Given a positive odd integer $n$, show that the arithmetic mean of fractional parts $\{\frac{k^{2n}}{p}\}, k=1,..., \frac{p-1}{2}$ is the same for infinitely many primes $p$ .

[b]p1.[/b] Warren interrogates the $25$ members of his cabinet, each of whom always lies or always tells the truth. He asks them all, “How many of you always lie?” He receives every integer answer from $1$ to $25$ exactly once. Find the actual number of liars in his cabinet. [b]p2.[/b] Abraham thinks of distinct nonzero digits $E$, $M$, and $C$ such that $E +M = \overline{CC}$. Help him evaluate the sum of the two digit numbers $\overline{EC}$ and $\overline{MC}$. (Note that $\overline{CC}$, $\overline{EC}$, and $\overline{MC}$ are read as two-digit numbers.) [b]p3.[/b] Let $\omega$, $\Omega$, $\Gamma$ be concentric circles such that $\Gamma$ is inside $\Omega$ and $\Omega$ is inside $\omega$. Points $A,B,C$ on $\omega$ and $D,E$ on $\Omega$ are chosen such that line $AB$ is tangent to $\Omega$, line $AC$ is tangent to $\Gamma$, and line $DE$ is tangent to $\Gamma$. If $AB = 21$ and $AC = 29$, find $DE$. [b]p4.[/b] Let $a$, $b$, and $c$ be three prime numbers such that $a + b = c$. If the average of two of the three primes is four less than four times the fourth power of the last, find the second-largest of the three primes. [b]p5.[/b] At Stillwells Ice Cream, customers must choose one type of scoop and two different types of toppings. There are currently $630$ different combinations a customer could order. If another topping is added to the menu, there would be $840$ different combinations. If, instead, another type of scoop were added to the menu, compute the number of different combinations there would be. [b]p6.[/b] Eleanor the ant takes a path from $(0, 0)$ to $(20, 24)$, traveling either one unit right or one unit up each second. She records every lattice point she passes through, including the starting and ending point. If the sum of all the $x$-coordinates she records is $271$, compute the sum of all the $y$-coordinates. (A lattice point is a point with integer coordinates.) [b]p7.[/b] Teddy owns a square patch of desert. He builds a dam in a straight line across the square, splitting the square into two trapezoids. The perimeters of the trapezoids are$ 64$ miles and $76$ miles, and their areas differ by $135$ square miles. Find, in miles, the length of the segment that divides them. [b]p8.[/b] Michelle is playing Spot-It with a magical deck of $10$ cards. Each card has $10$ distinct symbols on it, and every pair of cards shares exactly $1$ symbol. Find the minimum number of distinct symbols on all of the cards in total. [b]p9.[/b] Define the function $f(n) = \frac{1}{2^n} + \frac{1}{3^n} + \frac{1}{4^n} + ...$ for integers $n \ge 2$. Find $$f(2) + f(4) + f(6) + ... .$$ [b]p10.[/b] There are $9$ indistinguishable ants standing on a $3\times 3$ square grid. Each ant is standing on exactly one square. Compute the number of different ways the ants can stand so that no column or row contains more than $3$ ants. [b]p11.[/b] Let $s(N)$ denote the sum of the digits of $N$. Compute the sum of all two-digit positive integers $N$ for which $s(N^2) = s(N)^2$. [b]p12.[/b] Martha has two square sheets of paper, $A$ and $B$. With each sheet, she repeats the following process four times: fold bottom side to top side, fold right side to left side. With sheet $A$, she then makes a cut from the top left corner to the bottom right. With sheet $B$, she makes a cut from the bottom left corner to the top right. Find the total number of pieces of paper yielded from sheets $A$ and sheets $B$. [img]https://cdn.artofproblemsolving.com/attachments/f/6/ff3a459a135562002aa2c95067f3f01441d626.png[/img] [b]p13.[/b] Let $x$ and $y$ be positive integers such that gcd $(x^y, y^x) = 2^{28}$. Find the sum of all possible values of min $(x, y)$. [b]p14.[/b] Convex hexagon $TRUMAN$ has opposite sides parallel. If each side has length $3$ and the area of this hexagon is $5$, compute $$TU \cdot RM \cdot UA \cdot MN \cdot AT \cdot NR.$$ [b]p15.[/b] Let $x$, $y$, and $z$ be positive real numbers satisfying the system $$\begin{cases} x^2 + xy + y^2 = 25\\ y^2 + yz + z^2 = 36 \\ z^2 + zx + x^2 = 49 \end{cases}$$ Compute $x^2 + y^2 + z^2$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Given $k \in \mathbb{Z}$ prove that there exist infinite pairs of distinct natural numbers such that \begin{align*} n+s(2n)=m+s(2m) \\ kn+s(n^2)=km+s(m^2). \end{align*} ($s(n)$ denotes the sum of digits of $n$.) [i]Proposed by Mohammadamin Sharifi[/i]

Are there $10$ consecutive positive integers, such that if we consider the digits that appear in the decimal representations of those numbers as a multiset, every digit appears the same number of times in this multiset?

Given a natural number $n$. Prove that you can choose $ \phi (n)+1 $ (not necessarily different) divisors $n$ with the sum $n$. Here $ \phi (n)$ denotes the number of natural numbers less than $n$ that are coprime with $n$. (Fedir Yudin)

Given an integer $k\geq 2$, determine all functions $f$ from the positive integers into themselves such that $f(x_1)!+f(x_2)!+\cdots f(x_k)!$ is divisibe by $x_1!+x_2!+\cdots x_k!$ for all positive integers $x_1,x_2,\cdots x_k$. $Albania$

Find the number of ways that a positive integer $n$ can be represented as a sum of one or more consecutive positive integers.

Find all positive integers \(n\) such that among the \(n\) numbers \[ 2n + 1, \, 2^2 n + 1, \, \ldots, \, 2^n n + 1 \] there are \(n\), \(n - 1\), or \(n - 2\) primes.

Three athletes ran at different constant speeds along a track of length $1$. They started moving at the same time at one end of the track. Having reached one of the ends of the track, the athlete immediately turned around and continued running in the opposite direction. After a while, all three athletes met at the start and finished training. At what maximum $S$ can we knowingly say that at some point the sum of the pairwise distances between athletes was at least $S$? [i]Proposed by A. Golovanov, I. Rubanov[/i]

For a given positive integer $n$ one has to choose positive integers $a_0, a_1,...$ so that the following conditions hold: (1) $a_i = a_{i+n}$ for any $i$, (2) $a_i$ is not divisible by $n$ for any $i$, (3) $a_{i+a_i}$ is divisible by $a_i$ for any $i$. For which positive integers $n > 1$ is this possible only if the numbers $a_0, a_1, ...$ are all equal?

Let $p$ be an arbitrary odd prime and $\sigma(n)$ for $1 \le n \le p-1$ denote the inverse of $n \pmod p$. Show that the number of pairs $(a,b) \in \{1,2,\cdots,p-1\}^2$ with $a<b$ but $\sigma(a) > \sigma(b)$ is at least $$\left \lfloor \left(\frac{p-1}{4}\right)^2 \right \rfloor$$ [i]usjl[/i] Note: Partial credits may be awarded if the $4$ in the statement is replaced with some larger constant

For a positive integer $n$, let $S_n$ be the set of positive integers that do not exceed $n$ and are coprime to $n$. Define $f(n)$ as the smallest positive integer that allows $S_n$ to be partitioned into $f(n)$ disjoint subsets, each forming an arithmetic progression. Prove that there exist infinitely many pairs $(a, b)$ satisfying $a, b > 2025$, $a \mid b$, and $f(a) \nmid f(b)$.

Let $p$ be a prime number. Prove that there exists a prime number $q$ such that for every integer $n$, the number $n^p-p$ is not divisible by $q$.

Ana and Luca play the following game. Ana writes a list of $n$ different integer numbers. Luca wins if he can choose four different numbers, $a, b, c$ and $d$, so that the number $a+b-(c+d)$ is multiple of $20$. Determine the minimum value of $n$ for which, whatever Ana's list, Luca can win.

Take the number $2^{2004}$ and calculate the sum $S$ of all its digits. Then the sum of all the digits of $S$ is calculated to obtain $R$. Next, the sum of all the digits of $R$is calculated and so on until a single digit number is reached. Find it. (For example if we take $2^7=128$, we find that $S=11,R=2$. So in this case of $2^7$ the searched digit will be $2$).

Define the sequence of integers $\langle a_n\rangle$ as; \[a_0=1, \quad a_1=-1, \quad \text{ and } \quad a_n=6a_{n-1}+5a_{n-2} \quad \forall n\geq 2.\] Prove that $a_{2012}-2010$ is divisible by $2011.$

The numbers from 1 to 37 are written in a line so that the sum of any first several numbers is divided by the number following them. What number is worth in third place, if the number 37 is written in the first place, and in the second, 1?

Suppose $p > 3$ is a prime number and \[S = \sum_{2 \le i < j < k \le p-1} ijk\] Prove that $S+1$ is divisible by $p$.

Let $p \equiv 2 \pmod 3$ be a prime, $k$ a positive integer and $P(x) = 3x^{\frac{2p-1}{3}}+3x^{\frac{p+1}{3}}+x+1$. For any integer $n$, let $R(n)$ denote the remainder when $n$ is divided by $p$ and let $S = \{0,1,\cdots,p-1\}$. At each step, you can either (a) replaced every element $i$ of $S$ with $R(P(i))$ or (b) replaced every element $i$ of $S$ with $R(i^k)$. Determine all $k$ such that there exists a finite sequence of steps that reduces $S$ to $\{0\}$. [i]Proposed by fattypiggy123[/i]

The sequence $a_n$ for $n \in \mathbb{N}$ is defined as follows: \[ a_0 = 6, \quad a_1 = 7, \quad a_{n+2} = 3a_{n+1} - 2a_n \] Find all values of $n$ such that $n^2 = a_n$.

A pair of positive integer numbers \((a, b)\) is given. It turns out that for every positive integer number \(n\), for which the numbers \((n - a)(n + b)\) and \(n^2 - ab\) are positive, they have the same number of divisors. Is it necessarily true that \(a = b\)? [i]Proposed by Oleksii Masalitin[/i]

Let $n$ be a positive integer, such that $125n+22$ is a power of $3$. Prove that $125n+29$ has a prime factor greater than $100$.

In a class, the total numbers of boys and girls are in the ratio $4 : 3$. On one day it was found that $8$ boys and $14$ girls were absent from the class, and that the number of boys was the square of the number of girls. What is the total number of students in the class?

Find all positive integers $a$, $b$ and $c$ such that $ab$ is a square, and \[a+b+c-3\sqrt[3]{abc}=1.\] [i]Proposed by usjl[/i]

A number of $N$ children are at a party and they sit in a circle to play a game of Pass and Parcel. Because the host has no other form of entertainment, the parcel has infinitely many layers. On turn $i$, starting with $i=1$, the following two things happen in order: [b]$(1)$[/b] The parcel is passed $i^2$ positions clockwise; and [b]$(2)$[/b] The child currently holding the parcel unwraps a layer and claims the prize inside. For what values of $N$ will every chidren receive a prize? $Patrick \ Winter \, United \ Kingdom$