Found problems: 85335
2019 Regional Olympiad of Mexico Southeast, 1
Found the smaller multiple of $2019$ of the form $abcabc\dots abc$, where $a,b$ and $c$ are digits.
2023 Romania National Olympiad, 2
Prove that:
a) There are infinitely many pairs $(x,y)$ of real numbers from the interval $[0,\sqrt{3}]$ which satisfy the equation $x\sqrt{3-y^2}+y\sqrt{3-x^2}=3$.
b) There do not exist any pairs $(x,y)$ of rational numbers from the interval $[0,\sqrt{3}]$ that satisfy the equation $x\sqrt{3-y^2}+y\sqrt{3-x^2}=3$.
2019 Finnish National High School Mathematics Comp, 4
Define a sequence $ a_n = n^n + (n - 1)^{n+1}$ when $n$ is a positive integer.
Define all those positive integer $m$ , for which this sequence of numbers is eventually periodic modulo $m$, e.g. there are such positive integers $K$ and $s$ such that $a_k \equiv a_{k+s}$ ($mod \,m$), where $k$ is an integer with $k \ge K$.
2023 Macedonian Team Selection Test, Problem 2
Let $ABC$ be an acute triangle such that $AB<AC$ and $AB<BC$. Let $P$ be a point on the segment $BC$ such that $\angle APB = \angle BAC$. The tangent to the circumcircle of triangle $ABC$ at $A$ meets the circumcircle of triangle $APB$ at $Q \neq A$. Let $Q'$ be the reflection of $Q$ with respect to the midpoint of $AB$. The line $PQ$ meets the segment $AQ'$ at $S$. Prove that
$$\frac{1}{AB}+\frac{1}{AC} > \frac{1}{CS}.$$
[i]Authored by Nikola Velov[/i]
1995 AMC 8, 5
Find the smallest whole number that is larger than the sum
\[2\dfrac{1}{2}+3\dfrac{1}{3}+4\dfrac{1}{4}+5\dfrac{1}{5}.\]
$\text{(A)}\ 14 \qquad \text{(B)}\ 15 \qquad \text{(C)}\ 16 \qquad \text{(D)}\ 17 \qquad \text{(E)}\ 18$
2014 Danube Mathematical Competition, 3
Given any integer $n \ge 2$, show that there exists a set of $n$ pairwise coprime composite integers in arithmetic progression.
2006 IMO Shortlist, 5
If $a,b,c$ are the sides of a triangle, prove that
\[\frac{\sqrt{b+c-a}}{\sqrt{b}+\sqrt{c}-\sqrt{a}}+\frac{\sqrt{c+a-b}}{\sqrt{c}+\sqrt{a}-\sqrt{b}}+\frac{\sqrt{a+b-c}}{\sqrt{a}+\sqrt{b}-\sqrt{c}}\leq 3 \]
[i]Proposed by Hojoo Lee, Korea[/i]
2011 Bosnia And Herzegovina - Regional Olympiad, 4
Prove that among any $6$ irrational numbers you can pick three numbers $a$, $b$ and $c$ such that numbers $a+b$, $b+c$ and $c+a$ are irrational
2023 Bosnia and Herzegovina Junior BMO TST, 1.
Determine all real numbers $a, b, c, d$ for which
$ab+cd=6$
$ac+bd=3$
$ad+bc=2$
$a+b+c+d=6$
2025 International Zhautykov Olympiad, 6
$\indent$ For a positive integer $n$, let $S_n$ be the set of bijective functions from $\{1,2,\dots ,n\}$ to itself. For a pair of positive integers $(a,b)$ such that $1 \leq a <b \leq n$, and for a permutation $\sigma \in S_n$, we say the pair $(a,b)$ is [i][u]expanding[/u][/i] for $\sigma$ if $|\sigma (a)- \sigma(b)| \geq |a-b|$
$\indent$ [b](a)[/b] Is it true that for all integers $n > 1$, there exists $\sigma \in S_n$ so that the number of pairs $(a,b)$ that are expanding for permutation $\sigma$ is less than $1000n\sqrt n$ ?
$\indent$ [b](b)[/b] Does there exist a positive integer $n>1$ and a permutation $\sigma \in S_n$ so that the number of pairs $(a,b)$ that are expanding for the permutation $\sigma$ is less than $\frac{n\sqrt n}{1000}$?
2016 District Olympiad, 4
[b]a)[/b] Prove that not all functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that satisfy the equality
$$ f(x-1)+f(x+1) =\sqrt 5f(x) ,\quad\forall x\in\mathbb{R} , $$
are periodic.
[b]b)[/b] Prove that that all functions $ g:\mathbb{R}\longrightarrow\mathbb{R} $ that satisfy the equality
$$ g(x-1)+g(x+1)=\sqrt 3g(x) ,\quad\forall x\in\mathbb{R} , $$
are periodic.
2020 Silk Road, 3
A polynomial $ Q (x) = k_n x ^ n + k_ {n-1} x ^ {n-1} + \ldots + k_1 x + k_0 $ with real coefficients is called [i]powerful[/i] if the equality $ | k_0 | = | k_1 | + | k_2 | + \ldots + | k_ {n-1} | + | k_n | $, and [i]non-increasing[/i] , if $ k_0 \geq k_1 \geq \ldots \geq k_ {n-1} \geq k_n $.
Let for the polynomial $ P (x) = a_d x ^ d + a_ {d-1} x ^ {d-1} + \ldots + a_1 x + a_0 $ with nonzero real coefficients, where $ a_d> 0 $, the polynomial $ P (x) (x-1) ^ t (x + 1) ^ s $ is [i]powerful[/i] for some non-negative integers $ s $ and $ t $ ($ s + t> 0 $). Prove that at least one of the polynomials $ P (x) $ and $ (- 1) ^ d P (-x) $ is [i]nonincreasing[/i].
2025 Azerbaijan IZhO TST, 3
Find all natural numbers $a$ and $b$ such that \[a|b^2, \quad b|a^2 \mbox{ and } a+1|b^2+1.\]
2014 ELMO Shortlist, 10
We are given triangles $ABC$ and $DEF$ such that $D\in BC, E\in CA, F\in AB$, $AD\perp EF, BE\perp FD, CF\perp DE$. Let the circumcenter of $DEF$ be $O$, and let the circumcircle of $DEF$ intersect $BC,CA,AB$ again at $R,S,T$ respectively. Prove that the perpendiculars to $BC,CA,AB$ through $D,E,F$ respectively intersect at a point $X$, and the lines $AR,BS,CT$ intersect at a point $Y$, such that $O,X,Y$ are collinear.
[i]Proposed by Sammy Luo[/i]
2014 Iran MO (2nd Round), 1
Find all positive integers $(m,n)$ such that
\[n^{n^{n}}=m^{m}.\]
2014 Tuymaada Olympiad, 6
Each of $n$ black squares and $n$ white squares can be obtained by a translation from each other. Every two squares of different colours have a common point. Prove that ther is a point belonging at least to $n$ squares.
[i](V. Dolnikov)[/i]
2005 China Team Selection Test, 1
Find all positive integers $m$ and $n$ such that the inequality:
\[ [ (m+n) \alpha ] + [ (m+n) \beta ] \geq [ m \alpha ] + [n \beta] + [ n(\alpha+\beta)] \]
is true for any real numbers $\alpha$ and $\beta$. Here $[x]$ denote the largest integer no larger than real number $x$.
2008 JBMO Shortlist, 2
Kostas and Helene have the following dialogue:
Kostas: I have in my mind three positive real numbers with product $1$ and sum equal to the sum of all their pairwise products.
Helene: I think that I know the numbers you have in mind. They are all equal to $1$.
Kostas: In fact, the numbers you mentioned satisfy my conditions, but I did not think of these numbers. The numbers you mentioned have the minimal sum between all possible solutions of the problem.
Can you decide if Kostas is right? (Explain your answer).
2008 AMC 8, 5
Barney Schwinn notices that the odometer on his bicycle reads $1441$, a palindrome, because it reads the same forward and backward. After riding $4$ more hours that day and $6$ the next, he notices that the odometer shows another palindrome, $1661$. What was his average speed in miles per hour?
$\textbf{(A)}\ 15\qquad
\textbf{(B)}\ 16\qquad
\textbf{(C)}\ 18\qquad
\textbf{(D)}\ 20\qquad
\textbf{(E)}\ 22$
2016 CHKMO, 2
Find all integral ordered triples $(x,y,z)$ such that $\displaystyle\sqrt{\frac{2015}{x+y}}+\sqrt{\frac{2015}{y+z}}+\sqrt{\frac{2015}{x+z}}$ are positive integers
2014 Contests, 2.
Distinct points $A$, $B$ and $C$ lie on a line in this order. Point $D$ lies on the perpendicular bisector of the segment $BC$. Denote by $M$ the midpoint of the segment $BC$. Let $r$ be the radius of the incircle of the triangle $ABD$ and let $R$ be the radius of the circle with center lying outside the triangle $ACD$, tangent to $CD$, $AC$ and $AD$. Prove that $DM=r+R$.
2016 Sharygin Geometry Olympiad, 4
The Devil and the Man play a game. Initially, the Man pays some cash $s$ to the Devil. Then he lists some $97$ triples $\{i,j,k\}$ consisting of positive integers not exceeding $100$. After that, the Devil draws some convex polygon $A_1A_2...A_{100}$ with area $100$ and pays to the Man, the sum of areas of all triangles $A_iA_jA_k$. Determine the maximal value of $s$ which guarantees that the Man receives at least as much cash as he paid.
[i]Proposed by Nikolai Beluhov, Bulgaria[/i]
1999 Slovenia National Olympiad, Problem 1
What is the smallest possible value of $\left|12^m-5^n\right|$, where $m$ and $n$ are positive integers?
2015 Iran MO (3rd round), 3
Let $ABC$ be a triangle. consider an arbitrary point $P$ on the plain of $\triangle ABC$. Let $R,Q$ be the reflections of $P$ wrt $AB,AC$ respectively. Let $RQ\cap BC=T$. Prove that $\angle APB=\angle APC$ if and if only $\angle APT=90^{\circ}$.
2015 Balkan MO Shortlist, G3
A set of points of the plane is called [i] obtuse-angled[/i] if every three of it's points are not collinear and every triangle with vertices inside the set has one angle $ >91^o$. Is it correct that every finite [i] obtuse-angled[/i] set can be extended to an infinite [i]obtuse-angled[/i] set?
(UK)