Find all functions $g:\mathbb{N}\rightarrow\mathbb{N}$ such that \[\left(g(m)+n\right)\left(g(n)+m\right)\] is a perfect square for all $m,n\in\mathbb{N}.$ [i]Proposed by Gabriel Carroll, USA[/i]

Find all integers $a,b,c $ such that $a+b+c=abc$

Xenia and Sergey play the following game. Xenia thinks of a positive integer $N$ not exceeding $5000$. Then she fixes $20$ distinct positive integers $a_1, a_2, \cdots, a_{20}$ such that, for each $k = 1,2,\cdots,20$, the numbers $N$ and $a_k$ are congruent modulo $k$. By a move, Sergey tells Xenia a set $S$ of positive integers not exceeding $20$, and she tells him back the set $\{a_k : k \in S\}$ without spelling out which number corresponds to which index. How many moves does Sergey need to determine for sure the number Xenia thought of? [i]Sergey Kudrya, Russia[/i]

Let $n=p_1^{\alpha_1}\cdots p_s^{\alpha_s}\ge2$. If for any $\alpha\in\mathbb N$, $p_i-1\nmid\alpha$, where $i=1,2,\ldots,s$, prove that $n\mid\sum_{\alpha\in\mathbb Z^*_n}\alpha^{\alpha}$ where $\mathbb Z^*_n=\{a\in\mathbb Z_n:\gcd(a,n)=1\}$.

Find all integer solutions of the equation \[x^2+y^2=(x-y)^3.\]

For any given non-negative integer $m$, let $f(m)$ be the number of $1$'s in the base $2$ representation of $m$. Let $n$ be a positive integer. Prove that the integer $$\sum^{2^n - 1}_{m = 0} \Big( (-1)^{f(m)} \cdot 2^m \Big)$$ contains at least $n!$ positive divisors.

Find all solutions of the equation $(a^a)^5 = b^b$ in positive integers.

Let $p$ be a prime greater than $2$. Prove that $\frac{2}{p}$ can be expressed in exactly one way in the form $$\frac{1}{x}+\frac{1}{y}$$ where $x$ and $y$ are positive integers with $x > y$.

Find the number of ordered pairs of integers $(x, y)$ such that $2167$ divides $3x^2 + 27y^2 + 2021$ with $0 \le x, y \le 2166$.

Prove whether or not there exist natural numbers $n,k$ where $1\le k\le n-2$ such that \[\binom{n}{k}^2+\binom{n}{k+1}^2=\binom{n}{k+2}^4 \]

For any set $A = \{x_1, x_2, x_3, x_4, x_5\}$ of five distinct positive integers denote by $S_A$ the sum of its elements, and denote by $T_A$ the number of triples $(i, j, k)$ with $1 \le i < j < k \le 5$ for which $x_i + x_j + x_k$ divides $S_A$. Find the largest possible value of $T_A$.

Let $F$ be the set of all fractions $m/n$ where $m$ and $n$ are positive integers with $m+n\le 2005$. Find the largest number $a$ in $F$ such that $a < 16/23$.

Let $d(n)$ be the number of divisors of a nonnegative integer $n$ (we set $d(0)=0$). Find all positive integers $d$ such that there exists a two-variable polynomial $P(x,y)$ of degree $d$ with integer coefficients such that: [list] [*] for any positive integer $y$, there are infinitely many positive integers $x$ such that $\gcd(x,y)=1$ and $d(|P(x,y)|) \mid x$, and [*] for any positive integer $x$, there are infinitely many positive integers $y$ such that $\gcd(x,y)=1$ and $d(|P(x,y)|) \mid y$. [/list] [i]Allen Wang[/i]

Prove that if $n$ is a positive integer then $2 (n^2 + 1) - n$ is not a square of an integer.

Prove that for some positive integer \(N\), \(N\) points can be chosen on a circle such that there are at least \(1000N^2\) unordered quadruples \((A,B,C,D)\) of distinct selected points for which \(\displaystyle \frac{AC}{BC} = \frac{AD}{BD}\).

A positive integer $n$ is called [i]Olympic[/i], if there exists a quadratic trinomial with integer coeffecients $f(x)$ satisfying $f(f(\sqrt{n}))=0$. Determine, with proof, the largest Olympic number not exceeding $2015$. [i]A. Khrabrov[/i]

$R$ is a ring such that $xy=yx$ for every $x,y\in R$ and if $ab=0$ then $a=0$ or $b=0$. if for every Ideal $I\subset R$ there exist $x_1,x_2,..,x_n$ in $R$ ($n$ is not constant) such that $I=(x_1,x_2,...,x_n)$, prove that every element in $R$ that is not $0$ and it's not a unit, is the product of finite irreducible elements.($\frac{100}{6}$ points)

Let $m,n$ be integers greater than 1,$m \geq n$,$a_1,a_2,\dots,a_n$ are $n$ distinct numbers not exceed $m$,which are relatively primitive.Show that for any real $x$,there exists $i$ for which $||a_ix|| \geq \frac{2}{m(m+1)} ||x||$,where $||x||$ denotes the distance between $x$ and the nearest integer to $x$ .

[b]p1.[/b] What is the slope of the line perpendicular to the the graph $\frac{x}{4}+\frac{y}{9}= 1$ at $(0, 9)$? [b]p2.[/b] A boy is standing in the middle of a very very long staircase and he has two pogo sticks. One pogo stick allows him to jump $220$ steps up the staircase. The second pogo stick allows him to jump $125$ steps down the staircase. What is the smallest positive number of steps that he can reach from his original position by a series of jumps? [b]p3.[/b] If you roll three fair six-sided dice, what is the probability that the product of the results will be a multiple of $3$? [b]p4.[/b] Right triangle $ABC$ has squares $ABXY$ and $ACWZ$ drawn externally to its legs and a semicircle drawn externally to its hypotenuse $BC$. If the area of the semicircle is $18\pi$ and the area of triangle $ABC$ is $30$, what is the sum of the areas of squares $ABXY$ and $ACWZ$? [img]https://cdn.artofproblemsolving.com/attachments/5/1/c9717e7731af84e5286335420b73b974da2643.png[/img] [b]p5.[/b] You have a bag containing $3$ types of pens: red, green, and blue. $30\%$ of the pens are red pens, and $20\%$ are green pens. If, after you add $10$ blue pens, $60\%$ of the pens are blue pens, how many green pens did you start with? [b]p6.[/b] Canada gained partial independence from the United Kingdom in $1867$, beginning its long role as the headgear of the United States. It gained its full independence in $1982$. What is the last digit of $1867^{1982}$? [b]p7.[/b] Bacon, Meat, and Tomato are dealing with paperwork. Bacon can fill out $5$ forms in $3$ minutes, Meat can fill out $7$ forms in $5$ minutes, and Tomato can staple $3$ forms in $1$ minute. A form must be filled out and stapled together (in either order) to complete it. How long will it take them to complete $105$ forms? [b]p8.[/b] Nice numbers are defined to be $7$-digit palindromes that have no $3$ identical digits (e.g., $1234321$ or $5610165$ but not $7427247$). A pretty number is a nice number with a $7$ in its decimal representation (e.g., $3781873$). What is the $7^{th}$ pretty number? [b]p9.[/b] Let $O$ be the center of a semicircle with diameter $AD$ and area $2\pi$. Given square $ABCD$ drawn externally to the semicircle, construct a new circle with center $B$ and radius $BO$. If we extend $BC$, this new circle intersects $BC$ at $P$. What is the length of $CP$? [img]https://cdn.artofproblemsolving.com/attachments/b/1/be15e45cd6465c7d9b45529b6547c0010b8029.png[/img] [b]p10.[/b] Derek has $10$ American coins in his pocket, summing to a total of $53$ cents. If he randomly grabs $3$ coins from his pocket, what is the probability that they're all different? [b]p11.[/b] What is the sum of the whole numbers between $6\sqrt{10}$ and $7\pi$ ? [b]p12.[/b] What is the volume of a cylinder whose radius is equal to its height and whose surface area is numerically equal to its volume? [b]p13.[/b] $15$ people, including Luke and Matt, attend a Berkeley Math meeting. If Luke and Matt sit next to each other, a fight will break out. If they sit around a circular table, all positions equally likely, what is the probability that a fight doesn't break out? [b]p14.[/b] A non-degenerate square has sides of length $s$, and a circle has radius $r$. Let the area of the square be equal to that of the circle. If we have a rectangle with sides of lengths $r$, $s$, and its area has an integer value, what is the smallest possible value for $s$? [b]p15.[/b] How many ways can you arrange the letters of the word "$BERKELEY$" such that no two $E$'s are next to each other? [b]p16.[/b] Kim, who has a tragic allergy to cake, is having a birthday party. She invites $12$ people but isn't sure if $11$ or $12$ will show up. However, she needs to cut the cake before the party starts. What is the least number of slices (not necessarily of equal size) that she will need to cut to ensure that the cake can be equally split among either $11$ or $12$ guests with no excess? [b]p17.[/b] Tom has $2012$ blue cards, $2012$ red cards, and $2012$ boxes. He distributes the cards in such a way such that each box has at least $1$ card. Sam chooses a box randomly, then chooses a card randomly. Suppose that Tom arranges the cards such that the probability of Sam choosing a blue card is maximized. What is this maximum probability? [b]p18.[/b] Allison wants to bake a pie, so she goes to the discount market with a handful of dollar bills. The discount market sells different fruit each for some integer number of cents and does not add tax to purchases. She buys $22$ apples and $7$ boxes of blueberries, using all of her dollar bills. When she arrives back home, she realizes she needs more fruit, though, so she grabs her checkbook and heads back to the market. This time, she buys $31$ apples and $4$ boxes of blueberries, for a total of $60$ cents more than her last visit. Given she spent less than $100$ dollars over the two trips, how much (in dollars) did she spend on her first trip to the market? [b]p19.[/b] Consider a parallelogram $ABCD$. Let $k$ be the line passing through A and parallel to the bisector of $\angle ABC$, and let $\ell$ be the bisector of $\angle BAD$. Let $k$ intersect line $CD$ at $E$ and $\ell$ intersect line $CD$ at $F$. If $AB = 13$ and $BC = 37$, find the length $EF$. [b]p20.[/b] Given for some real $a, b, c, d,$ $$P(x) = ax^4 + bx^3 + cx^2 + dx$$ $$P(-5) = P(-2) = P(2) = P(5) = 1$$ Find $P(10).$ PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all prime numbers $ p $ for which $ 1 + p\cdot 2^{p} $ is a perfect square.

Determine the number of pairs of positive integers $x,y$ such that $x\le y$, $\gcd (x,y)=5!$ and $\text{lcm}(x,y)=50!$.

Let $k$ be a positive integer and let $n$ be the smallest number with exactly $k$ divisors. Given $n$ is a cube, is it possible that $k$ is divisible by a prime factor of the form $3j+2$?

Proove that if for a positive integer $n$ , both $3n + 1$ and $10n + 1$ are perfect squares , then $29n + 11$ is not a prime number.

Find the highest power of 2 that is a factor of the number $$ L_n = (n+1)(n+2)... 2n,$$ where $n$is a natural number.

Let $\mathbb{N}$ and $\mathbb{N}^*$ be the sets containing the natural numbers/positive integers respectively. We define a binary relation on $\mathbb{N}$ by $a\acute{\in}b$ iff the $a$-th bit in the binary representation of $b$ is $1$. We define a binary relation on $\mathbb{N}^*$ by $a\tilde{\in}b$ iff $b$ is a multiple of the $a$-th prime number $p_a$. i) Prove that there is no bijection $f:\mathbb{N}\to \mathbb{N}^*$ such that $a\acute{\in}b\Leftrightarrow f(a)\tilde{\in}f(b)$. ii) Prove that there is a bijection $g:\mathbb{N}\to \mathbb{N}^*$ such that $(a\acute{\in}b \vee b\acute{\in}a)\Leftrightarrow (g(a)\tilde{\in}g(b) \vee g(b)\tilde{\in}g(a))$.