[u]Round 1[/u] [b]p1.[/b] Tom and Jerry stole a chain of $7$ sausages and are now trying to divide the bounty. They take turns biting the sausages at one of the connections. When one of them breaks a connection, he may eat any single sausages that may fall out. Tom takes the first bite. Each of them is trying his best to eat more sausages than his opponent. Who will succeed? [b]p2. [/b]The King of the Mountain Dwarves wants to light his underground throne room by placing several torches so that the whole room is lit. The king, being very miserly, wants to use as few torches as possible. What is the least number of torches he could use? (You should show why he can't do it with a smaller number of torches.) This is the shape of the throne room: [img]https://cdn.artofproblemsolving.com/attachments/b/2/719daafd91fc9a11b8e147bb24cb66b7a684e9.png[/img] Also, the walls in all rooms are lined with velvet and do not reflect the light. For example, the picture on the right shows how another room in the castle is partially lit. [img]https://cdn.artofproblemsolving.com/attachments/5/1/0f6971274e8c2ff3f2d0fa484b567ff3d631fb.png[/img] [b]p3.[/b] In the Hundred Acre Wood, all the animals are either knights or liars. Knights always tell the truth and liars always lie. One day in the Wood, Winnie-the-Pooh, a knight, decides to visit his friend Rabbit, also a noble knight. Upon arrival, Pooh finds his friend sitting at a round table with $5$ other guests. One-by-one, Pooh asks each person at the table how many of his two neighbors are knights. Surprisingly, he gets the same answer from everybody! "Oh bother!" proclaims Pooh. "I still don't have enough information to figure out how many knights are at this table." "But it's my birthday," adds one of the guests. "Yes, it's his birthday!" agrees his neighbor. Now Pooh can tell how many knights are at the table. Can you? [b]p4.[/b] Several girls participate in a tennis tournament in which each player plays each other player exactly once. At the end of the tournament, it turns out that each player has lost at least one of her games. Prove that it is possible to find three players $A$, $B$, and $C$ such that $A$ defeated $B$, $B$ defeated $C$, and $C$ defeated $A$. [b]p5.[/b] There are $40$ piles of stones with an equal number of stones in each. Two players, Ann and Bob, can select any two piles of stones and combine them into one bigger pile, as long as this pile would not contain more than half of all the stones on the table. A player who can’t make a move loses. Ann goes first. Who wins? [u]Round 2[/u] [b]p6.[/b] In a galaxy far, far away, there is a United Galactic Senate with $100$ Senators. Each Senator has no more than three enemies. Tired of their arguments, the Senators want to split into two parties so that each Senator has no more than one enemy in his own party. Prove that they can do this. (Note: If $A$ is an enemy of $B$, then $B$ is an enemy of $A$.) [b]p7.[/b] Harry has a $2012$ by $2012$ chessboard and checkers numbered from $1$ to $2012 \times 2012$. Can he place all the checkers on the chessboard in such a way that whatever row and column Professor Snape picks, Harry will be able to choose three checkers from this row and this column such that the product of the numbers on two of the checkers will be equal to the number on the third? [img]https://cdn.artofproblemsolving.com/attachments/b/3/a87d559b340ceefee485f41c8fe44ae9a59113.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that there do not exist natural numbers $ n\ge 10$ having all digits different from zero, and such that all numbers which are obtained by permutations of its digits are perfect squares.

Let $n = 2188 = 3^7+1$ and let $A_0^{(0)}, A_1^{(0)}, ..., A_{n-1}^{(0)}$ be the vertices of a regular $n$-gon (in that order) with center $O$ . For $i = 1, 2, \dots, 7$ and $j=0,1,\dots,n-1$, let $A_j^{(i)}$ denote the centroid of the triangle \[ \triangle A_j^{(i-1)} A_{j+3^{7-i}}^{(i-1)} A_{j+2 \cdot 3^{7-i}}^{(i-1)}. \] Here the subscripts are taken modulo $n$. If \[ \frac{|OA_{2014}^{(7)}|}{|OA_{2014}^{(0)}|} = \frac{p}{q} \] for relatively prime positive integers $p$ and $q$, find $p+q$. [i]Proposed by Yang Liu[/i]

Decide whether or not there exists a nonconstant polynomial $Q(x)$ with integer coefficients with the following property: for every positive integer $n > 2$, the numbers \[ Q(0), \; Q(1), Q(2), \; \dots, \; Q(n-1) \] produce at most $0.499n$ distinct residues when taken modulo $n$. [i]Proposed by Yang Liu[/i]

Let $p>13$ be a prime of the form $2q+1$, where $q$ is prime. Find the number of ordered pairs of integers $(m,n)$ such that $0\le m<n<p-1$ and \[3^m+(-12)^m\equiv 3^n+(-12)^n\pmod{p}.\] [i]Alex Zhu.[/i] [hide="Note"]The original version asked for the number of solutions to $2^m+3^m\equiv 2^n+3^n\pmod{p}$ (still $0\le m<n<p-1$), where $p$ is a Fermat prime.[/hide]

Find the minimal positive integer $m$, so that there exist positive integers $n>k>1$, which satisfy $11...1=11...1.m$, where the first number has $n$ digits $1$, and the second has $k$ digits $1$.

Find all positive integers $n$ such that $n!$ can be written as the product of two Fibonacci numbers.

A positive number $a$ is given, such that $a$ could be expressed as difference of two inverses of perfect squares ($a=\frac1{n^2}-\frac1{m^2}$). Is it possible for $2a$ to be expressed as difference of two perfect squares?

Let $M$ be the set of all integers in the form of $a^2+13b^2$, where $a$ and $b$ are distinct itnegers. i) Prove that the product of any two elements of $M$ is also an element of $M$. ii) Determine, reasonably, if there exist infinite pairs of integers $(x,y)$ so that $x+y\not\in M$ but $x^{13}+y^{13}\in M$.

Find an odd natural number not exceeding $1000$ if you know that the sum of the last digits of all its divisors (including $1$ and the number itself) is $33$.

If $a_1$ is a positive integer, form the sequence $a_1,a_2,a_3,\dots$ by letting $a_2$ be the product of the digits of $a_1$, etc.. If $a_k$ consists of a single digit, for some $k\ge 1$, $a_k$ is called a [i]digital root[/i] of $a_1$. It is easy to check that every positive integer has a unique root. $($For example, if $a_1=24378$, then $a_2=1344$, $a_3=48$, $a_4=32$, $a_5=6$, and thus $6$ is the digital root of $24378.)$ Prove that the digital root of a positive integer $n$ equals $1$ if, and only if, all the digits of $n$ equal $1$.

Is it possible to arrange the numbers $1, 2, . . . , 60$ in that order, so that the sum of any two numbers between which there is one number, divisible by $2$, the sum of any two numbers between which there are two numbers divisible by $3$, . . . , the sum of any two numbers between which there is are there six numbers, divisible by $7$?

Let $S$ be the sum of all numbers of the form $a/b,$ where $a$ and $b$ are relatively prime positive divisors of $1000.$ What is the greatest integer that does not exceed $S/10?$

Given $S$ be the finite lattice (with integer coordinate) set in the $xy$-plane. $A$ is the subset of $S$ with most elements such that the line connecting any two points in $A$ is not parallel to $x$-axis or $y$-axis. $B$ is the subset of integer with least elements such that for any $(x,y)\in S$, $x \in B$ or $y \in B$ holds. Prove that $|A| \geq |B|$.

Find all positive integers $1 \le n \le 2008$ so that there exist a prime number $p \ge n$ such that $$\frac{2008^p + (n -1)!}{n}$$ is a positive integer.

Determine all real number $(x,y)$ pairs that satisfy the equation. $$2x^2+y^2+7=2(x+1)(y+1)$$

Solve the following problems - A) Find any $158$ consecutive integers such that the sum of digits for any of the numbers is not divisible by $17.$ B) Prove that, among any $159$ consecutive integers there will always be at least one integer whose sum of digits is divisible by $17.$

Prove that there exists a real number $\varepsilon>0$ such that there are infinitely many sequences of integers $0<a_1<a_2<\hdots<a_{2025}$ satisfying \[\gcd(a_1^2+1, a_2^2+1,\hdots, a_{2025}^2+1) > a_{2025}^{1+\varepsilon}.\] [i]Pitchayut Saengrungkongka[/i]

Find all prime numbers $p$ for which there exist positive integers $x$, $y$, and $z$ such that the number $x^p + y^p + z^p - x - y - z$ is a product of exactly three distinct prime numbers.

For a positive integer $n$, let $g(n) = \left[ \displaystyle \frac{2024}{n} \right]$. Find the value of $$\sum_{n = 1}^{2024}\left(1 - (-1)^{g(n)}\right)\phi(n).$$

Let $a,b,c,d$ be positive integers such that $ab\equal{}cd$. Prove that $a\plus{}b\plus{}c\plus{}d$ is a composite number.

The positive $n > 3$ called ‘nice’ if and only if $n +1$ and $8n + 1$ are both perfect squares. How many positive integers $k \le 15$ such that $4n + k$ are composites for all nice numbers $n$?

Find all positive integers $n$ for which the largest prime divisor of $n^2+3$ is equal to the least prime divisor of $n^4+6.$

The Serpent Gorynych has $1976$ heads. The fabulous hero can cut down $33, 21, 17$ or $1$ head with one blow of the sword, but at the same time, the Serpent grows, respectively, $48, 0, 14$ or $349$ heads. If all the heads are cut off, then no new heads will grow. Will the hero be able to defeat the Serpent?

Let $ n$ be a positive integer, let $ A$ be a subset of $ \{1, 2, \cdots, n\}$, satisfying for any two numbers $ x, y\in A$, the least common multiple of $ x$, $ y$ not more than $ n$. Show that $ |A|\leq 1.9\sqrt {n} \plus{} 5$.