a) Replace each letter in the following sum by a digit from $0$ to $9$, in such a way that the sum is correct. $\tab$ $\tab$ $ABC$ $\tab$ $\tab$ $DEF$ [u]$+GHI$[/u] $\tab$ $\tab$ $\tab$ $J J J$ Different letters must be replaced by different digits, and equal letters must be replaced by equal digits. Numbers $ABC$, $DEF$, $GHI$ and $JJJ$ cannot begin by $0$. b) Determine how many triples of numbers $(ABC,DEF,GHI)$ can be formed under the conditions given in a).

Find all pairs of positive integers $(x,y)$ such that $2^x + 3^y$ is a perfect square.

Let $n\ge 2$ be a positive integer. For any integer $a$, let $P_a(x)$ denote the polynomial $x^n+ax$. Let $p$ be a prime number and define the set $S_a$ as the set of residues mod $p$ that $P_a(x)$ attains. That is, $$S_a=\{b\mid 0\le b\le p-1,\text{ and there is }c\text{ such that }P_a(c)\equiv b \pmod{p}\}.$$Show that the expression $\frac{1}{p-1}\sum\limits_{a=1}^{p-1}|S_a|$ is an integer. [i]Proposed by fattypiggy123[/i]

Let $ a, b, c$ be integers satisfying $ 0 < a < c \minus{} 1$ and $ 1 < b < c$. For each $ k$, $ 0\leq k \leq a$, Let $ r_k,0 \leq r_k < c$ be the remainder of $ kb$ when divided by $ c$. Prove that the two sets $ \{r_0, r_1, r_2, \cdots , r_a\}$ and $ \{0, 1, 2, \cdots , a\}$ are different.

Let $x, y$ be distinct positive integers. Show that the number $$\frac{(x + y)^2}{x^3 + xy^2- x^2y -y^3}$$ is not an integer.

Suppose Alan, Michael, Kevin, Igor, and Big Rahul are in a running race. It is given that exactly one pair of people tie (for example, two people both get second place), so that no other pair of people end in the same position. Each competitor has equal skill; this means that each outcome of the race, given that exactly two people tie, is equally likely. The probability that Big Rahul gets first place (either by himself or he ties for first) can be expressed in the form $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$.

Prove that there are no integers $m$ and $n$ such that \[19m^2+95mn+2000n^2=1995.\]

[list=a] [*]Is there a non-linear integer-coefficient polynomial $P(x)$ and an integer $N{}$ such that all integers greater than $N{}$ may be written as the greatest common divisor of $P(a){}$ and $P(b){}$ for positive integers $a>b$? [*]Is there a non-linear integer-coefficient polynomial $Q(x)$ and an integer $M{}$ such that all integers greater than $M{}$ may be written as $Q(a) - Q(b)$ for positive integers $a>b$? [/list][i]Proposed by Dylan Toh[/i]

Let the sequences $(a_n)_{n=1}^{\infty}$ and $(b_n)_{n=1}^{\infty}$ satisfy $a_0 = b_0 = 1, a_n = 9a_{n-1} -2b_{n-1}$ and $b_n = 2a_{n-1} + 4b_{n-1}$ for all positive integers $n$. Let $c_n = a_n + b_n$ for all positive integers $n$. Prove that there do not exist positive integers $k, r, m$ such that $c^2_r = c_kc_m$.

Determine all functions $ f$ defined on the natural numbers that take values among the natural numbers for which \[ (f(n))^p \equiv n\quad {\rm mod}\; f(p) \] for all $ n \in {\bf N}$ and all prime numbers $ p$.

Let $X=\{1,2,\ldots,2001\}$. Find the least positive integer $m$ such that for each subset $W\subset X$ with $m$ elements, there exist $u,v\in W$ (not necessarily distinct) such that $u+v$ is of the form $2^{k}$, where $k$ is a positive integer.

Lisa writes a positive whole number in the decimal system on the blackboard and now makes in each turn the following: The last digit is deleted from the number on the board and then the remaining shorter number (or 0 if the number was one digit) becomes four times the number deleted number added. The number on the board is now replaced by the result of this calculation. Lisa repeats this until she gets a number for the first time was on the board. (a) Show that the sequence of moves always ends. (b) If Lisa begins with the number $53^{2022} - 1$, what is the last number on the board? Example: If Lisa starts with the number $2022$, she gets $202 + 4\cdot 2 = 210$ in the first move and overall the result $$2022 \to 210 \to 21 \to 6 \to 24 \to 18 \to 33 \to 15 \to 21$$. Since Lisa gets $21$ for the second time, the turn order ends. [i](Stephan Pfannerer)[/i]

Determine the number of positive divisors of $2008^8$ that are less than $2008^4$.

Show that for each natural number $n$, there exist $n$ distinct natural numbers whose sum is a square and whose product is a cube.

Let $n \geq 2$ be a natural. Define $$X = \{ (a_1,a_2,\cdots,a_n) | a_k \in \{0,1,2,\cdots,k\}, k = 1,2,\cdots,n \}$$. For any two elements $s = (s_1,s_2,\cdots,s_n) \in X, t = (t_1,t_2,\cdots,t_n) \in X$, define $$s \vee t = (\max \{s_1,t_1\},\max \{s_2,t_2\}, \cdots , \max \{s_n,t_n\} )$$ $$s \wedge t = (\min \{s_1,t_1 \}, \min \{s_2,t_2,\}, \cdots, \min \{s_n,t_n\})$$ Find the largest possible size of a proper subset $A$ of $X$ such that for any $s,t \in A$, one has $s \vee t \in A, s \wedge t \in A$.

Given a natural number $n$, call a divisor $d$ of $n$ to be $\textit{nontrivial}$ if $d>1$. A natural number $n$ is $\textit{good}$ if one or more distinct nontrivial divisors of $n$ sum up to $n-1$. Prove that every natural number $n$ has a multiple that is good.

A triangle of numbers is constructed as follows. The first row consists of the numbers from $1$ to $2000$ in increasing order, and under any two consecutive numbers their sum is written. (See the example corresponding to $5$ instead of $2000$ below.) What is the number in the lowermost row? 1 2 3 4 5 3 5 7 9 8 12 16 20 28 4

(a) Let $a$ and $b$ be rational numbers such that line $y = ax + b$ intersects the circle $x^2 + y^2 = 5$ at two different points. Show that if one of the intersections has two rational coordinates, so does the other intersection. (b) Show that there are infinitely many triples ($k, n, m$) that are such that $k^2 + n^2 = 5m^2$, where $k, n$ and $m$ are integers, and not all three have any in common prime factor.

Find the largest $k$ for which there exists a permutation $(a_1, a_2, \ldots, a_{2022})$ of integers from $1$ to $2022$ such that for at least $k$ distinct $i$ with $1 \le i \le 2022$ the number $\frac{a_1 + a_2 + \ldots + a_i}{1 + 2 + \ldots + i}$ is an integer larger than $1$. [i](Proposed by Oleksii Masalitin)[/i]

Suppose that $ k,l $ are natural numbers such that $ \gcd (11m-1,k)=\gcd (11m-1, l) , $ for any natural number $ m. $ Prove that there exists an integer $ n $ such that $ k=11^nl. $

Find all non-negative integers $m$ and $n$, such that $(2^n-1) \cdot (3^n-1)=m^2$.

Pedro writes all the numbers with four different digits that can be made with digits $a, b, c, d$, that meet the following conditions: $$ a\ne 0 \, , \, b=a+2 \, , \, c=b+2 \, , \, d=c+2$$ Find the sum of all the numbers Pedro wrote.

Let $M\subset \{1, 2, 3, . . . , 2007\}$ a set with the following property: Among every three numbers one can always choose two from $M$ such that one is divisible by the other. How many numbers can $M$ contain at most?

Find all pairs $x,y$ of natural numbers that satisfy the equation $$x^2-xy+2x-3y=1997$$

Find all sets comprising of 4 natural numbers such that the product of any 3 numbers in the set leaves a remainder of 1 when divided by the remaining number.