. Let $E$ be the intersection of $BP$ and $AC$. Prove that the bisector of angle $\angle BE A$ is perpendicular to segment $MR$.

In the set of equations $ z^x \equal{} y^{2x}, 2^z \equal{} 2\cdot4^x, x \plus{} y \plus{} z \equal{} 16$, the integral roots in the order $ x,y,z$ are: $ \textbf{(A)}\ 3,4,9 \qquad\textbf{(B)}\ 9, \minus{} 5 \minus{} ,12 \qquad\textbf{(C)}\ 12, \minus{} 5,9 \qquad\textbf{(D)}\ 4,3,9 \qquad\textbf{(E)}\ 4,9,3$

A crate of toys remains at rest on a sleigh as the sleigh is pulled up a hill with an increasing speed. The crate is not fastened down to the sleigh. What force is responsible for the crate’s increase in speed up the hill? $\textbf{(A)} \ \text{the force of static friction of the sleigh on the crate}$ $ \textbf{(B)} \ \text{the contact force (normal force) of the ground on the sleigh}$ $ \textbf{(C)} \ \text{the contact force (normal force) of the sleigh on the crate}$ $ \textbf{(D)} \ \text{the gravitational force acting on the sleigh}$ $ \textbf{(E)} \ \text{no force is needed}$

Let $a,b,c$ be positive reals such that $a+b+c=3$. Prove the inequality \[\frac{a^2}{a+b^2}+\frac{b^2}{b+c^2}+\frac{c^2}{c+a^2}\geq \frac{3}{2}.\]

Is it possible to colour all integers greater than $1{}$ in three colours (each integer in one colour, all three colours must be used) so that the colour of the product of any two differently coloured numbers is different from the colour of each of the factors?

Let $x,y,z$ be distinct positive integers such that $(y+z)(z+x)=(x+y)^2$ . Show that \[x^2+y^2>8(x+y)+2(xy+1).\] (Paolo Leonetti)

Let $n$ be a positive integer. Let $T_n$ be a set of positive integers such that: \[{T_n={ \{11(k+h)+10(n^k+n^h)| (1 \leq k,h \leq 10)}}\}\] Find all $n$ for which there don't exist two distinct positive integers $a, b \in T_n$ such that $a\equiv b \pmod{110}$

There is a unique continuous function $f$ over the positive real numbers satisfying $f(4) = 1$ and \[ 9 - (f(x))^4 = \frac{x^2}{(f(x))^2} - 2xf(x) \] for all positive $x$. Compute the value of $\int_{0}^{140} (f(x))^3\,dx$.

Consider a $(2m-1)\times(2n-1)$ rectangular region, where $m$ and $n$ are integers such that $m,n\ge 4.$ The region is to be tiled using tiles of the two types shown: \[ \begin{picture}(140,40) \put(0,0){\line(0,1){40}} \put(0,0){\line(1,0){20}} \put(0,40){\line(1,0){40}} \put(20,0){\line(0,1){20}} \put(20,20){\line(1,0){20}} \put(40,20){\line(0,1){20}} \multiput(0,20)(5,0){4}{\line(1,0){3}} \multiput(20,20)(0,5){4}{\line(0,1){3}} \put(80,0){\line(1,0){40}} \put(120,0){\line(0,1){20}} \put(120,20){\line(1,0){20}} \put(140,20){\line(0,1){20}} \put(80,0){\line(0,1){20}} \put(80,20){\line(1,0){20}} \put(100,20){\line(0,1){20}} \put(100,40){\line(1,0){40}} \multiput(100,0)(0,5){4}{\line(0,1){3}} \multiput(100,20)(5,0){4}{\line(1,0){3}} \multiput(120,20)(0,5){4}{\line(0,1){3}} \end{picture} \] (The dotted lines divide the tiles into $1\times 1$ squares.) The tiles may be rotated and reflected, as long as their sides are parallel to the sides of the rectangular region. They must all fit within the region, and they must cover it completely without overlapping. What is the minimum number of tiles required to tile the region?

Find the smallest constant $C > 0$ for which the following statement holds: among any five positive real numbers $a_1,a_2,a_3,a_4,a_5$ (not necessarily distinct), one can always choose distinct subscripts $i,j,k,l$ such that \[ \left| \frac{a_i}{a_j} - \frac {a_k}{a_l} \right| \le C. \]

For a table $n \times 9$ ($n$ rows and $9$ columns), determine the maximum of $n$ that we can write one number in the set $\left\{ {1,2,...,9} \right\}$ in each cell such that these conditions are satisfied: 1. Each row contains enough $9$ numbers of the set $\left\{ {1,2,...,9} \right\}$. 2. Any two rows are distinct. 3. For any two rows, we can find at least one column such that the two intersecting cells between it and the two rows contain the same number.

For positive integers $x$, let$$f(x)=\begin{cases} \frac{f\left(\frac{x}{2}\right)}{2} &\mbox{if }x\mbox{ is even,} \\ 2^{-x} &\mbox{if }x\mbox{ is odd.} \end{cases}$$Find $f(1)+f(2)+f(3)+\dots$.

Determine all triples of real numbers $(x, y, z)$ which satisfy the following system of equations: \[ \begin{cases} x+y+z=0 \\ x^3+y^3+z^3 = 90 \\ x^5+y^5+z^5=2850. \end{cases} \]

In the triangle ABC, D is the foot of the altitude from A to BC, and M is the midpoint of the line segment BC. The three angles ∠BAD, ∠DAM and ∠MAC are all equal. Find the angles of the triangle ABC.