Let $x,y,z$ be three positive integers with $\gcd(x,y,z)=1$. If \[x\mid yz(x+y+z),\] \[y\mid xz(x+y+z),\] \[z\mid xy(x+y+z),\] and \[x+y+z\mid xyz,\] show that $xyz(x+y+z)$ is a perfect square. [i]Proposed by usjl[/i]

Let $p$ be an odd prime. An integer $x$ is called a [i]quadratic non-residue[/i] if $p$ does not divide $x-t^2$ for any integer $t$. Denote by $A$ the set of all integers $a$ such that $1\le a<p$, and both $a$ and $4-a$ are quadratic non-residues. Calculate the remainder when the product of the elements of $A$ is divided by $p$. [i]Proposed by Richard Stong and Toni Bluher[/i]

The sequence $<a_n>$ is defined as follows, $a_1=a_2=1$, $a_3=2$, $$a_{n+3}=\frac{a_{n+2}a_{n+1}+n!}{a_n},$$ $n \ge 1$. Prove that all the terms in the sequence are integers.

Let $n$ be a natural number. Two numbers are called "unsociable" if their greatest common divisor is $1$. The numbers $\{1,2,...,2n\}$ are partitioned into $n$ pairs. What is the minimum number of "unsociable" pairs that are formed?

Show that the equation $ y^{37}\equiv x^3\plus{}11 \pmod p$ is solvable for every prime $ p$, where $ p\leq100$.

A positive integer $n$ is called [i]good[/i] if and only if there exist exactly $4$ positive integers $k_1, k_2, k_3, k_4$ such that $n+k_i|n+k_i^2$ ($1 \leq k \leq 4$). Prove that: [list] [*]$58$ is [i]good[/i]; [*]$2p$ is [i]good[/i] if and only if $p$ and $2p+1$ are both primes ($p>2$).[/list]

The sum of the base-$ 10$ logarithms of the divisors of $ 10^n$ is $ 792$. What is $ n$? $ \textbf{(A)}\ 11\qquad \textbf{(B)}\ 12\qquad \textbf{(C)}\ 13\qquad \textbf{(D)}\ 14\qquad \textbf{(E)}\ 15$

What is the largest positive integer that is not the sum of a positive integral multiple of 42 and a positive composite integer?

Prove or disprove the following hypotheses. a) For all $k \geq 2,$ each sequence of $k$ consecutive positive integers contains a number that is not divisible by any prime number less than $k.$ b) For all $k\geq 2,$ each sequence of $k$ consecutive positive integers contains a number that is relatively prime to all other members of the sequence.

Integers $a_1, a_2, \ldots, a_n$ satisfy $$1<a_1<a_2<\ldots < a_n < 2a_1.$$ If $m$ is the number of distinct prime factors of $a_1a_2\cdots a_n$, then prove that $$(a_1a_2\cdots a_n)^{m-1}\geq (n!)^m.$$

The infinite decimal notation of the real number $x$ contains all the digits. Let $u_n$ be the number of different $n$-digit segments encountered in $x$ notation. Prove that if for some $n$, $u_n \le (n+8)$, than $x$ is a rational number.

Let $a_1 \leq a_2 \leq \cdots$ be a non-decreasing sequence of positive integers. A positive integer $n$ is called [i]good[/i] if there is an index $i$ such that $n=\dfrac{i}{a_i}$. Prove that if $2013$ is [i]good[/i], then so is $20$.

Prove that for any positive integer $n$ there exists a positive integer $k$ such that $3^k+4^k-1 \vdots 12^n$

Let $x$ and $y$ be positive integers. If ${x^{2^n}}-1$ is divisible by $2^ny+1$ for every positive integer $n$, prove that $x=1$.

You are given a balance and one copy of each of ten weights of $1, 2, 4, 8, 16, 32, 64, 128, 256$ and $512$ grams. An object weighing $M$ grams, where $M$ is a positive integer, is put on one of the pans and may be balanced in different ways by placing various combinations of the given weights on either pan of the balance. (a) Prove that no object may be balanced in more than $89$ ways. (b) Find a value of $M$ such that an object weighing $M$ grams can be balanced in $89$ ways. (A Shapovalov, A Kulakov)

Three prime numbers $p,q,r$ and a positive integer $n$ are given such that the numbers \[ \frac{p+n}{qr}, \frac{q+n}{rp}, \frac{r+n}{pq} \] are integers. Prove that $p=q=r $. [i]Nazar Agakhanov[/i]

A number is [i]special[/i] if its tens digit is $9$. For example, $499$ and $1092$ are special, but $509$ is not. Diego has several cards. On each of them, he wrote a special number (he may write the same number on more than one card). When he adds up the numbers on the cards, the total is $2024$. What is the smallest number of cards Diego can have?

[b](a)[/b] The integers $x$, $x^2$ and $x^3$ begin with the same digit. Does it imply that this digit is $1$? [i]($2$ points) [/i] [b](b)[/b] The same question for the integers $x, x^2, x^3, \cdots, x^{2015}$ [i]($3$ points)[/i] .

For $n\ne 0$, let f(n) be the largest $k$ such that $3^k$ divides $n$. If $M$ is a set of $n > 1$ integers, show that the number of possible values for $f(m-n)$, where $m, n$ belong to $M$ cannot exceed $n-1$.

Given the number $720$, Juan must choose $4$ numbers that are divisors of $720$. He wins if none of the four chosen numbers is a divisor of the product of the other three. Decide whether Juan can win.

$(FRA 6)$ Consider the integer $d = \frac{a^b-1}{c}$, where $a, b$, and $c$ are positive integers and $c \le a.$ Prove that the set $G$ of integers that are between $1$ and $d$ and relatively prime to $d$ (the number of such integers is denoted by $\phi(d)$) can be partitioned into $n$ subsets, each of which consists of $b$ elements. What can be said about the rational number $\frac{\phi(d)}{b}?$

Prove that the equation $x^8 = n! + 1$ has finitely many solutions in positive integers.

Find all pairs of integers $(x, y)$ satisfying the condition $12x^2 + 6xy + 3y^2 = 28(x + y)$.

[b]p1.[/b] Find all $x$ such that $(\ln (x^4))^2 = (\ln (x))^6$. [b]p2.[/b] On a piece of paper, Alan has written a number $N$ between $0$ and $2010$, inclusive. Yiwen attempts to guess it in the following manner: she can send Alan a positive number $M$, which Alan will attempt to subtract from his own number, which we will call $N$. If $M$ is less than or equal $N$, then he will erase $N$ and replace it with $N -M$. Otherwise, Alan will tell Yiwen that $M > N$. What is the minimum number of attempts that Yiwen must make in order to determine uniquely what number Alan started with? [b]p3.[/b] How many positive integers between $1$ and $50$ have at least $4$ distinct positive integer divisors? (Remember that both $1$ and $n$ are divisors of $n$.) [b]p4.[/b] Let $F_n$ denote the $n^{th}$ Fibonacci number, with $F_0 = 0$ and $F_1 = 1$. Find the last digit of $$\sum^{97!+4}_{i=0}F_i.$$ [b]p5.[/b] Find all prime numbers $p$ such that $2p + 1$ is a perfect cube. [b]p6.[/b] What is the maximum number of knights that can be placed on a $9\times 9$ chessboard such that no two knights attack each other? [b]p7.[/b] $S$ is a set of $9$ consecutive positive integers such that the sum of the squares of the $5$ smallest integers in the set is the sum of the squares of the remaining $4$. What is the sum of all $9$ integers? [b]p8.[/b] In the following infinite array, each row is an arithmetic sequence, and each column is a geometric sequence. Find the sum of the infinite sequence of entries along the main diagonal. [img]https://cdn.artofproblemsolving.com/attachments/5/1/481dd1e496fed6931ee2912775df630908c16e.png[/img] [b]p9.[/b] Let $x > y > 0$ be real numbers. Find the minimum value of $\frac{x}{y} + \frac{4x}{x-y}$ . [b]p10.[/b] A regular pentagon $P = A_1A_2A_3A_4A_5$ and a square $S = B_1B_2B_3B_4$ are both inscribed in the unit circle. For a given pentagon $P$ and square $S$, let $f(P, S)$ be the minimum length of the minor arcs $A_iB_j$ , for $1 \le i \le 5$ and $1 \le j \le4$. Find the maximum of $f(P, S)$ over all pairs of shapes. [b]p11.[/b] Find the sum of the largest and smallest prime factors of $9^4 + 3^4 + 1$. [b]p12.[/b] A transmitter is sending a message consisting of $4$ binary digits (either ones or zeros) to a receiver. Unfortunately, the transmitter makes errors: for each digit in the message, the probability that the transmitter sends the correct digit to the receiver is only $80\%$. (Errors are independent across all digits.) To avoid errors, the receiver only accepts a message if the sum of the first three digits equals the last digit modulo $2$. If the receiver accepts a message, what is the probability that the message was correct? [b]p13.[/b] Find the integer $N$ such that $$\prod^{8}_{i=0}\sec \left( \frac{\pi}{9}2^i \right)= N.$$ PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In the sequence of digits $2,0,2,9,3,...$ any digit it equal to the last digit in the decimal representation of the sum of four previous digits. Do the four numbers $2,0,1,5$ in that order occur in the sequence? Folklore