Found problems: 85335
2016 LMT, 3
The squares of two positive integers differ by 2016. Find the maximum possible sum of the two integers.
[i]Proposed by Clive Chan
2004 Switzerland Team Selection Test, 2
Find the largest natural number $n$ for which $4^{995} +4^{1500} +4^n$ is a square.
1959 Poland - Second Round, 5
In the plane, $ n \geq 3 $ segments are placed in such a way that every $ 3 $ of them have a common point. Prove that there is a common point for all the segments.
2014 Harvard-MIT Mathematics Tournament, 5
Prove that there exists a nonzero complex number $c$ and a real number $d$ such that \[\left|\left|\dfrac1{1+z+z^2}\right|-\left|\dfrac1{1+z+z^2}-c\right|\right|=d\] for all $z$ with $|z|=1$ and $1+z+z^2\neq 0$. (Here, $|z|$ denotes the absolute value of the complex number $z$, so that $|a+bi|=\sqrt{a^2+b^2}$ for real numbers $a,b$.)
2003 Hong kong National Olympiad, 4
Find all integer numbers $a,b,c$ such that $\frac{(a+b)(b+c)(c+a)}{2}+(a+b+c)^{3}=1-abc$.
1990 Irish Math Olympiad, 1
Given a natural number $n$, calculate the number of rectangles in the plane, the coordinates of whose vertices are integers in the range $0$ to $n$, and whose sides are parallel to the axes.
2007 Baltic Way, 19
Let $r$ and $k$ be positive integers such that all prime divisors of $r$ are greater than $50$.
A positive integer, whose decimal representation (without leading zeroes) has at least $k$ digits, will be called [i]nice[/i] if every sequence of $k$ consecutive digits of this decimal representation forms a number (possibly with leading zeroes) which is a multiple of $r$.
Prove that if there exist infinitely many nice numbers, then the number $10^k-1$ is nice as well.
2003 AMC 12-AHSME, 22
Objects $A$ and $B$ move simultaneously in the coordinate plane via a sequence of steps, each of length one. Object $A$ starts at $(0,0)$ and each of its steps is either right or up, both equally likely. Object $B$ starts at $(5,7)$ and each of its steps is either left or down, both equally likely. Which of the following is closest to the probability that the objects meet?
$ \textbf{(A)}\ 0.10 \qquad
\textbf{(B)}\ 0.15 \qquad
\textbf{(C)}\ 0.20 \qquad
\textbf{(D)}\ 0.25 \qquad
\textbf{(E)}\ 0.30$
2011 Dutch IMO TST, 1
Find all pairs $(x, y)$ of integers that satisfy $x^2 + y^2 + 3^3 = 456\sqrt{x - y}$.
2013 IPhOO, 4
The Iphoon particle, of charge $q$, is accelerated from rest by a potential difference of $V$. This strange particle then enters a region with a uniform magnetic field, $B$, which is perpendicular to the particle's velocity. The Iphoon follows a circular path with radius $R$. If $ q = 1 \, \mu\text{C} $, $ V = 1 \, \text{kV} $, $ B = 1 \, \text{mT} $, and $ R = 2 \, \text{ft} $, let the weight of an Iphoon, in Newtons, be $ w $. If $ w \approx 10^p $, where $p$ is an integer, find $p$. That is, what is the order of magnitude of the weight?
[i](Proposed by Ahaan Rungta)[/i]
1995 All-Russian Olympiad Regional Round, 10.1
Given function $f(x) = \dfrac{1}{\sqrt[3]{1-x^3}}$, find $\underbrace{f(... f(f(19))...)}_{95}$.
.
2000 Croatia National Olympiad, Problem 1
Let $a>0$ and $x_1,x_2,x_3$ be real numbers with $x_1+x_2+x_3=0$. Prove that
$$\log_2\left(1+a^{x_1}\right)+\log_2\left(1+a^{x_2}\right)+\log_2\left(1+a^{x_3}\right)\ge3.$$
2024 Turkey Team Selection Test, 5
In a scalene triangle $ABC$, $H$ is the orthocenter, and $G$ is the centroid. Let $A_b$ and $A_c$ be points on $AB$ and $AC$, respectively, such that $B$, $C$, $A_b$, $A_c$ are cyclic, and the points $A_b$, $A_c$, $H$ are collinear. $O_a$ is the circumcenter of the triangle $AA_bA_c$. $O_b$ and $O_c$ are defined similarly. Prove that the centroid of the triangle $O_aO_bO_c$ lies on the line $HG$.
2010 South africa National Olympiad, 6
Write either $1$ or $-1$ in each of the cells of a $(2n) \times (2n)$-table, in such a way that there are exactly $2n^2$ entries of each kind. Let the minimum of the absolute values of all row sums and all column sums be $M$. Determine the largest possible value of $M$.
2007 Stanford Mathematics Tournament, 5
Two disks of radius 1 are drawn so that each disk's circumference passes through the center of the other disk. What is the circumference of the region in which they overlap?
1993 IberoAmerican, 3
Two nonnegative integers $a$ and $b$ are [i]tuanis[/i] if the decimal expression of $a+b$ contains only $0$ and $1$ as digits. Let $A$ and $B$ be two infinite sets of non negative integers such that $B$ is the set of all the [i]tuanis[/i] numbers to elements of the set $A$ and $A$ the set of all the [i]tuanis[/i] numbers to elements of the set $B$. Show that in at least one of the sets $A$ and $B$ there is an infinite number of pairs $(x,y)$ such that $x-y=1$.
2024 Iran MO (3rd Round), 3
An integer number $n\geq 2$ and real numbers $x_1<x_2<\cdots < x_n$ are given. $f: \mathbb R \to \mathbb R$ is a function defined as
$$
f(x) = \left |
\dfrac{(x-x_2)(x-x_3)\cdots (x-x_n)}{(x_1-x_2)(x_1-x_3)\cdots (x_1-x_n)}
\right | + \cdots +
\left |
\dfrac{(x-x_1)(x-x_2)\cdots (x-x_{n-1})}{(x_n-x_1)(x_n-x_2)\cdots (x_n-x_{n-1})}
\right |.
$$
Prove that there exists $i\in \{1,2,\cdots,n-1\}$ such that for all $x\in (x_i,x_{i+1})$ one has $f(x)< \sqrt n$.
Proposed by [i]Navid Safaei[/i]
2018 Moscow Mathematical Olympiad, 3
$O$ is circumcircle and $AH$ is the altitude of $\triangle ABC$. $P$ is the point on line $OC$ such that $AP \perp OC$. Prove, that midpoint of $AB$ lies on the line $HP$.
2016 India Regional Mathematical Olympiad, 1
Given are two circles $\omega_1,\omega_2$ which intersect at points $X,Y$. Let $P$ be an arbitrary point on $\omega_1$. Suppose that the lines $PX,PY$ meet $\omega_2$ again at points $A,B$ respectively. Prove that the circumcircles of all triangles $PAB$ have the same radius.
2011 ELMO Shortlist, 3
Let $n>1$ be a fixed positive integer, and call an $n$-tuple $(a_1,a_2,\ldots,a_n)$ of integers greater than $1$ [i]good[/i] if and only if $a_i\Big|\left(\frac{a_1a_2\cdots a_n}{a_i}-1\right)$ for $i=1,2,\ldots,n$. Prove that there are finitely many good $n$-tuples.
[i]Mitchell Lee.[/i]
2014 Abels Math Contest (Norwegian MO) Final, 3b
Nine points are placed on a circle. Show that it is possible to colour the $36$ chords connecting them using four colours so that for any set of four points, each of the four colours is used for at least one of the six chords connecting the given points
2010 IFYM, Sozopol, 7
Let $M$ be a convex polygon. Externally, on its sides are built squares. It is known that the vertices of these squares, that don’t lie on $M$, lie on a circle $k$. Determine $M$ (its type).
2020 Mexico National Olympiad, 1
A set of five different positive integers is called [i]virtual[/i] if the greatest common divisor of any three of its elements is greater than $1$, but the greatest common divisor of any four of its elements is equal to $1$. Prove that, in any virtual set, the product of its elements has at least $2020$ distinct positive divisors.
[i]Proposed by Víctor Almendra[/i]
2023 Irish Math Olympiad, P3
Let $A, B, C, D, E$ be five points on a circle such that $|AB| = |CD|$ and $|BC| = |DE|$. The segments $AD$ and $BE$ intersect at $F$. Let $M$ denote the midpoint of segment $CD$. Prove that the circle of center $M$ and radius $ME$ passes through the midpoint of segment $AF$.
2022 Math Prize for Girls Problems, 10
An algal cell population is found to have $a_k$ cells on day $k$. Each day, the number of cells at least doubles. If $a_0 \ge 1$ and $a_3 \le 60$, how many quadruples of integers $(a_0, a_1, a_2, a_3)$ could represent the algal cell population size on the first $4$ days?