Let $S$ be the set of all 9-digit natural numbers, which are written only with the digits 1, 2, and 3. Find all functions $f:S\rightarrow \{1,2,3\}$ which satisfy the following conditions: (1) $f(111111111)=1$, $f(222222222)=2$, $f(333333333)=3$, $f(122222222)=1$; (2) If $x,y\in S$ differ in each digit position, then $f(x)\neq f(y)$.

For any positive integer $n$, show that there exists a polynomial $P(x)$ of degree $n$ with integer coefficients such that $P(0),P(1), \ldots, P(n)$ are all distinct powers of $2$.

Prove that there are infinitely many positive integers $m$, such that $n^4+m$ is not prime for any positive integer $n$.

Prove that there exist infinitely many triples of positive integers $(a,b,c)$ so that $a,b,c$ are pairwise coprime and $$\bigg \lfloor \frac{a^2}{2021} \bigg \rfloor + \bigg \lfloor \frac{b^2}{2021} \bigg \rfloor = \bigg \lfloor \frac{c^2}{2021} \bigg \rfloor.$$

Let $a$ and $b$ be integers such that $a - b = a^2c - b^2d$ for some consecutive integers $c$ and $d$. Prove that $|a - b|$ is a perfect square.

Let $S \subset Z^+$ be a set of positive integers with the following property: for any $a, b \in S$, if $a \ne b$ then $a + b$ is a perfect square. Given that $2009 \in S$ and $2087 \in S$, what is the maximum number of elements in $S$?

Positive integers $a,b,c$ satisfy that $a+b=b(a-c)$ and c+1 is a square of a prime. Prove that $a+b$ or $ab$ is a square.

Three members of the Euclid Middle School girls' softball team had the following conversation. Ashley: I just realized that our uniform numbers are all $2$-digit primes. Bethany: And the sum of your two uniform numbers is the date of my birthday earlier this month. Caitlin: That's funny. The sum of your two uniform numbers is the date of my birthday later this month. Ashley: And the sum of you two uniform numbers is today's date. What number does Caitlin wear? $\textbf{(A) }11\qquad\textbf{(B) }13\qquad\textbf{(C) }17\qquad\textbf{(D) }19\qquad \textbf{(E) }23$

Find all positive integers $ x$ and $ y$ such that $ 5^{x}-3^{y}= 16$.

Each square on an $8\times 8$ checkers board contains either one or zero checkers. The number of checkers in each row is a multiple of $3$, the number of checkers in each column is a multiple of $5$. Assuming the top left corner of the board is shown below, how many checkers are used in total? [img]https://cdn.artofproblemsolving.com/attachments/0/8/e46929e7ec3fff9be4892ef954ae299e0cb8c7.png[/img]

Let be a natural number $ a\ge 2. $ [b]a)[/b] Show that there is no infinite sequence $ \left( k_n \right)_{n\ge 1} $ of pairwise distinct natural numbers greater than $ 1 $ having the property that the sequence $ \left( a^{1/k_n} \right)_{n\ge 1} $ is a geometric progression. [b]b)[/b] Show that there are finite sequences $ \left( l_i \right)_i, $ of any length, of pairwise distinct natural numbers greater than $ 1 $ with the property that $ \left( a^{1/l_i} \right)_{i} $ is a geometric progression. [i]Bogdan Enescu[/i]

Let $f$ be a non-constant function from the set of positive integers into the set of positive integer, such that $a-b$ divides $f(a)-f(b)$ for all distinct positive integers $a$, $b$. Prove that there exist infinitely many primes $p$ such that $p$ divides $f(c)$ for some positive integer $c$. [i]Proposed by Juhan Aru, Estonia[/i]

The decimal representation of $m/n$, where $m$ and $n$ are relatively prime positive integers and $m < n$, contains the digits 2, 5, and 1 consecutively, and in that order. Find the smallest value of $n$ for which this is possible.

Find $5$ last digits of the number $M = 5^{2010}$ . (A): $65625$, (B): $45625$, (C): $25625$, (D): $15625$, (E) None of the above.

$n$ is an odd positive integer and $x,y$ are two rational numbers satisfying $$x^n+2y=y^n+2x.$$Prove that $x=y$.

Is it possible to arrange on a circle all composite positive integers not exceeding $ 10^6$, so that no two neighbouring numbers are coprime? [i]Author: L. Emelyanov[/i] [hide="Tuymaada 2008, Junior League, First Day, Problem 2."]Prove that all composite positive integers not exceeding $ 10^6$ may be arranged on a circle so that no two neighbouring numbers are coprime. [/hide]

Find all integers $x,y,z$ such that $x^3 +5y^3 = 9z^3$.

Let $p$ be an odd prime number and $c$ an integer for which $2c -1$ is divisible by $p$. Prove that $$(-1)^{\frac{p+1}{2}}+\sum_{n=0}^{\frac{p-1}{2}} {2n \choose n}c^n$$ is divisible by $p$.

Show that for any positive integer $n > 2$ we can find $n$ distinct positive integers such that the sum of their reciprocals is $1$.

[b]p1.[/b] Farmer John wants to bring some cows to a pasture with grass that grows at a constant rate. Initially, the pasture has some nonzero amount of grass and it will stop growing if there is no grass left. The pasture sustains $100$ cows for ten days. The pasture can also sustain $100$ cows for five days, and then $120$ cows for three more days. If cows eat at a constant rate, fund the maximum number of cows Farmer John can bring to the pasture so that they can be sustained indefinitely. [b]p2.[/b] Sam is learning basic arithmetic. He may place either the operation $+$ or $-$ in each of the blank spots between the numbers below: $$5\,\, \_ \,\, 8\,\, \_ \,\,9\,\, \_ \,\,7\,\,\_ \,\,2\,\,\_ \,\,3$$ In how many ways can he place the operations so the result is divisible by $3$? [b]p3.[/b] Will loves the color blue, but he despises the color red. In the $5\times 6$ rectangular grid below, how many rectangles are there containing at most one red square and with sides contained in the gridlines? [img]https://cdn.artofproblemsolving.com/attachments/1/7/7ce55bdc9e05c7c514dddc7f8194f3031b93c4.png[/img] [b]p4.[/b] Let $r_1, r_2, r_3$ be the three roots of a cubic polynomial $P(x)$. Suppose that $$\frac{P(2) + P(-2)}{P(0)}= 200.$$ If $\frac{1}{r_1r_2}+ \frac{1}{r_2r_3}+\frac{1}{r_3r_1}= \frac{m}{n}$ for relatively prime positive integers $m$ and $n$, compute $m + n$. [b]p5.[/b] Consider a rectangle $ABCD$ with $AB = 3$ and $BC = 1$. Let $O$ be the intersection of diagonals $AC$ and $BD$. Suppose that the circumcircle of $ \vartriangle ADO$ intersects line $AB$ again at $E \ne A$. Then, the length $BE$ can be written as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find $m + n$. [b]p6.[/b] Let $ABCD$ be a square with side length $100$ and $M$ be the midpoint of side $AB$. The circle with center $M$ and radius $50$ intersects the circle with center $D$ and radius $100$ at point $E$. $CE$ intersects $AB$ at $F$. If $AF = \frac{m}{n}$ for relatively prime positive integers $m$ and $n$, find $m + n$. [b]p7.[/b] How many pairs of real numbers $(x, y)$, with $0 < x, y < 1$ satisfy the property that both $3x + 5y$ and $5x + 2y$ are integers? [b]p8.[/b] Sebastian is coloring a circular spinner with $4$ congruent sections. He randomly chooses one of four colors for each of the sections. If two or more adjacent sections have the same color, he fuses them and considers them as one section. (Sections meeting at only one point are not adjacent.) Suppose that the expected number of sections in the final colored spinner is equal to $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m + n$. [b]p9.[/b] Let $ABC$ be a triangle and $D$ be a point on the extension of segment $BC$ past $C$. Let the line through $A$ perpendicular to $BC$ be $\ell$. The line through $B$ perpendicular to $AD$ and the line through $C$ perpendicular to $AD$ intersect $\ell$ at $H_1$ and $H_2$, respectively. If $AB = 13$, $BC = 14$, $CA = 15$, and $H_1H_2 = 1001$, find $CD$. [b]p10.[/b] Find the sum of all positive integers $k$ such that $$\frac21 -\frac{3}{2 \times 1}+\frac{4}{3\times 2\times 1} + ...+ (-1)^{k+1} \frac{k+1}{k\times (k - 1)\times ... \times 2\times 1} \ge 1 + \frac{1}{700^3}$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $n$ be an odd positive integer with $n > 1$ and let $a_1, a_2,... , a_n$ be positive integers such that gcd $(a_1, a_2,... , a_n) = 1$. Let $d$ = gcd $(a_1^n + a_1\cdot a_2 \cdot \cdot \cdot a_n, a_2^n + a_1\cdot a_2 \cdot \cdot \cdot a_n, ... , a_n^n + a_1\cdot a_2 \cdot \cdot \cdot a_n)$. Show that the possible values of $d$ are $d = 1, d = 2$

Determine all increasing sequences $\{a_n\}_{n=1}^\infty$ of natural numbers with the following property: for each two natural numbers $i$ and $j$ (not necessarily different), the numbers $i+j$ and $a_i+a_j$ have an equal number of distinct natural divisors.

How many ten-digit positive integers consist of ten different digits and are divisible by $99$?

Consider all $100$-digit numbers divisible by $19$. Prove that the number of such numbers not containing the digits $4, 5$, and $6$ is the number of such numbers that do not contain the digits $1, 4$ and $7$.

For each prime $p$, construct a graph $G_p$ on $\{1,2,\ldots p\}$, where $m\neq n$ are adjacent if and only if $p$ divides $(m^{2} + 1-n)(n^{2} + 1-m)$. Prove that $G_p$ is disconnected for infinitely many $p$