Find all the triples of positive integers $(a,b,c)$ for which the number \[\frac{(a+b)^4}{c}+\frac{(b+c)^4}{a}+\frac{(c+a)^4}{b}\] is an integer and $a+b+c$ is a prime.

Is the following set of prime numbers $p$ finite or infinite, where each $p$ [b]doesn't[/b] divide the numbers that can be expressed as $n^{2016}+2016^{2016}$ for $n\in \mathbb{N}$, if: a) $p=4k+3$; b) $p=4k+1$?

For any positive integer $n$ we define $n!!=\prod_{k=0}^{\lceil n/2\rceil -1}(n-2k)$. Prove that if the positive integers $a,b,c$ satisfy $a!=b!!+c!!$, then $b$ and $c$ are odd. [i]Proposed by Mihai Cipu[/i]

Find all prime numbers $p_1,…,p_n$ (not necessarily different) such that : $$ \prod_{i=1}^n p_i=10 \sum_{i=1}^n p_i$$

Let $d$ be a positive divisor of a positive integer $m$ and $(a_l), (b_l)$ two arithmetic sequences of positive integers. It is given that $gcd(a_i, b_j) = 1$ and $gcd(a_k, b_n) = m$ for some positive integers $i,j,k,$ and $n$. Prove that there exist positive integers $t$ and $s$ such that $gcd(a_t, b_s) = d$.

Given \(n, k \in \mathbb{N}\), prove that \((n-1)^2\) divides \(n^k - 1\) if and only if \(n-1 \mid k\).

The function is given $f(x) = \frac{2x^3 -6x^2 + 13x + 10}{2x^2 - 9x}$. Determine all positive integers $x$ for which $f(x)$ is an integer

Let $p \geq 3$ be a prime number. Prove that there is a permutation $(x_1,\ldots, x_{p-1})$ of $(1,2,\ldots,p-1)$ such that $x_1x_2 + x_2x_3 + \cdots + x_{p-2}x_{p-1} \equiv 2 \pmod p$.

Let $a > 1$ be a positive integer and $d > 1$ be a positive integer coprime to $a$. Let $x_1=1$, and for $k\geq 1$, define $$x_{k+1} = \begin{cases} x_k + d &\text{if } a \text{ does not divide } x_k \\ x_k/a & \text{if } a \text{ divides } x_k \end{cases}$$ Find, in terms of $a$ and $d$, the greatest positive integer $n$ for which there exists an index $k$ such that $x_k$ is divisible by $a^n$.

For a positive integer $n$, let $\omega(n)$ denote the number of positive prime divisors of $n$. Find the smallest positive tinteger $k$ such that $2^{\omega(n)}\leq k\sqrt[4]{n}\forall n\in\mathbb{N}$.

Find all triples of integers $(x, y, z)$ such that $x^4 + 5y^4 = z^4$.

The digits of a $3$-digit number are interchanged so that none of the digits retain their original position. The difference of the two numbers is a $2$-digit number and is a perfect square. Find the difference.

Find all primes $p$ such that $\dfrac{2^{p+1}-4}{p}$ is a perfect square.

Positive integers $a, b$ satisfy equality $b^2 = a^2 + ab + b$. Prove that $b$ is a square of a positive integer. (Patrik Bak)

[u]Set 5[/u] [b]p13.[/b] An equiangular $12$-gon has side lengths that alternate between $2$ and $\sqrt3$. Find the area of the circumscribed circle of this $12$-gon. [b]p14.[/b] For positive integers $n$, let $\sigma(n)$ denote the number of positive integer factors of $n$. Then $\sigma(17280) = \sigma (2^7 \cdot 3^3 \cdot 5)= 64$. Let $S$ be the set of factors $k$ of $17280$ such that $\sigma(k) = 32$. If $p$ is the product of the elements of $S$, find $\sigma(p)$. [b]p15.[/b] How many odd $3$-digit numbers have exactly four $1$’s in their binary (base $2$) representation? For example, $225_{10} = 11100001_2$ would be valid. PS. You should use hide for answers.

[u]Set 4[/u] [b]p16.[/b] Albert, Beatrice, Corey, and Dora are playing a card game with two decks of cards numbered $1-50$ each. Albert, Beatrice, and Corey draw cards from the same deck without replacement, but Dora draws from the other deck. What is the probability that the value of Corey’s card is the highest value or is tied for the highest value of all $4$ drawn cards? [b]p17.[/b] Suppose that $s$ is the sum of all positive values of $x$ that satisfy $2016\{x\} = x+[x]$. Find $\{s\}$. (Note: $[x]$ denotes the greatest integer less than or equal to $x$ and $\{x\}$ denotes $x - [x]$.) [b]p18.[/b] Let $ABC$ be a triangle such that $AB = 41$, $BC = 52$, and $CA = 15$. Let H be the intersection of the $B$ altitude and $C$ altitude. Furthermore let $P$ be a point on $AH$. Both $P$ and $H$ are reflected over $BC$ to form $P'$ and $H'$ . If the area of triangle $P'H'C$ is $60$, compute $PH$. [b]p19.[/b] A random integer $n$ is chosen between $1$ and $30$, inclusive. Then, a random positive divisor of $n, k$, is chosen. What is the probability that $k^2 > n$? [b]p20.[/b] What are the last two digits of the value $3^{361}$? [u]Set 5[/u] [b]p21.[/b] Let $f(n)$ denote the number of ways a $3 \times n$ board can be completely tiled with $1 \times 3$ and $1 \times 4$ tiles, without overlap or any tiles hanging over the edge. The tiles may be rotated. Find $\sum^9_{i=0} f(i) = f(0) + f(1) + ... + f(8) + f(9)$. By convention, $f(0) = 1$. [b]p22.[/b] Find the sum of all $5$-digit perfect squares whose digits are all distinct and come from the set $\{0, 2, 3, 5, 7, 8\}$. [b]p23.[/b] Mary is flipping a fair coin. On average, how many flips would it take for Mary to get $4$ heads and $2$ tails? [b]p24.[/b] A cylinder is formed by taking the unit circle on the $xy$-plane and extruding it to positive infinity. A plane with equation $z = 1 - x$ truncates the cylinder. As a result, there are three surfaces: a surface along the lateral side of the cylinder, an ellipse formed by the intersection of the plane and the cylinder, and the unit circle. What is the total surface area of the ellipse formed and the lateral surface? (The area of an ellipse with semi-major axis $a$ and semi-minor axis $b$ is $\pi ab$.) [b]p25.[/b] Let the Blair numbers be defined as follows: $B_0 = 5$, $B_1 = 1$, and $B_n = B_{n-1} + B_{n-2}$ for all $n \ge 2$. Evaluate $$\sum_{i=0}^{\infty} \frac{B_i}{51^i}= B_0 +\frac{B_1}{51} +\frac{B_2}{51^2} +\frac{B_3}{51^3} +...$$ [u]Estimation[/u] [b]p26.[/b] Choose an integer between $1$ and $10$, inclusive. Your score will be the number you choose divided by the number of teams that chose your number. [b]p27.[/b] $2016$ blind people each bring a hat to a party and leave their hat in a pile at the front door. As each partier leaves, they take a random hat from the ones remaining in a pile. Estimate the probability that at least $1$ person gets their own hat back. [b]p28.[/b] Estimate how many lattice points lie within the graph of $|x^3| + |y^3| < 2016$. [b]p29.[/b] Consider all ordered pairs of integers $(x, y)$ with $1 \le x, y \le 2016$. Estimate how many such ordered pairs are relatively prime. [b]p30.[/b] Estimate how many times the letter “e” appears among all Guts Round questions. PS. You should use hide for answers. First sets have been posted [url=https://artofproblemsolving.com/community/c3h2779594p24402189]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all pairs $(p,q)$ of prime numbers such that $$ p(p^2 - p - 1) = q(2q + 3) .$$

Let $M$ be the set of natural numbers $k$ for which there exists a natural number $n$ such that $$3^n \equiv k\pmod n.$$ Prove that $M$ has infinitely many elements.

Find all pairs of prime numbers $p,\,q(p\le q)$ satisfying the following condition: There exists a natural number $n$ such that $2^{n}+3^{n}+\cdots+(2pq-1)^{n}$ is a multiple of $2pq$.

Find all quadruples $(a, b, c, d)$ of non-negative integers such that $ab =2(1 + cd)$ and there exists a non-degenerate triangle with sides of length $a - c$, $b - d$, and $c + d$.

a) Prove that there are infinitely many pairs $(m, n)$ of positive integers satisfying the following equality $[(4 + 2\sqrt3)m] = [(4 -2\sqrt3)n]$ b) Prove that if $(m, n)$ satisfies the equality, then the number $(n + m)$ is odd. (I. Voronovich)

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. $p_i|k+i$).

Let $x$ be a natural number. We call $\{x_0,x_1,\dots ,x_l\}$ a [i]factor link [/i]of $x$ if the sequence $\{x_0,x_1,\dots ,x_l\}$ satisfies the following conditions: (1) $x_0=1, x_l=x$; (2) $x_{i-1}<x_i, x_{i-1}|x_i, i=1,2,\dots,l$ . Meanwhile, we define $l$ as the length of the [i]factor link [/i] $\{x_0,x_1,\dots ,x_l\}$. Denote by $L(x)$ and $R(x)$ the length and the number of the longest [i]factor link[/i] of $x$ respectively. For $x=5^k\times 31^m\times 1990^n$, where $k,m,n$ are natural numbers, find the value of $L(x)$ and $R(x)$.

The digits $2,3,4,5,6,7,8,9$ are written down in some order. When read in that order, the digits form an $8$-digit, base $10$ positive integer. if this integer is divisible by $44$, how many ways could the digits have been initially ordered? [i]Proposed by Evan Chang (squareman), USA[/i]

Octavio writes an integer $n \geq 1$ on a blackboard and then he starts a process in which, at each step he erases the integer $k$ written on the blackboard and replaces it with one of the following numbers: $$3k-1, \quad 2k+1, \quad \frac{k}{2}.$$ provided that the result is an integer. Show that for any integer $n \geq 1$, Octavio can write on the blackboard the number $3^{2023}$ after a finite number of steps.