Found problems: 85335
1999 Polish MO Finals, 2
Prove that for any $ 2n$ real numbers $ a_{1}$, $ a_{2}$, ..., $ a_{n}$, $ b_{1}$, $ b_{2}$, ..., $ b_{n}$, we have $ \sum_{i < j}{\left|a_{i}\minus{}a_{j}\right|}\plus{}\sum_{i < j}{\left|b_{i}\minus{}b_{j}\right|}\leq\sum_{i,j\in\left[1,n\right]}{\left|a_{i}\minus{}b_{j}\right|}$.
1997 AIME Problems, 9
Given a nonnegative real number $x,$ let $\langle x\rangle$ denote the fractional part of $x;$ that is, $\langle x\rangle=x-\lfloor x\rfloor,$ where $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x.$ Suppose that $a$ is positive, $\langle a^{-1}\rangle=\langle a^2\rangle,$ and $2<a^2<3.$ Find the value of $a^{12}-144a^{-1}.$
1995 Israel Mathematical Olympiad, 5
Let $n$ be an odd positive integer and let $x_1,x_2,...,x_n$ be n distinct real numbers that satisfy $|x_i -x_j| \le 1$ for $1 \le i < j \le n$. Prove that
$$\sum_{i<j} |x_i -x_j| \le \left[\frac{n}{2} \right] \left(\left[\frac{n}{2} \right]-1 \right)$$
2013 Romanian Master of Mathematics, 5
Given a positive integer $k\geq2$, set $a_1=1$ and, for every integer $n\geq 2$, let $a_n$ be the smallest solution of equation
\[x=1+\sum_{i=1}^{n-1}\left\lfloor\sqrt[k]{\frac{x}{a_i}}\right\rfloor\]
that exceeds $a_{n-1}$. Prove that all primes are among the terms of the sequence $a_1,a_2,\ldots$
1969 IMO Longlists, 19
$(FRA 2)$ Let $n$ be an integer that is not divisible by any square greater than $1.$ Denote by $x_m$ the last digit of the number $x^m$ in the number system with base $n.$ For which integers $x$ is it possible for $x_m$ to be $0$? Prove that the sequence $x_m$ is periodic with period $t$ independent of $x.$ For which $x$ do we have $x_t = 1$. Prove that if $m$ and $x$ are relatively prime, then $0_m, 1_m, . . . , (n-1)_m$ are different numbers. Find the minimal period $t$ in terms of $n$. If n does not meet the given condition, prove that it is possible to have $x_m = 0 \neq x_1$ and that the sequence is periodic starting only from some number $k > 1.$
1973 Bundeswettbewerb Mathematik, 1
A positive integer has 1000 digits (decimal system), all but at most one of them being the digit $5$. Show that this number isn't a perfect square.
2015 HMNT, 6
Marcus and four of his relatives are at a party. Each pair of the five people are either $\textit{friends}$ or $\textit{enemies}$. For any two enemies, there is no person that they are both friends with. In how many ways is this possible?
1996 Baltic Way, 12
Let $S$ be a set of integers containing the numbers $0$ and $1996$. Suppose further that any integer root of any non-zero polynomial with coefficients in $S$ also belongs to $S$. Prove that $-2$ belongs to $S$.
1998 Kurschak Competition, 2
Prove that for every positive integer $n$, there exists a polynomial with integer coefficients whose values at points $1,2,\dots,n$ are pairwise different powers of $2$.
2016 Kosovo National Mathematical Olympiad, 1
Find all three digit numbers such that the square of that number is equal to the sum of their digits in power of $5$ .
2022 USA TSTST, 5
Let $A_1$, $\ldots$, $A_{2022}$ be the vertices of a regular $2022$-gon in the plane. Alice and Bob play a game. Alice secretly chooses a line and colors all points in the plane on one side of the line blue, and all points on the other side of the line red. Points on the line are colored blue, so every point in the plane is either red or blue. (Bob cannot see the colors of the points.)
In each round, Bob chooses a point in the plane (not necessarily among $A_1, \ldots, A_{2022}$) and Alice responds truthfully with the color of that point. What is the smallest number $Q$ for which Bob has a strategy to always determine the colors of points $A_1, \ldots, A_{2022}$ in $Q$ rounds?
2015 ASDAN Math Tournament, 6
A circle $A$ is circumscribed about a unit square and a circle $B$ is inscribed inside the same unit square. Compute the ratio of the area of $A$ to the area of $B$.
2024 Serbia National Math Olympiad, 3
Let $n$ be a positive integer and let $a_1, a_2, \ldots, a_n$ and $b_1, b_2, \ldots, b_n$ be reals. Show that for any positive integer $1 \leq m \leq n$, there exist two distinct reals $\alpha, \beta$, $\alpha^2+\beta^2>0$, such that $p_m=\min\{p_1, p_2, \ldots, p_n\}$, where $$p_j=\sum_{i=1}^n|\alpha(a_i-a_j)+\beta(b_i-b_j)|$$ for $1\leq j \leq n$.
2021 Princeton University Math Competition, A1 / B3
Select two distinct diagonals at random from a regular octagon. What is the probability that the two diagonals intersect at a point strictly within the octagon? Express your answer as $a + b$, where the probability is $\tfrac{a}{b}$ and $a$ and $b$ are relatively prime positive integers.
1993 All-Russian Olympiad Regional Round, 9.2
Find the largest natural number which cannot be turned into a multiple of $11$ by reordering its (decimal) digits.
2009 All-Russian Olympiad, 5
Prove that \[ \log_ab\plus{}\log_bc\plus{}\log_ca\le \log_ba\plus{}\log_cb\plus{}\log_ac\] for all $ 1<a\le b\le c$.
1982 Czech and Slovak Olympiad III A, 3
In the plane with coordinates $x,y$, find an example of a convex set $M$ that contains infinitely many lattice points (i.e. points with integer coordinates), but at the same time only finitely many lattice points from $M$ lie on each line in that plane.
LMT Speed Rounds, 2010.6
Al has three red marbles and four blue marbles. He draws two different marbles at the same time. What is the probability that one is red and the other is blue?
2021 Winter Stars of Mathematics, 3
Determine the primes $p$ for which the numbers $2\lfloor p/k\rfloor - 1, \ k = 1,2,\ldots, p,$ are all quadratic residues modulo $p.$
[i]Vlad Matei[/i]
2024 Thailand TST, 2
Let $n\geqslant 2$ be a positive integer. Paul has a $1\times n^2$ rectangular strip consisting of $n^2$ unit squares, where the $i^{\text{th}}$ square is labelled with $i$ for all $1\leqslant i\leqslant n^2$. He wishes to cut the strip into several pieces, where each piece consists of a number of consecutive unit squares, and then [i]translate[/i] (without rotating or flipping) the pieces to obtain an $n\times n$ square satisfying the following property: if the unit square in the $i^{\text{th}}$ row and $j^{\text{th}}$ column is labelled with $a_{ij}$, then $a_{ij}-(i+j-1)$ is divisible by $n$.
Determine the smallest number of pieces Paul needs to make in order to accomplish this.
2018 Iran MO (3rd Round), 3
Find all functions $f:\mathbb{N}\to \mathbb{N}$ so that for every natural numbers $m,n$ :$f(n)+2mn+f(m)$ is a perfect square.
2008 Junior Balkan MO, 4
A $ 4\times 4$ table is divided into $ 16$ white unit square cells. Two cells are called neighbors if they share a common side. A [i]move[/i] consists in choosing a cell and the colors of neighbors from white to black or from black to white. After exactly $ n$ moves all the $ 16$ cells were black. Find all possible values of $ n$.
2012 India IMO Training Camp, 3
Determine the greatest positive integer $k$ that satisfies the following property: The set of positive integers can be partitioned into $k$ subsets $A_1, A_2, \ldots, A_k$ such that for all integers $n \geq 15$ and all $i \in \{1, 2, \ldots, k\}$ there exist two distinct elements of $A_i$ whose sum is $n.$
[i]Proposed by Igor Voronovich, Belarus[/i]
2012 VJIMC, Problem 2
Determine all $2\times2$ integer matrices $A$ having the following properties:
$1.$ the entries of $A$ are (positive) prime numbers,
$2.$ there exists a $2\times2$ integer matrix $B$ such that $A=B^2$ and the determinant of $B$ is the square of a prime number.
2003 Argentina National Olympiad, 3
Let $a\geq 4$ be a positive integer. Determine the smallest value of $n\geq 5$, $n\neq a$, such that $a$ can be represented in the form$$a=\frac{x_1^2+x_2^2+\cdots + x_n^ 2}{x_1x_2\cdots x_n}$$for a suitable choice of the $n$ positive integers $x_1,x_2,\ldots ,x_n$.