Find all pairs of positive integers $(m, n)$ satisfying the equation $$m!+n!=m^n+1.$$

Find the $2019$th strictly positive integer $n$ such that $\binom{2n}{n}$ is not divisible by $5$.

Let $n$ be a positive integer. Let $S$ be a set of ordered pairs $(x, y)$ such that $1\leq x \leq n$ and $0 \leq y \leq n$ in each pair, and there are no pairs $(a, b)$ and $(c, d)$ of different elements in $S$ such that $a^2+b^2$ divides both $ac+bd$ and $ad - bc$. In terms of $n$, determine the size of the largest possible set $S$.

Let $p > 3$ be a prime. For each $k\in \{1,2, \ldots , p-1\}$, define $x_k$ to be the unique integer in $\{1, \ldots, p-1\}$ such that $kx_k\equiv 1 \pmod{p}$ and set $kx_k = 1+ pn_k$. Prove that : \[\sum_{k=1}^{p-1}kn_k \equiv \frac{p-1}{2} \pmod{p}\]

Determine all finite nonempty sets $S$ of positive integers satisfying \[ {i+j\over (i,j)}\qquad\mbox{is an element of S for all i,j in S}, \] where $(i,j)$ is the greatest common divisor of $i$ and $j$.

$a,b,m,k\in \mathbb{Z}$, $a,b,m>2,k>1$, for which $k^n a+b$ is an $m$-th power of a natural number for $\forall n\in \mathbb{N}$. Prove that $b$ is an $m$-th power of a non-negative integer.

[b]p1.[/b] Suppose $a_1 \cdot 2 = a_2 \cdot 3 = a_3$ and $a_1 + a_2 + a_3 = 66$. What is $a_3$? [b]p2.[/b] Ankit buys a see-through plastic cylindrical water bottle. However, in coming home, he accidentally hits the bottle against a wall and dents the top portion of the bottle (above the $7$ cm mark). Ankit now wants to determine the volume of the bottle. The area of the base of the bottle is $20$ cm$^2$ . He fills the bottle with water up to the $5$ cm mark. After flipping the bottle upside down, he notices that the height of the empty space is at the $7$ cm mark. Find the total volume (in cm$^3$) of this bottle. [img]https://cdn.artofproblemsolving.com/attachments/1/9/f5735c77b056aaf31b337ea1b777a591807819.png[/img] [b]p3.[/b] If $P$ is a quadratic polynomial with leading coefficient $ 1$ such that $P(1) = 1$, $P(2) = 2$, what is $P(10)$? [b]p4.[/b] Let ABC be a triangle with $AB = 1$, $AC = 3$, and $BC = 3$. Let $D$ be a point on $BC$ such that $BD =\frac13$ . What is the ratio of the area of $BAD$ to the area of $CAD$? [b]p5.[/b] A coin is flipped $ 12$ times. What is the probability that the total number of heads equals the total number of tails? Express your answer as a common fraction in lowest terms. [b]p6.[/b] Moor pours $3$ ounces of ginger ale and $ 1$ ounce of lime juice in cup $A$, $3$ ounces of lime juice and $ 1$ ounce of ginger ale in cup $B$, and mixes each cup well. Then he pours $ 1$ ounce of cup $A$ into cup $B$, mixes it well, and pours $ 1$ ounce of cup $B$ into cup $A$. What proportion of cup $A$ is now ginger ale? Express your answer as a common fraction in lowest terms. [b]p7.[/b] Determine the maximum possible area of a right triangle with hypotenuse $7$. Express your answer as a common fraction in lowest terms. [b]p8.[/b] Debbie has six Pusheens: $2$ pink ones, $2$ gray ones, and $2$ blue ones, where Pusheens of the same color are indistinguishable. She sells two Pusheens each to Alice, Bob, and Eve. How many ways are there for her to do so? [b]p9.[/b] How many nonnegative integer pairs $(a, b)$ are there that satisfy $ab = 90 - a - b$? [b]p10.[/b] What is the smallest positive integer $a_1...a_n$ (where $a_1, ... , a_n$ are its digits) such that $9 \cdot a_1 ... a_n = a_n ... a_1$, where $a_1$, $a_n \ne 0$? [b]p11.[/b] Justin is growing three types of Japanese vegetables: wasabi root, daikon and matsutake mushrooms. Wasabi root needs $2$ square meters of land and $4$ gallons of spring water to grow, matsutake mushrooms need $3$ square meters of land and $3$ gallons of spring water, and daikon need $ 1$ square meter of land and $ 1$ gallon of spring water to grow. Wasabi sell for $60$ per root, matsutake mushrooms sell for $60$ per mushroom, and daikon sell for $2$ per root. If Justin has $500$ gallons of spring water and $400$ square meters of land, what is the maximum amount of money, in dollars, he can make? [b]p12.[/b] A [i]prim [/i] number is a number that is prime if its last digit is removed. A [i]rime [/i] number is a number that is prime if its first digit is removed. Determine how many numbers between $100$ and $999$ inclusive are both prim and rime numbers. [b]p13.[/b] Consider a cube. Each corner is the intersection of three edges; slice off each of these corners through the midpoints of the edges, obtaining the shape below. If we start with a $2\times 2\times 2$ cube, what is the volume of the resulting solid? [img]https://cdn.artofproblemsolving.com/attachments/4/8/856814bf99e6f28844514158344477f6435a3a.png[/img] [b]p14.[/b] If a parallelogram with perimeter $14$ and area $ 12$ is inscribed in a circle, what is the radius of the circle? [b]p15.[/b] Take a square $ABCD$ of side length $1$, and draw $\overline{AC}$. Point $E$ lies on $\overline{BC}$ such that $\overline{AE}$ bisects $\angle BAC$. What is the length of $BE$? [b]p16.[/b] How many integer solutions does $f(x) = (x^2 + 1)(x^2 + 2) + (x^2 + 3)(x + 4) = 2017$ have? [b]p17.[/b] Alice, Bob, Carol, and Dave stand in a circle. Simultaneously, each player selects another player at random and points at that person, who must then sit down. What is the probability that Alice is the only person who remains standing? [b]p18.[/b] Let $x$ be a positive integer with a remainder of $2$ when divided by $3$, $3$ when divided by $4$, $4$ when divided by $5$, and $5$ when divided by $6$. What is the smallest possible such $x$? [b]p19[/b]. A circle is inscribed in an isosceles trapezoid such that all four sides of the trapezoid are tangent to the circle. If the radius of the circle is $ 1$, and the upper base of the trapezoid is $ 1$, what is the area of the trapezoid? [b]p20.[/b] Ray is blindfolded and standing $ 1$ step away from an ice cream stand. Every second, he has a $1/4$ probability of walking $ 1$ step towards the ice cream stand, and a $3/4$ probability of walking $ 1$ step away from the ice cream stand. When he is $0$ steps away from the ice cream stand, he wins. What is the probability that Ray eventually wins? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Determine whether there exist real numbers $x$, $y$, $z$, such that \[x+\frac{1}{y}=z,\quad y+\frac{1}{z}=x,\quad z+\frac{1}{x}=y.\]

Let m be a positive integer. Prove that there exist infinitely many prime numbers p such that m+p^3 is composite.

The prime number $p$ and a positive integer $k$ are given. Assume that $P(x)\in \mathbb Z[X]$ is a polynomial with coefficients in the set $\{0,1,\cdots,p-1\}$ with least degree which satisfies the following property: There exists a permutaion of numbers $1,2,\cdots,p-1$ around a circle such that for any $k$ consecutive numbers $a_1,a_2,\cdots,a_k$ one has $$ p | P(a_1)+P(a_2)+\cdots+ P(a_k). $$ Prove that $P(x)$ is of the form $ax^d+b$. Proposed by [i]Yahya Motevassel[/i]

Suppose that $a$ is an even positive integer and $A=a^{n}+a^{n-1}+\ldots +a+1,n\in \mathbb{N^{*}}$ is a perfect square.Prove that $8\mid a$.

Let $p$ be a prime number and $f$ be a bijection from $\left\{0,1,\ldots,p-1\right\}$ to itself. Suppose that for integers $a,b \in \left\{0,1,\ldots,p-1\right\}$, $|f(a) - f(b)|\leqslant 2024$ if $p \mid a^2 - b$. Prove that there exists infinite many $p$ such that there exists such an $f$ and there also exists infinite many $p$ such that there doesn't exist such an $f$.

Let $n$ be a positive integer. Prove that $n^2 + n + 1$ cannot be written as the product of two positive integers of which the difference is smaller than $2\sqrt{n}$.

Find all triples $(a, b, c)$ of positive integers for which $\frac{32a + 3b + 48c}{4abc}$ is also an integer.

Find eight positive integers $n_1, n_2, \dots , n_8$ with the following property: For every integer $k$, $-1985 \leq k \leq 1985$, there are eight integers $a_1, a_2, \dots, a_8$, each belonging to the set $\{-1, 0, 1\}$, such that $k=\sum_{i=1}^{8} a_i n_i .$

Find the sum of the four least positive integers each of whose digits add to $12$.

An integer $a$ is called friendly if the equation $(m^2+n)(n^2+m)=a(m-n)^3$ has a solution over the positive integers. [b]a)[/b] Prove that there are at least $500$ friendly integers in the set $\{ 1,2,\ldots ,2012\}$. [b]b)[/b] Decide whether $a=2$ is friendly.

Call a triple $(a, b, c)$ of positive integers a [b]nice[/b] triple if $a, b, c$ forms a non-decreasing arithmetic progression, $gcd(b, a) = gcd(b, c) = 1$ and the product $abc$ is a perfect square. Prove that given a nice triple, there exists some other nice triple having at least one element common with the given triple.

Let $ p$ be an odd prime. Determine positive integers $ x$ and $ y$ for which $ x \leq y$ and $ \sqrt{2p} \minus{} \sqrt{x} \minus{} \sqrt{y}$ is non-negative and as small as possible.

Find the smallest natural number which is a multiple of $2009$ and whose sum of (decimal) digits equals $2009$ [i]Proposed by Milos Milosavljevic[/i]

A set $S$ of positive integers is said to be [i]channeler[/i] if for any three distinct numbers $a,b,c \in S$, we have $a\mid bc$, $b\mid ca$, $c\mid ab$. a) Prove that for any finite set of positive integers $ \{ c_1, c_2, \ldots, c_n \} $ there exist infinitely many positive integers $k$, such that the set $ \{ kc_1, kc_2, \ldots, kc_n \} $ is a channeler set. b) Prove that for any integer $n \ge 3$ there is a channeler set who has exactly $n$ elements, and such that no integer greater than $1$ divides all of its elements.

[u]Set 1[/u] [b]G1.[/b] The Daily Challenge office has a machine that outputs the number $2.75$ when operated. If it is operated $12$ times, then what is the sum of all $12$ of the machine outputs? [b]G2.[/b] A car traveling at a constant velocity $v$ takes $30$ minutes to travel a distance of $d$. How long does it take, in minutes, for it travel $10d$ with a constant velocity of $2.5v$? [b]G3.[/b] Andy originally has $3$ times as many jelly beans as Andrew. After Andrew steals 15 of Andy’s jelly beans, Andy now only has $2$ times as many jelly beans as Andrew. Find the number of jelly beans Andy originally had. [u]Set 2[/u] [b]G4.[/b] A coin is weighted so that it is $3$ times more likely to come up as heads than tails. How many times more likely is it for the coin to come up heads twice consecutively than tails twice consecutively? [b]G5.[/b] There are $n$ students in an Areteem class. When 1 student is absent, the students can be evenly divided into groups of $5$. When $8$ students are absent, the students can evenly be divided into groups of $7$. Find the minimum possible value of $n$. [b]G6.[/b] Trapezoid $ABCD$ has $AB \parallel CD$ such that $AB = 5$, $BC = 4$ and $DA = 2$. If there exists a point $M$ on $CD$ such that $AM = AD$ and $BM = BC$, find $CD$. [u]Set 3[/u] [b]G7.[/b] Angeline has $10$ coins (either pennies, nickels, or dimes) in her pocket. She has twice as many nickels as pennies. If she has $62$ cents in total, then how many dimes does she have? [b]G8.[/b] Equilateral triangle $ABC$ has side length $6$. There exists point $D$ on side $BC$ such that the area of $ABD$ is twice the area of $ACD$. There also exists point $E$ on segment $AD$ such that the area of $ABE$ is twice the area of $BDE$. If $k$ is the area of triangle $ACE$, then find $k^2$. [b]G9.[/b] A number $n$ can be represented in base $ 6$ as $\underline{aba}_6$ and base $15$ as $\underline{ba}_{15}$, where $a$ and $b$ are not necessarily distinct digits. Find $n$. PS. You should use hide for answers. Sets 4-6 have been posted [url=https://artofproblemsolving.com/community/c3h3131305p28367080]here[/url] and 7-9 [url=https://artofproblemsolving.com/community/c3h3131308p28367095]here[/url].Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Positive integer numbers $k$ and $n$ satisfy the inequality $k > n!$. Prove that there exist pairwisely different prime numbers $p_1, p_2, \ldots, p_n$ which are divisors of the numbers $k+1, k+2, \ldots, k+n$ respectively (i.e. $p_i|k+i$).

Determine all pairs of prime numbers $(p; q)$ such that $p^2 + 5pq + 4q^2$ is the square of an integer.

Find all odd positive integers $ n > 1$ such that if $ a$ and $ b$ are relatively prime divisors of $ n$, then $ a\plus{}b\minus{}1$ divides $ n$.