Let $\mathbb{F}_{13} = {\overline{0}, \overline{1}, \cdots, \overline{12}}$ be the finite field with $13$ elements (with sum and product modulus $13$). Find how many matrix $A$ of size $5$ x $5$ with entries in $\mathbb{F}_{13}$ exist such that $$A^5 = I$$ where $I$ is the identity matrix of order $5$

The integers $a_1,a_2,...,a_{2012}$ are given. Exactly $29$ of them are divisible by $3$. Prove that the sum $a_1^2+a_2^2+...+a_{2012}^2$ is divisible by $3$.

[u]Set 1[/u] [b]p1.[/b] Assume the speed of sound is $343$ m/s. Anastasia and Bananastasia are standing in a field in front of you. When they both yell at the same time, you hear Anastasia’s yell $5$ seconds before Bananastasia’s yell. If Bananastasia yells first, and then Anastasia yells when she hears Bananastasia yell, you hear Anastasia’s yell $5$ seconds after Bananastasia’s yell. What is the distance between Anastasia and Bananastasia in meters? [b]p2.[/b] Michelle picks a five digit number with distinct digits. She then reverses the digits of her number and adds that to her original number. What is the largest possible sum she can get? [b]p3.[/b] Twain is trying to crack a $4$-digit number combination lock. They know that the second digit must be even, the third must be odd, and the fourth must be different from the previous three. If it takes Twain $10$ seconds to enter a combination, how many hours would it take them to try every possible combination that satisfies these rules? PS. You should use hide for answers.

Find all the finite sequences with $ n$ consecutive natural numbers $ a_1, a_2,..., a_n$, with $ n\geq3$ such that $ a_1\plus{} a_2\plus{}...\plus{} a_n\equal{}2009$.

Prove that the equation $3y^2 = x^4 + x$ has no positive integer solutions.

Prove that there doesn’t exist any prime $p$ such that every power of $p$ is a palindrome (a palindrome is a number that is read the same from the left as it is from the right; in particular, a number that ends in one or more zeros cannot be a palindrome).

Let $m,n$ be positive integers. Find the minimum positive integer $N$ which satisfies the following condition. If there exists a set $S$ of integers that contains a complete residue system module $m$ such that $| S | = N$, then there exists a nonempty set $A \subseteq S$ so that $n\mid {\sum\limits_{x \in A} x }$.

Let $n\geq 5$ an integer and consider a regular $n$-gon. Initially, Nacho is situated in one of the vertices of the $n$-gon, in which he puts a flag. He will start moving clockwise. First, he moves one position and puts another flag, then, two positions and puts another flag, etcetera, until he finally moves $n-1$ positions and puts a flag, in such a way that he puts $n$ flags in total. ¿For which values of $n$, Nacho will have put a flag in each of the $n$ vertices?

Let $A$ be the sum of three consecutive integers and $B$ be the sum of the exact three consecutive integers. Is it possible to have $AB=33333$ ?

Sabine has a very large collection of shells. She decides to give part of her collection to her sister. On the first day, she lines up all her shells. She takes the shells that are in a position that is a perfect square (the first, fourth, ninth, sixteenth, etc. shell), and gives them to her sister. On the second day, she lines up her remaining shells. Again, she takes the shells that are in a position that is a perfect square, and gives them to her sister. She repeats this process every day. The $27$th day is the first day that she ends up with fewer than $1000$ shells. The $28$th day she ends up with a number of shells that is a perfect square for the tenth time. What are the possible numbers of shells that Sabine could have had in the very beginning?

Let $d$ be an even positive integer. John writes the numbers $1^2 ,3^2 ,\ldots,(2n-1)^2 $ on the blackboard and then chooses three of them, let them be ${a_1}, {a_2}, {a_3}$, erases them and writes the number $1+ \displaystyle\sum_{1\le i<j\leq 3} |{a_i} -{a_j}|$ He continues until two numbers remain written on on the blackboard. Prove that the sum of squares of those two numbers is different than the numbers $1^2 ,3^2 ,\ldots,(2n-1)^2$. (Albania)

Find all functions $f: \mathbb N \to \mathbb N$ such that for all positive integers $m,n$, [b](i)[/b] $mf(f(m))=\left( f(m) \right)^2$, [b](ii)[/b] If $\gcd(m,n)=d$, then $f(mn) \cdot f(d)=d \cdot f(m) \cdot f(n)$, [b](iii)[/b] $f(m)=m$ if and only if $m=1$.

Find the sum of all non-zero digits that can repeat at the end of a perfect square. (For example, if $811$ were a perfect square, $1$ would be one of these non-zero digits.)

Find all positive integers $a$ for which the equation $7an -3n! = 2020$ has a positive integer solution $n$. (Richard Henner)

Let $ f(x)\equal{}5x^{13}\plus{}13x^5\plus{}9ax$. Find the least positive integer $ a$ such that $ 65$ divides $ f(x)$ for every integer $ x$.

Let $k$ be a positive integer. Find all functions $f:\mathbb{N}\to \mathbb{N}$ satisfying the following two conditions:\\ • For infinitely many prime numbers $p$ there exists a positve integer $c$ such that $f(c)=p^k$.\\ • For all positive integers $m$ and $n$, $f(m)+f(n)$ divides $f(m+n)$.

Let $p$ be a prime greater than $2$. Prove that there is a prime $q < p$ such that $q^{p-1} - 1$ is not divisible by $p^2$

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 $p$

Find all polynomials $P$ with integer coefficients, for which there exists a number $N$, such that for every natural number $n \geq N$, all prime divisors of $n+2^{\lfloor \sqrt{n} \rfloor}$ are also divisors of $P(n)$.

Is it possible to assign every integral point $(x,y)$ of the plane a positive integer $a_{x,y}$ such that for every two integers $i$ and $j$ the following equality holds $$a_{i,j}=\gcd(a_{i-1,j},a_{i+1,j})+\gcd(a_{i,j-1},a_{i,j+1})$$ [i]M. Shutro[/i]

Let $p>5$ be a prime number. Show that $p-4$ cannot be the fourth power of a prime number.

[u]Round 1[/u] [b]p1.[/b] At a fortune-telling exam, $13$ witches are sitting in a circle. To pass the exam, a witch must correctly predict, for everybody except herself and her two neighbors, whether they will pass or fail. Each witch predicts that each of the $10$ witches she is asked about will fail. How many witches could pass? [b]p2.[/b] Out of $152$ coins, $7$ are counterfeit. All counterfeit coins have the same weight, and all real coins have the same weight, but counterfeit coins are lighter than real coins. How can you find $19$ real coins if you are allowed to use a balance scale three times? [b]p3.[/b] The digits of a number $N$ increase from left to right. What could the sum of the digits of $9 \times N$ be? [b]p4.[/b] The sides and diagonals of a pentagon are colored either blue or red. You can choose three vertices and flip the colors of all three lines that join them. Can every possible coloring be turned all blue by a sequence of such moves? [img]https://cdn.artofproblemsolving.com/attachments/5/a/644aa7dd995681fc1c813b41269f904283997b.png[/img] [b]p5.[/b] You have $100$ pancakes, one with a single blueberry, one with two blueberries, one with three blueberries, and so on. The pancakes are stacked in a random order. Count the number of blueberries in the top pancake and call that number $N$. Pick up the stack of the top $N$ pancakes and flip it upside down. Prove that if you repeat this counting-and-flipping process, the pancake with one blueberry will eventually end up at the top of the stack. [u]Round 2[/u] [b]p6.[/b] A circus owner will arrange $100$ fleas on a long string of beads, each flea on her own bead. Once arranged, the fleas start jumping using the following rules. Every second, each flea chooses the closest bead occupied by one or more of the other fleas, and then all fleas jump simultaneously to their chosen beads. If there are two places where a flea could jump, she jumps to the right. At the start, the circus owner arranged the fleas so that, after some time, they all gather on just two beads. What is the shortest amount of time it could take for this to happen? [b]p7.[/b] The faraway land of Noetheria has $2016$ cities. There is a nonstop flight between every pair of cities. The price of a nonstop ticket is the same in both directions, but flights between different pairs of cities have different prices. Prove that you can plan a route of $2015$ consecutive flights so that each flight is cheaper than the previous one. It is permissible to visit the same city several times along the way. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find the greatest prime number $p$ such that $p^3$ divides $$\frac{122!}{121}+ 123!:$$

Prove that among $39$ consecutive natural numbers, there is always a number that has sum of its digits divisible by $ 12$. Is it true if we replace $39$ with $38$?

[u]Round 1[/u] [b]p1.[/b] This year, the Mathathon consists of $7$ rounds, each with $3$ problems. Another math test, Aspartaime, consists of $3$ rounds, each with $5$ problems. How many more problems are on the Mathathon than on Aspartaime? [b]p2.[/b] Let the solutions to $x^3 + 7x^2 - 242x - 2016 = 0 $be $a, b$, and $c$. Find $a^2 + b^2 + c^2$. (You might find it helpful to know that the roots are all rational.) [b]p3.[/b] For triangle $ABC$, you are given $AB = 8$ and $\angle A = 30^o$ . You are told that $BC$ will be chosen from amongst the integers from $1$ to $10$, inclusive, each with equal probability. What is the probability that once the side length $BC$ is chosen there is exactly one possible triangle $ABC$? [u]Round 2 [/u] [b]p4.[/b] It’s raining! You want to keep your cat warm and dry, so you want to put socks, rain boots, and plastic bags on your cat’s four paws. Note that for each paw, you must put the sock on before the boot, and the boot before the plastic bag. Also, the items on one paw do not affect the items you can put on another paw. How many different orders are there for you to put all twelve items of rain footwear on your cat? [b]p5.[/b] Let $a$ be the square root of the least positive multiple of $2016$ that is a square. Let $b$ be the cube root of the least positive multiple of $2016$ that is a cube. What is $ a - b$? [b]p6.[/b] Hypersomnia Cookies sells cookies in boxes of $6, 9$ or $10$. You can only buy cookies in whole boxes. What is the largest number of cookies you cannot exactly buy? (For example, you couldn’t buy $8$ cookies.) [u]Round 3 [/u] [b]p7.[/b] There is a store that sells each of the $26$ letters. All letters of the same type cost the same amount (i.e. any ‘a’ costs the same as any other ‘a’), but different letters may or may not cost different amounts. For example, the cost of spelling “trade” is the same as the cost of spelling “tread,” even though the cost of using a ‘t’ may be different from the cost of an ‘r.’ If the letters to spell out $1$ cost $\$1001$, the letters to spell out $2$ cost $\$1010$, and the letters to spell out $11$ cost $\$2015$, how much do the letters to spell out $12$ cost? [b]p8.[/b] There is a square $ABCD$ with a point $P$ inside. Given that $PA = 6$, $PB = 9$, $PC = 8$. Calculate $PD$. [b]p9.[/b] How many ordered pairs of positive integers $(x, y)$ are solutions to $x^2 - y^2 = 2016$? [u]Round 4 [/u] [b]p10.[/b] Given a triangle with side lengths $5, 6$ and $7$, calculate the sum of the three heights of the triangle. [b]p11. [/b]There are $6$ people in a room. Each person simultaneously points at a random person in the room that is not him/herself. What is the probability that each person is pointing at someone who is pointing back to them? [b]p12.[/b] Find all $x$ such that $\sum_{i=0}^{\infty} ix^i =\frac34$. PS. You should use hide for answers. Rounds 5-7 have been posted [url=https://artofproblemsolving.com/community/c4h2782837p24446063]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].