Let $n$ be a positive integer. Let $a$ be an integer such that $\gcd (a,n)=1$. Prove that \[\frac{a^{\phi (n)}-1}{n}=\sum_{i\in R}\frac{1}{ai}\left[\frac{ai}{n}\right]\pmod{n}\] where $R$ is the reduced residue system of $n$ with each element a positive integer at most $n$.

Prove that there exist infinitely many positive integers $n$ such that the largest prime divisor of $n^4 + n^2 + 1$ is equal to the largest prime divisor of $(n+1)^4 + (n+1)^2 +1$.

A set $S$ of integers is Balearic, if there are two (not necessarily distinct) elements $s,s'\in S$ whose sum $s+s'$ is a power of two; otherwise it is called a non-Balearic set. Find an integer $n$ such that $\{1,2,\ldots,n\}$ contains a 99-element non-Balearic set, whereas all the 100-element subsets are Balearic.

Consider all pairs of positive integers $(a,b)$, with $a<b$, such that $\sqrt{a} +\sqrt{b} = \sqrt{2,160}$ Determine all possible values of $a$.

[u]Round 1[/u] [b]p1.[/b] A party is attended by ten people (men and women). Among them is Pat, who always lies to people of the opposite gender and tells the truth to people of the same gender. Pat tells five of the guests: “There are more men than women at the party.” Pat tells four of the guests: “There are more women than men at the party.” Is Pat a man or a woman? [b]p2.[/b] Once upon a time in a land far, far away there lived $100$ knights, $99$ princesses, and $101$ dragons. Over time, knights beheaded dragons, dragons ate princesses, and princesses poisoned knights. But they always obeyed an ancient law that prohibits killing any creature who has killed an odd number of others. Eventually only one creature remained alive. Could it have been a knight? A dragon? A princess? [b]p3.[/b] The numbers $1 \circ 2 \circ 3 \circ 4 \circ 5 \circ 6 \circ 7 \circ 8 \circ 9 \circ 10$ are written in a line. Alex and Vicky play a game, taking turns inserting either an addition or a multiplication symbol between adjacent numbers. The last player to place a symbol wins if the resulting expression is odd and loses if it is even. Alex moves first. Who wins? (Remember that multiplication is performed before addition.) [b]p4.[/b] A chess tournament had $8$ participants. Each participant played each other participant once. The winner of a game got $1$ point, the loser $0$ points, and in the case of a draw each got $1/2$ a point. Each participant scored a different number of points, and the person who got $2$nd prize scored the same number of points as the $5$th, $6$th, $7$th and $8$th place participants combined. Can you determine the result of the game between the $3$rd place player and the $5$th place player? [b]p5.[/b] One hundred gnomes sit in a circle. Each gnome gets a card with a number written on one side and a different number written on the other side. Prove that it is possible for all the gnomes to lay down their cards so that no two neighbors have the same numbers facing up. [u]Round 2[/u] [b]p6.[/b] A casino machine accepts tokens of $32$ different colors, one at a time. For each color, the player can choose between two fixed rewards. Each reward is up to $\$10$ cash, plus maybe another token. For example, a blue token always gives the player a choice of getting either $\$5$ plus a red token or $\$3$ plus a yellow token; a black token can always be exchanged either for $\$10$ (but no token) or for a brown token (but no cash). A player may keep playing as long as he has a token. Rob and Bob each have one white token. Rob watches Bob play and win $\$500$. Prove that Rob can win at least $\$1000$. [img]https://cdn.artofproblemsolving.com/attachments/6/6/e55614bae92233c9b2e7d66f5f425a18e6475a.png[/img] [b]p7.[/b] Each of the $100$ residents of Pleasantville has at least $30$ friends in town. A resident votes in the mayoral election only if one of her friends is a candidate. Prove that it is possible to nominate two candidates for mayor so that at least half of the residents will vote. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $f$ be a polynomial of degree $n$ with integer coefficients and $p$ a prime for which $f$, considered modulo $p$, is a degree-$k$ irreducible polynomial over $\mathbb{F}_p$. Show that $k$ divides the degree of the splitting field of $f$ over $\mathbb{Q}$.

Prove that for all primes \( p \) such that \( p \equiv 3 \pmod{4} \) or \( p \equiv 5 \pmod{8} \), there exist integers \[ 1 \leq a_1 < a_2 < \cdots < a_{(p-1)/2} < p \] such that \[ \prod_{\substack{1 \leq i < j \leq (p-1)/2}} (a_i + a_j)^2 \equiv 1 \pmod{p}. \]

Prove that the sequence defined by: $$ y_ {n + 1} = \frac {1} {2} (3y_ {n} + \sqrt {5y_ {n} ^ {2} -4}) , \,\, \forall n \ge 0$$ with $ y_ {0} = 1$ consists only of integers.

Let $n$ be an integer greater than $6$.Show that if $n+1$ is a prime number,than $\left\lceil \frac{(n-1)!}{n(n+1)}\right \rceil$ is $ODD.$

Let $n \geqslant 100$ be an integer. Ivan writes the numbers $n, n+1, \ldots, 2 n$ each on different cards. He then shuffles these $n+1$ cards, and divides them into two piles. Prove that at least one of the piles contains two cards such that the sum of their numbers is a perfect square.

If $a$ and $b$ are positive integers, we say that $a$ [i]almost divides[/i] $b$ if $a$ divides at least one of $b - 1$ and $b + 1$. We call a positive integer $n$ [i]almost prime[/i] if the following holds: for any positive integers $a, b$ such that $n$ almost divides $ab$, we have that $n$ almost divides at least one of $a$ and $b$. Determine all almost prime numbers. [hide = original link][url]https://mathematical.olympiad.ch/fileadmin/user_upload/Archiv/Intranet/Olympiads/Mathematics/deploy/exams/2024/FinalRound/Exam/englishFinalRound2024.pdf[/url]!![/hide]

Find all $x;y\in\mathbb{Z}$ satisfying the following condition: $$x^3=y^4+9x^2$$

Let $a,b,c$ be positive integers such that $a^2 - bc$ is a square. Prove that $2a + b + c$ is not prime. [i]Evan o'Dorney[/i]

Find all polynomials $f(x)$ with integer coefficients such that $f(n)$ and $f(2^{n})$ are co-prime for all natural numbers $n$.

A positive number $\dfrac{m}{n}$ has the property that it is equal to the ratio of $7$ plus the number’s reciprocal and $65$ minus the number’s reciprocal. Given that $m$ and $n$ are relatively prime positive integers, find $2m + n$.

Prove that between every $27$ different positive integers , less than $100$, there exist some two which are[color=red] NOT [/color]relative prime. [u]babis[/u]

[u]Round 1[/u] [b]p1.[/b] Alex is on a week-long mining quest. Each morning, she mines at least $1$ and at most $10$ diamonds and adds them to her treasure chest (which already contains some diamonds). Every night she counts the total number of diamonds in her collection and finds that it is divisible by either $22$ or $25$. Show that she miscounted. [b]p2.[/b] Hermione set out a row of $11$ Bertie Bott’s Every Flavor Beans for Ron to try. There are $5$ chocolateflavored beans that Ron likes and $6$ beans flavored like earwax, which he finds disgusting. All beans look the same, and Hermione tells Ron that a chocolate bean always has another chocolate bean next to it. What is the smallest number of beans that Ron must taste to guarantee he finds a chocolate one? [b]p3.[/b] There are $101$ pirates on a pirate ship: the captain and $100$ crew. Each pirate, including the captain, starts with $1$ gold coin. The captain makes proposals for redistributing the coins, and the crew vote on these proposals. The captain does not vote. For every proposal, each crew member greedily votes “yes” if he gains coins as a result of the proposal, “no” if he loses coins, and passes otherwise. If strictly more crew members vote “yes” than “no,” the proposal takes effect. The captain can make any number of proposals, one after the other. What is the largest number of coins the captain can accumulate? [b]p4.[/b] There are $100$ food trucks in a circle and $10$ gnomes who sample their menus. For the first course, all the gnomes eat at different trucks. For each course after the first, gnome #$1$ moves $1$ truck left or right and eats there; gnome #$2$ moves $2$ trucks left or right and eats there; ... gnome #$10$ moves $10$ trucks left or right and eats there. All gnomes move at the same time. After some number of courses, each food truck had served at least one gnome. Show that at least one gnome ate at some food truck twice. [b]p5.[/b] The town of Lumenville has $100$ houses and is preparing for the math festival. The Tesla wiring company lays lengths of power wire in straight lines between the houses so that power flows between any two houses, possibly by passing through other houses.The Edison lighting company hangs strings of lights in straight lines between pairs of houses so that each house is connected by a string to exactly one other. Show that however the houses are arranged, the Edison company can always hang their strings of lights so that the total length of the strings is no more than the total length of the power wires the Tesla company used. [img]https://cdn.artofproblemsolving.com/attachments/9/2/763de9f4138b4dc552247e9316175036c649b6.png[/img] [u]Round 2[/u] [b]p6.[/b] What is the largest number of zeros that could appear at the end of $1^n + 2^n + 3^n + 4^n$, where n can be any positive integer? [b]p7.[/b] A tennis academy has $2023$ members. For every group of 1011 people, there is a person outside of the group who played a match against everyone in it. Show there is someone who has played against all $2022$ other members. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Suppose that the positive integers $ a$ and $ b$ satisfy the equation $ a^b\minus{}b^a\equal{}1008$ Prove that $ a$ and $ b$ are congruent modulo 1008.

Let $S=\{1,2,\ldots ,98\}$. Find the least natural number $n$ such that we can pick out $10$ numbers in any $n$-element subset of $S$ satisfying the following condition: no matter how we equally divide the $10$ numbers into two groups, there exists a number in one group such that it is coprime to the other numbers in that group, meanwhile there also exists a number in the other group such that it is not coprime to any of the other numbers in the same group.

Let $a,b,c$ be positive integers. Prove that $a^2+b^2+c^2$ is divisible by $4$, if and only if $a,b,c$ are even.

Compute \[\sum_{k=1}^{420} \gcd(k,420).\] [i]2016 CCA Math Bonanza Individual #14[/i]

Let all positive integers $n$ satisfy the following condition: for each non-negative integers $k, m$ with $k + m \le n$, the numbers $\binom{n-k}{m}$ and $\binom{n-m}{k}$ leave the same remainder when divided by $2$. (Poland) PS. The translation was done using Google translate and in case it is not right, there is the original text in Slovak

Show that if $n \ge 4, n \in N$ and $\big [ \frac{2^n}{n} ]$ is a power of $2$, then $n$ is a power of $2$.

What is the last digit in the number $2007^{2007}$?

Call integer number $x$ as far from squares and cubes, if for every integer $k$ it is true : $|x-k^2|>10^6,|x-k^3|>10^6$. Prove, that there are infinitely many far from squares and cubes degrees of $2$