(a) Let $x$ and $y$ be integers. Prove that $x = y$ if $x^n \equiv y^n$ mod $n$ for all positive integers $n$. (b) For which pairs of integers $(x, y)$ are there infinitely many positive integers $n$ such that $x^n \equiv y^n$ mod $n$?

Find the smallest multiple of $81$ that only contains the digit $1$. How many $ 1$’s does it contain?

A quadruple of integers $(a, b, c, d)$ is said good if $ad-bc=2020$. Two good quadruplets are said to be dissimilar if it is not possible to obtain one from the other using a finite number of applications of the following operations: $$(a,b,c,d) \rightarrow (-c,-d,a,b)$$ $$(a,b,c,d) \rightarrow (a,b,c+a,d+b)$$ $$(a,b,c,d) \rightarrow (a,b,c-a,d-b)$$ Let $A$ be a set of $k$ good quadruples, two by two dissimilar. Show that $k \leq 4284$.

Determine whether there exists an infinite sequence of nonzero digits $a_1 , a_2 , a_3 , \cdots $ and a positive integer $N$ such that for every integer $k > N$, the number $\overline{a_k a_{k-1}\cdots a_1 }$ is a perfect square.

Let $ n$ be a natural number, such that $ (n,2(2^{1386}\minus{}1))\equal{}1$. Let $ \{a_{1},a_{2},\dots,a_{\varphi(n)}\}$ be a reduced residue system for $ n$. Prove that:\[ n|a_{1}^{1386}\plus{}a_{2}^{1386}\plus{}\dots\plus{}a_{\varphi(n)}^{1386}\]

Let $s(n)$ be the number of positive integer divisors of $n$. Find the all positive values of $k$ that is providing $k=s(a)=s(b)=s(2a+3b)$.

We say that a subset $A$ of $\mathbb N$ is good if for some positive integer $n$, the equation $x-y=n$ admits infinitely many solutions with $x,y\in A$. If $A_1,A_2,\ldots,A_{100}$ are sets whose union is $\mathbb N$, prove that at least one of the $A_i$s is good.

Let's consider all possible quadratic trinomials of the form $x^2 + ax + b$, where $a$ and $b$ are positive integers not exceeding some positive integer $N$. Prove that the number of pairs of such trinomials having a common root does not exceed $N^2$.

Show that if $a$ and $b$ are integer numbers, and $a^2 + b^2 + 9ab$ is divisible by $11$, then $a^2-b^2$ divisible by $11$.

Prove that there infinitely many numbers of the form $18^{m}+45^{m}+50^{m}+125^{m}$, divisible by 2006. $m\in N$

Does there exist positive integers $a, b, c$ such that $4(ab - a - c^2) = b$?

[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].

If the representation of a positive number as a product of powers of distinct prime numbers contains no even powers other than $0$s, we will call the number singular. At most how many consequtive singular numbers are there? $ \textbf{(A)}\ 6 \qquad \textbf{(B)}\ 7 \qquad \textbf{(C)}\ 8 \qquad \textbf{(D)}\ 9 \qquad \textbf{(E)}\ \text{None}$

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]

What digit must be put in place of the "$?$" in the number $888...88?999...99$ (where the $8$ and $9$ are each written $50$ times) in order that the resulting number is divisible by $7$? (M . I. Gusarov)

Determine the number of ordered quadruples of integers $(a,b,c,d)$ for which $0\leq a,b,c,d\leq 36$ and $37|a^2+b^2-c^3-d^3$.

Show that there exist infinitely many integers that cannot be written as the difference between two perfect squares.

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.

Find all positive integers $ m$ with the property that the fourth power of the number of (positive) divisors of $ m$ equals $ m$.

Determine the integers $n\geqslant 2$ for which the equation $x^2-\hat{3}\cdot x+\hat{5}=\hat{0}$ has a unique solution in $(\mathbb{Z}_n,+,\cdot).$

Given two positives integers $m$ and $n$, prove that there exists a positive integer $k$ and a set $S$ of at least $m$ multiples of $n$ such that the numbers $\frac {2^k{\sigma({s})}} {s}$ are odd for every $s \in S$. $\sigma({s})$ is the sum of all positive integers of $s$ (1 and $s$ included).

Find all primes $p,q$ and even $n>2$ such that $p^n+p^{n-1}+...+1=q^2+q+1$.

Find all positive integers $n$ for which there exist two distinct numbers of $n$ digits, $\overline{a_1a_2\ldots a_n}$ and $\overline{b_1b_2\ldots b_n}$, such that the number of $2n$ digits $\overline{a_1a_2\ldots a_nb_1b_2\ldots b_n}$ is divisible by $\overline{b_1b_2\ldots b_na_1a_2\ldots a_n}$.

Let $n$ be a positive integer. Determine, in terms of $n$, the greatest integer which divides every number of the form $p + 1$, where $p \equiv 2$ mod $3$ is a prime number which does not divide $n$.

Find all pairs $\left(m,n\right)$ of positive integers, with $m,n\geq2$, such that $a^n-1$ is divisible by $m$ for each $a\in \left\{1,2,3,\ldots,n\right\}$.