Let $p$ be a prime number. At a school of $p^{2020}$ students it is required that each club consist of exactly $p$ students. Is it possible for each pair of students to have exactly one club in common?

Let $n$ be a prime number. Show that there is a permutation $a_1,a_2,...,a_n$ of $1,2,...,n$ so that $a_1,a_1a_2,...,a_1a_2...a_n$ leave distinct remainders when divided by $n$

Find all pairs $(m,n)$ of nonnegative integers for which \[m^2 + 2 \cdot 3^n = m\left(2^{n+1} - 1\right).\] [i]Proposed by Angelo Di Pasquale, Australia[/i]

Prove that there exist infinitely many positive integers $n$ such that $3^n+2$ and $5^n+2$ are all composite numbers.

Find the ordered quadruple of digits $(A,B,C,D)$ with $A > B > C > D$, such that $$\begin{tabular}{ccccc} & A & B & C & D \\ - & D & C & B & A \\ \hline = & B & D & A & C \\ \end{tabular}$$

* Prove that a number of the form $2^n$ for a positive integer $n$ may begin with any given combination of digits.

Find all positive integers $k \geq 2$ for which there exists some positive integer $n$ such that the last $k$ digits of the decimal representation of $10^{10^n} - 9^{9^n}$ are the same. [i]Proposed by Andrew Wen[/i]

[b]p1.[/b] At a conference a mathematician and a chemist were talking. They were amazed to find that they graduated from the same high school. One of them, the chemist, mentioned that he had three sons and asked the other to calculate the ages of his sons given the following facts: (a) their ages are integers, (b) the product of their ages is $36$, (c) the sum of their ages is equal to the number of windows in the high school of the chemist and the mathematician. The mathematician considered this problem and noted that there was not enough information to obtain a unique solution. The chemist then noted that his oldest son had red hair. The mathematician then announced that he had determined the ages of the three sons. Please (aspiring mathematicians) determine the ages of the chemists three sons and explain your solution. [b]p2.[/b] A square is inscribed in a triangle. Two vertices of this square are on the base of the triangle and two others are on the lateral sides. Prove that the length of the side of the square is greater than and less than $2r$, where $r$ is a radius of the circle inscribed in the triangle. [b]p3.[/b] You are given any set of $100$ integers in which none of the integers is divisible by $100$. Prove that it is possible to select a subset of this set of $100$ integers such that their sum is a multiple of $100$. [b]p4.[/b] Find all prime numbers $a$ and $b$ such that $a^b + b^a$ is a prime number. [b]p5.[/b] $N$ airports are connected by airlines. Some airports are directly connected and some are not. It is always possible to travel from one airport to another by changing planes as needed. The board of directors decided to close one of the airports. Prove that it is possible to select an airport to close so that the remaining airports remain connected. [b]p6.[/b] (A simplified version of the Fermat’s Last Theorem). Prove that there are no positive integers $x, y, z$ and $z \le n$ satisfying the following equation: $x^n + y^n = z^n$. PS. You should use hide for answers.

How many natural numbers $(a,b,n)$ with $ gcd(a,b)=1$ and $ n>1 $ such that the equation \[ x^{an} +y^{bn} = 2^{2010} \] has natural numbers solution $ (x,y) $

Show that all terms of the sequence $1,11,111,1111,...$ in base $9$ are triangular numbers, i.e. of the form $\frac{m(m+1)}{2} $for an integer $m$

Find the number of positive divisors of $20^{22}$ that are perfect squares or perfect cubes.

Find all primes $p,$ $q,$ $r$ satisfying $p^{2}+2q^{2}=r^{2}.$

Solve equation $\frac{1}{x}-\frac{1}{y}=\frac{1}{5}-\frac{1}{xy}$, where $x$ and $y$ are positive integers.

For which $n{}$ is it possible that a product of $n{}$ consecutive positive integers is equal to a sum of $n{}$ consecutive (not necessarily the same) positive integers? [i]Boris Frenkin[/i]

[b]p1.[/b] A boy has as many sisters as brothers. How ever, his sister has twice as many brothers as sisters. How many boys and girls are there in the family? [b]p2.[/b] Solve each of the following problems. (1) Find a pair of numbers with a sum of $11$ and a product of $24$. (2) Find a pair of numbers with a sum of $40$ and a product of $400$. (3) Find three consecutive numbers with a sum of $333$. (4) Find two consecutive numbers with a product of $182$. [b]p3.[/b] $2008$ integers are written on a piece of paper. It is known that the sum of any $100$ numbers is positive. Show that the sum of all numbers is positive. [b]p4.[/b] Let $p$ and $q$ be prime numbers greater than $3$. Prove that $p^2 - q^2$ is divisible by $24$. [b]p5.[/b] Four villages $A,B,C$, and $D$ are connected by trails as shown on the map. [img]https://cdn.artofproblemsolving.com/attachments/4/9/33ecc416792dacba65930caa61adbae09b8296.png[/img] On each path $A \to B \to C$ and $B \to C \to D$ there are $10$ hills, on the path $A \to B \to D$ there are $22$ hills, on the path $A \to D \to B$ there are $45$ hills. A group of tourists starts from $A$ and wants to reach $D$. They choose the path with the minimal number of hills. What is the best path for them? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $p$ be a prime. We say that a sequence of integers $\{z_n\}_{n=0}^\infty$ is a [i]$p$-pod[/i] if for each $e \geq 0$, there is an $N \geq 0$ such that whenever $m \geq N$, $p^e$ divides the sum \[\sum_{k=0}^m (-1)^k {m \choose k} z_k.\] Prove that if both sequences $\{x_n\}_{n=0}^\infty$ and $\{y_n\}_{n=0}^\infty$ are $p$-pods, then the sequence $\{x_ny_n\}_{n=0}^\infty$ is a $p$-pod.

For finite sets $A,M$ such that $A \subseteq M \subset \mathbb{Z}^+$, we define $$f_M(A)=\{x\in M \mid x\text{ is divisible by an odd number of elements of }A\}.$$ Given a positive integer $k$, we call $M$ [i]k-colorable[/i] if it is possible to color the subsets of $M$ with $k$ colors so that for any $A \subseteq M$, if $f_M(A)\neq A$ then $f_M(A)$ and $A$ have different colors. Determine the least positive integer $k$ such that every finite set $M \subset\mathbb{Z}^+$ is k-colorable.

Prove that $1$ plus the product of any four consecutive integers is a perfect square.

Find all possible values of the number $$\frac{a + b}{c}+\frac{a + c}{b}+\frac{b + c}{a},$$ where $a, b, c$ are positive integers, and $\frac{a + b}{c},\frac{a + c}{b},\frac{b + c}{a}$ are also positive integers.

Set $S_n = \sum_{p=1}^n (p^5+p^7)$. Determine the greatest common divisor of $S_n$ and $S_{3n}.$

Let a, b, c, d, and e be positive integers. Are there any solutions to $a^2+b^3+c^5+d^7=e^{11}$?

Prove that for any integer the number $2n^3+3n^2+7n$ is divisible by $6$.

Solve the following equation in positive integers: $x^{3} = 2y^{2} + 1 $

A circular track with diameter $500$ is externally tangent at a point A to a second circular track with diameter $1700.$ Two runners start at point A at the same time and run at the same speed. The first runner runs clockwise along the smaller track while the second runner runs clockwise along the larger track. There is a first time after they begin running when their two positions are collinear with the point A. At that time each runner will have run a distance of $\frac{m\pi}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n. $

[b]p1.[/b] a) There are barrels weighing $1, 2, 3, 4, ..., 19, 20$ pounds. Is it possible to distribute them equally (by weight) into three trucks? b) The same question for barrels weighing $1, 2, 3, 4, ..., 9, 10$ pounds. [b]p2.[/b] There are apples and pears in the basket. If you add the same number of apples there as there are now pears (in pieces), then the percentage of apples will be twice as large as what you get if you add as many pears to the basket as there are now apples. What percentage of apples are in the basket now? [b]p3.[/b] What is the smallest number of integers from $1000$ to $1500$ that must be marked so that any number $x$ from $1000$ to $1500$ differs from one of the marked numbers by no more than $10\% $of the value of $x$? [b]p4.[/b] Draw a perpendicular from a given point to a given straight line, having a compass and a short ruler (the length of the ruler is significantly less than the distance from the point to the straight line; the compass reaches from the point to the straight line “with a margin”). [b]p5.[/b] There is a triangle on the chessboard (left figure). It is allowed to roll it around the sides (in this case, the triangle is symmetrically reflected relative to the side around which it is rolled). Can he, after a few steps, take the position shown in right figure? [img]https://cdn.artofproblemsolving.com/attachments/f/5/eeb96c92f30b837e7ed2cdf7cf77b0fbb8ceda.png[/img] [b]p6.[/b] The natural number $a$ is less than the natural number $b$. In this case, the sum of the digits of number $a$ is $100$ less than the sum of the digits of number $b$. Prove that between the numbers $ a$ and $b$ there is a number whose sum of digits is $43$ more than the sum of the digits of $a$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.[/url]