Found problems: 85335
2020 Thailand Mathematical Olympiad, 10
Determine all polynomials $P(x)$ with integer coefficients which satisfies $P(n)\mid n!+2$ for all postive integer $n$.
2014 Grand Duchy of Lithuania, 4
Determine all positive integers $n > 1$ for which $n + D(n)$ is a power of $10$, where $D(n)$ denotes the largest integer divisor of $n$ satisfying $D(n) < n$.
2004 Croatia National Olympiad, Problem 3
The sequences $(x_n),(y_n),(z_n),n\in\mathbb N$, are defined by the relations
$$x_{n+1}=\frac{2x_n}{x_n^2-1},\qquad y_{n+1}=\frac{2y_n}{y_n^2-1},\qquad z_{n+1}=\frac{2z_n}{z_n^2-1},$$where $x_1=2$, $y_1=4$, and $x_1y_1z_1=x_1+y_1+z_1$.
(a) Show that $x_n^2\ne1$, $y_n^2\ne1$, $z_n^2\ne1$ for all $n$;
(b) Does there exist a $k\in\mathbb N$ for which $x_k+y_k+z_k=0$?
2021 Harvard-MIT Mathematics Tournament., 3
Let $N$ be a positive integer. Brothers Michael and Kylo each select a positive integer less than or equal to $N$, independently and uniformly at random. Let $p_N$ denote the probability that the product of these two integers has a units digit of $0$. The maximum possible value of $p_N$ over all possible choices of $N$ can be written as $\tfrac ab,$ where $a$ and $b$ are relatively prime positive integers. Compute $100a + b$.
LMT Guts Rounds, 2020 F18
Given that $\sqrt{x+2y}-\sqrt{x-2y}=2,$ compute the minimum value of $x+y.$
[i]Proposed by Alex Li[/i]
2012 IMO Shortlist, N3
Determine all integers $m \geq 2$ such that every $n$ with $\frac{m}{3} \leq n \leq \frac{m}{2}$ divides the binomial coefficient $\binom{n}{m-2n}$.
1998 VJIMC, Problem 3
Give an example of a sequence of continuous functions on $\mathbb R$ converging pointwise to $0$ which is not uniformly convergent on any nonempty open set.
2018 PUMaC Live Round, 4.3
Let $0\leq a,b,c,d\leq 10$. For how many ordered quadruples $(a,b,c,d)$ is $ad-bc$ a multiple of $11?$
2010 Postal Coaching, 3
Find all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ such that
$\boxed{1} \ f(1) = 1$
$\boxed{2} \ f(m+n)(f(m)-f(n)) = f(m-n)(f(m)+f(n)) \ \forall \ m,n \in \mathbb{Z}$
2004 Turkey MO (2nd round), 5
The excircle of a triangle $ABC$ corresponding to $A$ touches the lines $BC,CA,AB$ at $A_1,B_1,C_1$, respectively. The excircle corresponding to $B$ touches $BC,CA,AB$ at $A_2,B_2,C_2$, and the excircle corresponding to $C$ touches $BC,CA,AB$ at $A_3,B_3,C_3$, respectively. Find the maximum possible value of the ratio of the sum of the perimeters of $\triangle A_1B_1C_1$, $\triangle A_2B_2C_2$ and $\triangle A_3B_3C_3$ to the circumradius of $\triangle ABC$.
1993 Flanders Math Olympiad, 4
Define the sequence $oa_n$ as follows: $oa_0=1, oa_n= oa_{n-1} \cdot cos\left( \dfrac{\pi}{2^{n+1}} \right)$.
Find $\lim\limits_{n\rightarrow+\infty} oa_n$.
2016 Junior Balkan Team Selection Tests - Romania, 2
Given three colors and a rectangle m × n dice unit, we want to color each segment constituting one side of a square drive with one of three colors so that each square unit have two sides of one color and two sides another color. How many colorings we have?
2017 Greece Team Selection Test, 1
Let $ABC$ be an acute-angled triangle inscribed in circle $c(O,R)$ with $AB<AC<BC$,
and $c_1$ be the inscribed circle of $ABC$ which intersects $AB, AC, BC$ at
$F, E, D$ respectivelly. Let $A', B', C'$ be points which lie on $c$ such that the quadrilaterals
$AEFA', BDFB', CDEC'$ are inscribable.
(1) Prove that $DEA'B'$ is inscribable.
(2) Prove that $DA', EB', FC'$ are concurrent.
1997 Croatia National Olympiad, Problem 4
In the plane are given $1997$ points. Show that among the pairwise distances between these points, there are at least $32$ different values.
2021 IMO, 2
Show that the inequality \[\sum_{i=1}^n \sum_{j=1}^n \sqrt{|x_i-x_j|}\leqslant \sum_{i=1}^n \sum_{j=1}^n \sqrt{|x_i+x_j|}\]holds for all real numbers $x_1,\ldots x_n.$
2014 BMT Spring, 10
Suppose that $x^3-x+10^{-6}=0$. Suppose that $x_1<x_2<x_3$ are the solutions for $x$. Find the integers $(a,b,c)$ closest to $10^8x_1$, $10^8x_2$, and $10^8x_3$ respectively.
2022 Korea Winter Program Practice Test, 1
Let $ABCD$ be a cyclic quadrilateral inscribed in a circle $\Omega$ such that $AB<CD$. Suppose that $AC$ meets $BD$ at $E$, $AD$ meets $BC$ at $F$, and $\Omega$ meets $(FAE)$, $(FBE)$ at $X$, $Y$, respectively. Prove that if $XY$ is diameter of $\Omega$, then $XY$ is perpendicular to $EF$.
2006 China Girls Math Olympiad, 3
Show that for any $i=1,2,3$, there exist infinity many positive integer $n$, such that among $n$, $n+2$ and $n+28$, there are exactly $i$ terms that can be expressed as the sum of the cubes of three positive integers.
LMT Team Rounds 2010-20, B7
Zachary tries to simplify the fraction $\frac{2020}{5050}$ by dividing the numerator and denominator by the same integer to get the fraction $\frac{m}{n}$ , where $m$ and $n$ are both positive integers. Find the sum of the (not necessarily distinct) prime factors of the sum of all the possible values of $m +n$
2022 New Zealand MO, 6
Let a positive integer $n$ be given. Determine, in terms of $n$, the least positive integer $k$ such that among any $k$ positive integers, it is always possible to select a positive even number of them having sum divisible by $n$.
PEN M Problems, 25
Let $\{a_{n}\}_{n \ge 1}$ be a sequence of positive integers such that \[0 < a_{n+1}-a_{n}\le 2001 \;\; \text{for all}\;\; n \in \mathbb{N}.\] Show that there are infinitely many pairs $(p, q)$ of positive integers such that $p>q$ and $a_{q}\; \vert \; a_{p}$.
2014 Indonesia MO Shortlist, N3
Find all pairs of natural numbers $(a, b)$ that fulfill $a^b=(a+b)^a$.
2003 Iran MO (3rd Round), 19
An integer $ n$ is called a good number if and only if $ |n|$ is not square of another intger. Find all integers $ m$ such that they can be written in infinitely many ways as sum of three different good numbers and product of these three numbers is square of an odd number.
VMEO IV 2015, 12.3
Find all integes $a,b,c,d$ that form an arithmetic progression satisfying $d-c+1$ is prime number and $a+b^2+c^3=d^2b$
2005 National Olympiad First Round, 6
Which of the following divides $3^{3n+1} + 5^{3n+2}+7^{3n+3}$ for every positive integer $n$?
$
\textbf{(A)}\ 3
\qquad\textbf{(B)}\ 5
\qquad\textbf{(C)}\ 7
\qquad\textbf{(D)}\ 11
\qquad\textbf{(E)}\ 53
$