Find all pairs of positive integers $(a,b)$ such that $$a!+b!=a^b + b^a.$$

Let $m$ be an positive odd integer not divisible by $3$. Prove that $\left[4^m -(2+\sqrt 2)^m\right]$ is divisible by $112.$

Inside the convex quadrilateral $ABCD$ lies the point $'M$. Reflect it symmetrically with respect to the midpoints of the sides of the quadrilateral and connect the obtained points so that they form a convex quadrilateral. Prove that the area of ​​this quadrilateral does not depend on the choice of the point $M$.

A coin is tossed $10$ times. Compute the probability that two heads will turn up in succession somewhere in the sequence of throws.

In two bowls there are in total ${N}$ balls, numbered from ${1}$ to ${N}$. One ball is moved from one of the bowls into the other. The average of the numbers in the bowls is increased in both of the bowls by the same amount, ${x}$. Determine the largest possible value of ${x}$.

An ant is on the bottom edge of a right circular cone with base area $\pi$ and slant length $6$. What is the shortest distance that the ant has to travel to loop around the cone and come back to its starting position?

Prove that if X is a compact $T_2$ space, and X has density d(X), then $X^3$ contains a discrete subspace of cardinality $d(X)$. note: $d(X)$ is the smallest cardinality of a dense subspace of X.

Let all the edges of tetrahedron \(A_1A_2A_3A_4\) be tangent to sphere \(S\). Let \(\displaystyle a_i\) denote the length of the tangent from \(A_i\) to \(S\). Prove that \[\bigg(\sum_{i=1}^4 \frac 1{a_i}\bigg)^{\!\!2}> 2\bigg(\sum_{i=1}^4 \frac1{a_i^2}\bigg). \] [i]Submitted by Viktor Vígh, Szeged[/i]

The numbers 1, 2, ... , 36 are written in the cells of a $6 \times 6$ grid. Two cells are called neighbors if they have a common side or vertex. A frog is located at the cell with the number 1 written on it. Every minute, if a neighboring cell has a bigger number than the cell where the frog is located, the frog jumps to the neighboring cell that has the biggest number written on it. The frog continues like this until there are no neighboring cells with a bigger number than the cell where the frog is located. What is the biggest possible number of jumps the frog can make?

In each cell of a matrix $ n\times n$ a number from a set $ \{1,2,\ldots,n^2\}$ is written --- in the first row numbers $ 1,2,\ldots,n$, in the second $ n\plus{}1,n\plus{}2,\ldots,2n$ and so on. Exactly $ n$ of them have been chosen, no two from the same row or the same column. Let us denote by $ a_i$ a number chosen from row number $ i$. Show that: \[ \frac{1^2}{a_1}\plus{}\frac{2^2}{a_2}\plus{}\ldots \plus{}\frac{n^2}{a_n}\geq \frac{n\plus{}2}{2}\minus{}\frac{1}{n^2\plus{}1}\]

A circle is tangent to sides $AD,\text{ }DC,\text{ }CB$ of a convex quadrilateral $ABCD$ at $\text{K},\text{ L},\text{ M}$ respectively. A line $l$, passing through $L$ and parallel to $AD$, meets $KM$ at $N$ and $KC$ at $P$. Prove that $PL=PN$.

Bill is buying cans of soup. Cans come in $2$ shapes. Can $A$ is a rectangular prism shaped can with dimensions $20\times16\times10$, and can $B$ is a cylinder shaped can with radius $10$ and height $10$. Let $\alpha$ be the volume of the larger can, and $\beta$ be the volume of the smaller can. What is $\alpha-\beta$?

Suppose that each of the vertices of $ABC$ is a lattice point in the $xy$-plane and that there is exactly one lattice point $P$ in the interior of the triangle. The line $AP$ is extended to meet $BC$ at $E$. Determine the largest possible value for the ratio of lengths of segments $$\frac{|AP|}{|PE|}.$$

Consider a triangle $ABC$ such that $\angle A = 90^o, \angle C =60^o$ and $|AC|= 6$. Three circles with centers $A, B$ and $C$ are pairwise tangent in points on the three sides of the triangle. Determine the area of the region enclosed by the three circles (the grey area in the figure). [asy] unitsize(0.2 cm); pair A, B, C; real[] r; A = (6,0); B = (6,6*sqrt(3)); C = (0,0); r[1] = 3*sqrt(3) - 3; r[2] = 3*sqrt(3) + 3; r[3] = 9 - 3*sqrt(3); fill(arc(A,r[1],180,90)--arc(B,r[2],270,240)--arc(C,r[3],60,0)--cycle, gray(0.7)); draw(A--B--C--cycle); draw(Circle(A,r[1])); draw(Circle(B,r[2])); draw(Circle(C,r[3])); dot("$A$", A, SE); dot("$B$", B, NE); dot("$C$", C, SW); [/asy]

Let $A C$ and $C E$ be perpendicular line segments, each of length $18 .$ Suppose $B$ and $D$ are the midpoints of $A C$ and $C E$ respectively. If $F$ be the point of intersection of $E B$ and $A D,$ then the area of $\triangle B D F$ is? [list=1] [*] $27\sqrt{2}$ [*] $18\sqrt{2}$ [*] 13.5 [*] 18 [/list]

Triangle $\vartriangle ABC$ has sidelengths $AB = 10$, $AC = 14$, and, $BC = 16$. Circle $\omega_1$ is tangent to rays $\overrightarrow{AB}$, $\overrightarrow{AC}$ and passes through $B$. Circle $\omega_2$ is tangent to rays $\overrightarrow{AB}$, $\overrightarrow{AC}$ and passes through $C$. Let $\omega_1$, $\omega_2$ intersect at points $X, Y$ . The square of the perimeter of triangle $\vartriangle AXY$ is equal to $\frac{a+b\sqrt{c}}{d}$ , where $a, b, c$, and, $d$ are positive integers such that $a$ and $d$ are relatively prime, and $c$ is not divisible by the square of any prime. Find $a + b + c + d$.

We consider the following game for one person. The aim of the player is to reach a fixed capital $C>2$. The player begins with capital $0<x_0<C$. In each turn let $x$ be the player’s current capital. Define $s(x)$ as follows: $$s(x):=\begin{cases}x&\text{if }x<1\\C-x&\text{if }C-x<1\\1&\text{otherwise.}\end{cases}$$Then a fair coin is tossed and the player’s capital either increases or decreases by $s(x)$, each with probability $\frac12$. Find the probability that in a finite number of turns the player wins by reaching the capital $C$.

Suppose $P(x)$ is a non-constant polynomial with real coefficients, and even degree. Bob writes the polynomial $P(x)$ on a board. At every step, if the polynomial on the board is $f(x)$, he can replace it with 1. $f(x)+c$ for a real number $c$, or 2. the polynomial $P(f(x))$. Can he always find a finite sequence of steps so the final polynomial on the board has exactly $2020$ real roots? What about $2021$? [i]~Sutanay Bhattacharya[/i]

Let $a$ be a positive integer and let $\{a_n\}$ be defined by $a_0 = 0$ and \[a_{n+1 }= (a_n + 1)a + (a + 1)a_n + 2 \sqrt{a(a + 1)a_n(a_n + 1)} \qquad (n = 1, 2 ,\dots ).\] Show that for each positive integer $n$, $a_n$ is a positive integer.

Let $x: [0, \infty) \to\Bbb R$ be a differentiable function. Prove that if for all t>1 $$x'(t)=-x^3(t)+\frac{t-1}{t}x^3(t-1)$$ then $\lim_{t\to\infty} x(t) = 0$

The points $A, B, M$ and $N$ are on a circle with center $O$ such that the radii $OA$ and $OB$ are perpendicular to each other, and $MN$ is parallel to $AB$ and intersects the radius $OA$ at $P$. Find the radius of the circle if $|MP|= 12$ and $|P N| = 2 \sqrt{14}$

There are $ 2000 $ people, and some of them have called each other. Two people can call each other at most $1$ time. For any two groups of three people $ A$ and $ B $ which $ A \cap B = \emptyset $, there exist one person from each of $A$ and $B$ that haven't called each other. Prove that the number of two people called each other is less than $ 201000 $.

Find all polynomials $f(x)$ with integer coefficients such that the coefficients of both $f(x)$ and $[f(x)]^3$ lie in the set $\{0,1, -1\}$.

Let $ABCD$ be a parallelogram and let $E$ be a point in the plane such that $AE\perp AB$ and $BC\perp EC$. Show that either $\angle AED=\angle BEC$ or $\angle AED+\angle BEC=180^\circ$.

The first of an infinite triangular spreadsheet the line contains one number, the second line contains two numbers, the third line contains three numbers, and so on. In doing so is in any $k$-th row ($k = 1, 2, 3,...$) in the first and last place the number $k$, each other the number in the table is found, however, than in the previous row the least common of the two numbers above it multiple (the adjacent figure shows the first five rows of this table). We choose any two numbers from the table that are not in their row in the first or last place. Prove that one of the selected numbers is divisible by another. [img]https://cdn.artofproblemsolving.com/attachments/3/7/107d8999d9f04777719a0f1b1df418dbe00023.png[/img]