How many positive integers divide the number $10! = 1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8 \times 9 \times 10$ ?

Positive numbers $x,y$ satisfy equality $$x^3+y^3=x-y$$ Prove that $$x^2+y^2<1$$

The incircle of the triangle $ABC$ touches the sides $AC$ and $BC$ at points $K$ and $L$, respectively. the $B$-excircle touches the side $AC$ of this triangle at point $P$. The segment $AL$ intersects the inscribed circle for the second time at point $S$. Line $KL$ intersects the circumscribed circle of triangle $ASK$ for the second at point $M$. Prove that $PL = PM$.

Prove that, for each odd integer $n \ge 5$, the number $1^n+2^n+...+15^n$ is divisible by $480$.

Let $n$ and $c$ be coprime positive integers. For any integer $i$, denote by $i' $ the remainder of division of product $ci$ by $n$. Let $A_o.A_1,A_2,...,A_{n-1}$ be a regular $n$-gon. Prove that a) if $A_iA_j \parallel A_kA_i$ then $A_{i'}A_{j'} \parallel A_{k'}A_{i'}$ b) if $A_iA_j \perp A_kA_l$ then $A_{i'}A_{j'} \perp A_{k'}A_{l'}$

For a positive integer $n$, we say an $n$-[i]shuffling[/i] is a bijection $\sigma: \{1,2, \dots , n\} \rightarrow \{1,2, \dots , n\}$ such that there exist exactly two elements $i$ of $\{1,2, \dots , n\}$ such that $\sigma(i) \neq i$. Fix some three pairwise distinct $n$-shufflings $\sigma_1,\sigma_2,\sigma_3$. Let $q$ be any prime, and let $\mathbb{F}_q$ be the integers modulo $q$. Consider all functions $f:(\mathbb{F}_q^n)^n\to\mathbb{F}_q$ that satisfy, for all integers $i$ with $1 \leq i \leq n$ and all $x_1,\ldots x_{i-1},x_{i+1}, \dots ,x_n, y, z\in\mathbb{F}_q^n$, \[f(x_1, \ldots ,x_{i-1}, y, x_{i+1}, \ldots , x_n) +f(x_1, \ldots ,x_{i-1}, z, x_{i+1}, \ldots , x_n) = f(x_1, \ldots ,x_{i-1}, y+z, x_{i+1}, \ldots , x_n), \] and that satisfy, for all $x_1,\ldots,x_n\in\mathbb{F}_q^n$ and all $\sigma\in\{\sigma_1,\sigma_2,\sigma_3\}$, \[f(x_1,\ldots,x_n)=-f(x_{\sigma(1)},\ldots,x_{\sigma(n)}).\] For a given tuple $(x_1,\ldots,x_n)\in(\mathbb{F}_q^n)^n$, let $g(x_1,\ldots,x_n)$ be the number of different values of $f(x_1,\ldots,x_n)$ over all possible functions $f$ satisfying the above conditions. Pick $(x_1,\ldots,x_n)\in(\mathbb{F}_q^n)^n$ uniformly at random, and let $\varepsilon(q,\sigma_1,\sigma_2,\sigma_3)$ be the expected value of $g(x_1,\ldots,x_n)$. Finally, let \[\kappa(\sigma_1,\sigma_2,\sigma_3)=-\lim_{q \to \infty}\log_q\left(-\ln\left(\frac{\varepsilon(q,\sigma_1,\sigma_2,\sigma_3)-1}{q-1}\right)\right).\] Pick three pairwise distinct $n$-shufflings $\sigma_1,\sigma_2,\sigma_3$ uniformly at random from the set of all $n$-shufflings. Let $\pi(n)$ denote the expected value of $\kappa(\sigma_1,\sigma_2,\sigma_3)$. Suppose that $p(x)$ and $q(x)$ are polynomials with real coefficients such that $q(-3) \neq 0$ and such that $\pi(n)=\frac{p(n)}{q(n)}$ for infinitely many positive integers $n$. Compute $\frac{p\left(-3\right)}{q\left(-3\right)}$.

Let $O$ be the circumcentre, and $\Omega$ be the circumcircle of an acute-angled triangle $ABC$. Let $P$ be an arbitrary point on $\Omega$, distinct from $A$, $B$, $C$, and their antipodes in $\Omega$. Denote the circumcentres of the triangles $AOP$, $BOP$, and $COP$ by $O_A$, $O_B$, and $O_C$, respectively. The lines $\ell_A$, $\ell_B$, $\ell_C$ perpendicular to $BC$, $CA$, and $AB$ pass through $O_A$, $O_B$, and $O_C$, respectively. Prove that the circumcircle of triangle formed by $\ell_A$, $\ell_B$, and $\ell_C$ is tangent to the line $OP$.

The English alphabet, which has $26$ letters, is randomly permuted. Let $p_1$ be the probability that $\text{AB},$ $\text{CD},$ and $\text{EF}$ all appear as contiguous substrings. Let $p_2$ be the probability that $\text{ABC}$ and $\text{DEF}$ both appear as contiguous substrings. Compute $\tfrac{p_1}{p_2}.$

In the diagram below, the three circles and the three line segments are tangent as shown. Given that the radius of all of the three circles is $1$, compute the area of the triangle. [img]https://cdn.artofproblemsolving.com/attachments/b/e/8af4ea38d9a4c675edd0957aaa5336caec0ae2.png[/img]

Find all polynomials $p(x)\in\mathbb{R}[x]$ such that for all $x\in \mathbb{R}$: $p(5x)^2-3=p(5x^2+1)$ such that: $a) p(0)\neq 0$ $b) p(0)=0$

Are there any integers $a,b$ and $c$, not all of them $0$, such that $$a^2=2021b^2+2022c^2~~?$$ [i] (Cosmin Gavrilă)[/i]

Tanya wrote numbers in forms $n^7-1$ for $n=2,3,...$ and noticed that for $n=8$ she got number divisible by $337$. For what minimal $n$ did she get number divisible by $2022$?

Let $N$ be a positive integer. Define a sequence $a_0,a_1,\ldots$ by $a_0=0$, $a_1=1$, and $a_{n+1}+a_{n-1}=a_n(2-1/N)$ for $n\ge1$. Prove that $a_n<\sqrt{N+1}$ for all $n$. [i]Evan O'Dorney.[/i]

An object starting from rest can roll without slipping down an incline. Which of the following four objects, each with mass $M$ and radius $R$, would have the largest acceleration down the incline? A) a uniform solid sphere B) a uniform solid disk C) a hollow spherical shell D) a hoop E) All objects would have the same acceleration.

Let $n$ be a positive integer. Find all polynomials $Q(x)$ with integer coefficients so that the degree of $Q(x)$ is less than $n$ and there exists an integer $m\geq 1$ for which \[x^n-1\mid Q(x)^m-1\]

The smallest number greater than 2 that leaves a remainder of 2 when divided by 3, 4, 5, or 6 lies between what numbers? $\textbf{(A)}\hspace{.05in}40\text{ and }50 \qquad \textbf{(B)}\hspace{.05in}51\text{ and }55 \qquad \textbf{(C)}\hspace{.05in}56\text{ and }60 \qquad \textbf{(D)}\hspace{.05in} \text{61 and 65}\qquad \textbf{(E)}\hspace{.05in} \text{66 and 99}$

Show that, in a chessboard, it is possible to traverse to any given square from another given square using a knight. (A knight can move in a chessboard by going two steps in one direction and one step in a perpendicular direction as shown in the given figure)

Let $\mathcal{X}$ be the collection of all non-empty subsets (not necessarily finite) of the positive integer set $\mathbb{N}$. Determine all functions $f: \mathcal{X} \to \mathbb{R}^+$ satisfying the following properties: (i) For all $S$, $T \in \mathcal{X}$ with $S\subseteq T$, there holds $f(T) \le f(S)$. (ii) For all $S$, $T \in \mathcal{X}$, there hold \[f(S) + f(T) \le f(S + T),\quad f(S)f(T) = f(S\cdot T), \] where $S + T = \{s + t\mid s\in S, t\in T\}$ and $S \cdot T = \{s\cdot t\mid s\in S, t\in T\}$. [i]Proposed by Li4, Untro368, and Ming Hsiao.[/i]

Four points in space $A,B,C,D$ satisfy that $|AB|=3,|BC|=7,|CD|=11,|DA|=9$, then the number of values of $\overrightarrow{AC}\cdot\overrightarrow{BD}$ is $\text{(A)}$ Only one. $\text{(B)}$ Two. $\text{(C)}$ Three. $\text{(D)}$ Infinitely many.

Phillip is trying to make a two-dimensional donut, but in a fun way: He is trying to make a donut shaped in a way that the inner circle of the donut is inscribed inside a pentagon, and the outer circle of the donut circumscribes the same pentagon. This pentagon has a side length of $6$. The area of Phillip's donut is of the form $a \pi$. Find $a$. (Note that $\sin 54^o= \frac{\sqrt5+1}{4}$ )

Let $N$ be a positive integer. Initially, a positive integer $A$ is written on the board. At each step, we can perform one of the following two operations with the number written on the board: (i) Add $N$ to the number written on the board and replace that number with the sum obtained; (ii) If the number on the board is greater than $1$ and has at least one digit $1$, then we can remove the digit $1$ from that number, and replace the number initially written with this one (with removal of possible leading zeros) For example, if $N = 63$ and $A = 25$, we can do the following sequence of operations: $$25 \rightarrow 88 \rightarrow 151 \rightarrow 51 \rightarrow 5$$ And if $N = 143$ and $A = 2$, we can do the following sequence of operations: $$2 \rightarrow 145 \rightarrow 288 \rightarrow 431 \rightarrow 574 \rightarrow 717 \rightarrow 860 \rightarrow 1003 \rightarrow 3$$ For what values of $N$ is it always possible, regardless of the initial value of $A$ on the blackboard, to obtain the number $1$ on the blackboard, through a finite number of operations?

Find all triples of three-digit positive integers $x < y < z$ with $x,y,z$ in arithmetic progression and $x, y, z + 1000$ in geometric progression. [i]For this problem, you may use calculators or computers to gain an intuition about how to solve the problem. However, your final submission should include mathematical derivations or proofs and should not be a solution by exhaustive search.[/i]

Alphonse and Beryl play a game involving $n$ safes. Each safe can be opened by a unique key and each key opens a unique safe. Beryl randomly shuffles the $n$ keys, and after placing one key inside each safe, she locks all of the safes with her master key. Alphonse then selects $m$ of the safes (where $m < n$), and Beryl uses her master key to open just the safes that Alphonse selected. Alphonse collects all of the keys inside these $m$ safes and tries to use these keys to open up the other $n - m$ safes. If he can open a safe with one of the $m$ keys, he can then use the key in that safe to try to open any of the remaining safes, repeating the process until Alphonse successfully opens all of the safes, or cannot open any more. Let $P_m(n)$ be the probability that Alphonse can eventually open all $n$ safes starting from his initial selection of $m$ keys. (a) Show that $P_2(3) = \frac23$. (b) Prove that $P_1(n) = \frac1n$. (c) For all integers $n \geq 2$, prove that $$P_2(n) = \frac2n \cdot P_1(n-1) + \frac{n-2}{n} \cdot P_2(n-1).$$ (d) Determine a formula for $P_2 (n)$.

The critical temperature of water is the $ \textbf{(A)}\hspace{.05in}\text{temperature at which solid, liquid and gaseous water coexist} \qquad$ $\textbf{(B)}\hspace{.05in}\text{temperature at which water vapor condenses}\qquad$ $\textbf{(C)}\hspace{.05in}\text{maximum temperature at which liquid water can exist}\qquad$ $\textbf{(D)}\hspace{.05in}\text{minimum temperature at which water vapor can exist}\qquad$

A hexagon that is inscribed in a circle has side lengths $22$, $22$, $20$, $22$, $22$, and $20$ in that order. The radius of the circle can be written as $p+\sqrt{q}$, where $p$ and $q$ are positive integers. Find $p+q$.