Let $ p$ be a prime number and let $ 0\leq a_{1}< a_{2}<\cdots < a_{m}< p$ and $ 0\leq b_{1}< b_{2}<\cdots < b_{n}< p$ be arbitrary integers. Let $ k$ be the number of distinct residues modulo $ p$ that $ a_{i}\plus{}b_{j}$ give when $ i$ runs from 1 to $ m$, and $ j$ from 1 to $ n$. Prove that a) if $ m\plus{}n > p$ then $ k \equal{} p$; b) if $ m\plus{}n\leq p$ then $ k\geq m\plus{}n\minus{}1$.

For a nonnegative integer $n$ define $\operatorname{rad}(n)=1$ if $n=0$ or $n=1$, and $\operatorname{rad}(n)=p_1p_2\cdots p_k$ where $p_1<p_2<\cdots <p_k$ are all prime factors of $n$. Find all polynomials $f(x)$ with nonnegative integer coefficients such that $\operatorname{rad}(f(n))$ divides $\operatorname{rad}(f(n^{\operatorname{rad}(n)}))$ for every nonnegative integer $n$.

Let $\mathbb{Z} _{>0}$ be the set of positive integers. Find all functions $f: \mathbb{Z} _{>0}\rightarrow \mathbb{Z} _{>0}$ such that \[ m^2 + f(n) \mid mf(m) +n \] for all positive integers $m$ and $n$.

Show that the equation $2x^2-3x=3y^2$ has infinitely many solutions in positive integers.

Find the number of trailing zeros at the end of the base-$10$ representation of the integer $525^{25^2} \cdot 252^{52^5}$ .

Is there an infinite periodic sequence of digits for which the following condition condition is fulfilled: for any natural number $n{}$ a natural number divisible by $2^n{}$ can be cut from this sequence of digits (as a word)? [i]Proposed by P. Kozhevnikov[/i]

Let $n$ be a natural number. We have $n$ colors. Each of the numbers $1, 2, 3,... , 1000$ was colored with one of the $n$ colors. It is known that, for any two distinct numbers, if one divides the other then these two numbers have different colors. Determine the smallest possible value of $n$.

Prove that $27 195^8 - 10 887^8 + 10 152^8$ is divisible by $26 460$.

Kailey starts with the number $0$, and she has a fair coin with sides labeled $1$ and $2$. She repeatedly flips the coin, and adds the result to her number. She stops when her number is a positive perfect square. What is the expected value of Kailey’s number when she stops? If E is your estimate and A is the correct answer, you will receive $\left\lfloor 25e^{-\frac{5|E-A|}{2} }\right\rfloor$ points.

Find a positive integer $x$, with $x> 1$ such that all numbers in the sequence $$x + 1,x^x + 1,x^{x^x}+1,...$$ are divisible by $2009.$

For which real numbers $a$, there exist pairwise different real numbers $x, y, z$ satisfying $$\frac{x^3+a}{y+z}=\frac{y^3+a}{x+z}=\frac{z^3+a}{x+y}= -3.$$

Each of $27$ bricks (right rectangular prisms) has dimensions $a \times b \times c$, where $a$, $b$, and $c$ are pairwise relatively prime positive integers. These bricks are arranged to form a $3 \times 3 \times 3$ block, as shown on the left below. A $28$[sup]th[/sup] brick with the same dimensions is introduced, and these bricks are reconfigured into a $2 \times 2 \times 7$ block, shown on the right. The new block is $1$ unit taller, $1$ unit wider, and $1$ unit deeper than the old one. What is $a + b + c$? [img]https://cdn.artofproblemsolving.com/attachments/2/d/b18d3d0a9e5005c889b34e79c6dab3aaefeffd.png[/img] $ \textbf{(A) }88 \qquad \textbf{(B) }89 \qquad \textbf{(C) }90 \qquad \textbf{(D) }91 \qquad \textbf{(E) }92 \qquad $

Prove that the parity of each term of the sequence $ \left( \left\lfloor \left( \lfloor \sqrt q \rfloor +\sqrt{q} \right)^n \right\rfloor \right)_{n\ge 1} $ is opposite to the parity of its index, where $ q $ is a squarefree natural number.

Any natural number $n, n\ge 3$ can be presented in different ways as a sum several summands (not necessarily different). Find the greatest possible value of these summands. Folklore

Call admissible a set $A$ of integers that has the following property: If $x,y \in A$ (possibly $x=y$) then $x^2+kxy+y^2 \in A$ for every integer $k$. Determine all pairs $m,n$ of nonzero integers such that the only admissible set containing both $m$ and $n$ is the set of all integers. [i]Proposed by Warut Suksompong, Thailand[/i]

[b]p1.[/b] The English alphabet consists of $21$ consonants and $5$ vowels. (We count $y$ as a consonant.) (a) Suppose that all the letters are listed in an arbitrary order. Prove that there must be $4$ consecutive consonants. (b) Give a list to show that there need not be $5$ consecutive consonants. (c) Suppose that all the letters are arranged in a circle. Prove that there must be $5$ consecutive consonants. [b]p2.[/b] From the set $\{1,2,3,... , n\}$, $k$ distinct integers are selected at random and arranged in numerical order (lowest to highest). Let $P(i, r, k, n)$ denote the probability that integer $i$ is in position $r$. For example, observe that $P(1, 2, k, n) = 0$. (a) Compute $P(2, 1,6,10)$. (b) Find a general formula for $P(i, r, k, n)$. [b]p3.[/b] (a) Write down a fourth degree polynomial $P(x)$ such that $P(1) = P(-1)$ but $P(2) \ne P(-2)$ (b) Write down a fifth degree polynomial $Q(x)$ such that $Q(1) = Q(-1)$ and $Q(2) = Q(-2)$ but $Q(3) \ne Q(-3)$. (c) Prove that, if a sixth degree polynomial $R(x)$ satisfies $R(1) = R(-1)$, $R(2) = R(-2)$, and $R(3) = R(-3)$, then $R(x) = R(-x)$ for all $x$. [b]p4.[/b] Given five distinct real numbers, one can compute the sums of any two, any three, any four, and all five numbers and then count the number $N$ of distinct values among these sums. (a) Give an example of five numbers yielding the smallest possible value of $N$. What is this value? (b) Give an example of five numbers yielding the largest possible value of $N$. What is this value? (c) Prove that the values of $N$ you obtained in (a) and (b) are the smallest and largest possible ones. [b]p5.[/b] Let $A_1A_2A_3$ be a triangle which is not a right triangle. Prove that there exist circles $C_1$, $C_2$, and $C_3$ such that $C_2$ is tangent to $C_3$ at $A_1$, $C_3$ is tangent to $C_1$ at $A_2$, and $C_1$ is tangent to $C_2$ at $A_3$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Call two regular polygons supplementary if the sum of an internal angle from each polygon adds up to $180^o$. For instance, two squares are supplementary because the sum of the internal angles is $90^o + 90^o = 180^o$. Find the other pair of supplementary polygons. Write your answer in the form $(m, n)$ where m and n are the number of sides of the polygons and $m < n$.

Prove that the equation \[y^3=x^2+5\] doesn't have any solutions in $Z$.

Find all integers $a$ and $b$ satisfying the equality $3^a - 5^b = 2$. (I. Gorodnin)

Determine all positive integers $n$ such that $\frac{n(n-1)}{2}-1$ divides $1^7+2^7+\dots +n^7$.

Find all prime numbers $p$ for which $p^2 +11$ has exactly six positive divisors.

Let $\mathbb{Z}_{>0}$ denote the set of positive integers. For any positive integer $k$, a function $f: \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$ is called [i]$k$-good[/i] if $\gcd(f(m) + n, f(n) + m) \le k$ for all $m \neq n$. Find all $k$ such that there exists a $k$-good function. [i]Proposed by James Rickards, Canada[/i]

Let $p$ be a prime. Let $f(x)$ be the number of ordered pairs $(a, b)$ of positive integers less than $p$, such that $a^b \equiv x \pmod p$. Suppose that there do not exist positive integers $x$ and $y$, both less than $p$, such that $f(x) = 2f(y)$, and that the maximum value of $f$ is greater than $2018$. Find the smallest possible value of $p$.

Consider all the sums of the form \[\displaystyle\sum_{k=1}^{1985} e_kk^5=\pm 1^5\pm 2^5\pm\cdots\pm1985^5\] where $e_k=\pm 1$. What is the smallest nonnegative value attained by a sum of this type?

The difference between the product and the sum of two different integers is equal to the sum of their GCD (greatest common divisor) and LCM (least common multiple). Findall these pairs of numbers. Justify your answer.