Found problems: 85335
2022 Balkan MO Shortlist, G1
Let $ABC$ be an acute triangle such that $CA \neq CB$ with circumcircle $\omega$ and circumcentre $O$. Let $t_A$ and $t_B$ be the tangents to $\omega$ at $A$ and $B$ respectively, which meet at $X$. Let $Y$ be the foot of the perpendicular from $O$ onto the line segment $CX$. The line through $C$ parallel to line $AB$ meets $t_A$ at $Z$. Prove that the line $YZ$ passes through the midpoint of the line segment $AC$.
[i]Proposed by Dominic Yeo, United Kingdom[/i]
CIME II 2018, 2
Garfield and Odie are situated at $(0,0)$ and $(25,0)$, respectively. Suddenly, Garfield and Odie dash in the direction of the point $(9, 12)$ at speeds of $7$ and $10$ units per minute, respectively. During this chase, the minimum distance between Garfield and Odie can be written as $\frac{m}{\sqrt{n}}$ for relatively prime positive integers $m$ and $n$. Find $m+n$.
[i]Proposed by [b] Th3Numb3rThr33 [/b][/i]
2017 District Olympiad, 2
Let $ E(x,y)=\frac{x}{y} +\frac{x+1}{y+1} +\frac{x+2}{y+2} . $
[b]a)[/b] Solve in $ \mathbb{N}^2 $ the equation $ E(x,y)=3. $
[b]b)[/b] Show that there are infinitely many natural numbers $ n $ such that the equation $ E(x,y)=n $ has at least one solution in $ \mathbb{N}^2. $
2017 Nordic, 1
Let $n$ be a positive integer. Show that there exist positive integers $a$ and $b$ such that \[ \frac{a^2 + a + 1}{b^2 + b + 1} = n^2 + n + 1. \]
2000 ITAMO, 4
Let $n > 1$ be a fixed integer. Alberto and Barbara play the following game:
(i) Alberto chooses a positive integer,
(ii) Barbara chooses an integer greater than $1$ which is a multiple or submultiple of the number Alberto chose (including itself),
(iii) Alberto increases or decreases the Barbara’s number by $1$.
Steps (ii) and (iii) are alternatively repeated. Barbara wins if she succeeds to reach the number $n$ in at most $50$ moves. For which values of $n$ can she win, no matter how Alberto plays?
2009 QEDMO 6th, 12
Find all functions $f: R\to R$, which satisfy the equation $f (xy + f (x)) = xf (y) + f (x)$.
2024 Moldova EGMO TST, 9
Given a convex quadrilateral $ KLMN $, in which $ \angle NKL = {{90} ^ {\circ}} $. Let $ P $ be the midpoint of the segment $ LM $. It turns out that $ \angle KNL = \angle MKP $. Prove that $ \angle KNM = \angle LKP $.
1997 Irish Math Olympiad, 2
A circle $ \Gamma$ is inscribed in a quadrilateral $ ABCD$. If $ \angle A\equal{}\angle B\equal{}120^{\circ}, \angle D\equal{}90^{\circ}$ and $ BC\equal{}1$, find, with proof, the length of $ AD$.
2022 Switzerland - Final Round, 8
Let $ABC$ be a triangle and let $P$ be a point in the interior of the side $BC$. Let $I_1$ and $I_2$ be the incenters of the triangles $AP B$ and $AP C$, respectively. Let $X$ be the closest point to $A$ on the line $AP$ such that $XI_1$ is perpendicular to $XI_2$. Prove that the distance $AX$ is independent of the choice of $P$.
2013 Gheorghe Vranceanu, 2
Given two natural numbers $ n\ge 2,a, $ prove that there exists another natural number $ v\ge 2 $ such that:
$$ \frac{v+\sqrt{v^2-4}}{2} =\left( \frac{n+\sqrt{n^2-4}}{2} \right)^a $$
2017 QEDMO 15th, 9
Iskandar arranged $n \in N$ integer numbers in a circle, the sum of which is $2n-1$. Crescentia now selects one of these numbers and name the given numbers in clockwise direction with $a_1,a_2,...., a_n$. Show that she can choose the starting number such that for all $k \in \{1, 2,..., n\}$ the inequality $a_1 + a_2 +...+ a_k \le 2k -1$ holds.
2021 Bosnia and Herzegovina Junior BMO TST, 2
Let $p, q, r$ be prime numbers and $t, n$ be natural numbers such that $p^2 +qt =(p + t)^n$ and $p^2 + qr = t^4$ .
a) Show that $n < 3$.
b) Determine all the numbers $p, q, r, t, n$ that satisfy the given conditions.
2009 District Olympiad, 2
Let $n\in \mathbb{N}^*$ and a matrix $A\in \mathcal{M}_n(\mathbb{C}),\ A=(a_{ij})_{1\le i, j\le n}$ such that:
\[a_{ij}+a_{jk}+a_{ki}=0,\ (\forall)i,j,k\in \{1,2,\ldots,n\}\]
Prove that $\text{rank}\ A\le 2$.
2007 Iran MO (3rd Round), 4
Find all integer solutions of \[ x^{4}\plus{}y^{2}\equal{}z^{4}\]
2019 Danube Mathematical Competition, 3
We color some unit squares in a $ 99\times 99 $ square grid with one of $ 5 $ given distinct colors, such that each color appears the same number of times. On each row and on each column there are no differently colored unit squares. Find the maximum possible number of colored unit squares.
2006 Romania Team Selection Test, 2
Find all non-negative integers $m,n,p,q$ such that \[ p^mq^n = (p+q)^2 +1 . \]
2019 JBMO Shortlist, A1
Real numbers $a$ and $b$ satisfy $a^3+b^3-6ab=-11$. Prove that $-\frac{7}{3}<a+b<-2$.
[i]Proposed by Serbia[/i]
2020 Dutch IMO TST, 4
Let $a, b \ge 2$ be positive integers with $gcd (a, b) = 1$. Let $r$ be the smallest positive value that $\frac{a}{b}- \frac{c}{d}$ can take, where $c$ and $d$ are positive integers satisfying $c \le a$ and $d \le b$. Prove that $\frac{1}{r}$ is an integer.
2021 Brazil Team Selection Test, 4
Determine all functions $f$ defined on the set of all positive integers and taking non-negative integer values, satisfying the three conditions:
[list]
[*] $(i)$ $f(n) \neq 0$ for at least one $n$;
[*] $(ii)$ $f(x y)=f(x)+f(y)$ for every positive integers $x$ and $y$;
[*] $(iii)$ there are infinitely many positive integers $n$ such that $f(k)=f(n-k)$ for all $k<n$.
[/list]
2008 Bosnia Herzegovina Team Selection Test, 2
Let $ AD$ be height of triangle $ \triangle ABC$ and $ R$ circumradius. Denote by $ E$ and $ F$ feet of perpendiculars from point $ D$ to sides $ AB$ and $ AC$.
If $ AD\equal{}R\sqrt{2}$, prove that circumcenter of triangle $ \triangle ABC$ lies on line $ EF$.
2018 AMC 10, 5
Alice, Bob, and Charlie were on a hike and were wondering how far away the nearest town was. When Alice said, "We are at least 6 miles away," Bob replied, "We are at most 5 miles away." Charlie then remarked, "Actually the nearest town is at most 4 miles away." It turned out that none of the three statements were true. Let $d$ be the distance in miles to the nearest town. Which of the following intervals is the set of all possible values of $d$?
$\textbf{(A) } (0,4) \qquad \textbf{(B) } (4,5) \qquad \textbf{(C) } (4,6) \qquad \textbf{(D) } (5,6) \qquad \textbf{(E) } (5,\infty) $
2000 Harvard-MIT Mathematics Tournament, 3
Evaluate $\displaystyle\sum_{n=1}^\infty \dfrac{1}{n^2+2n}$.
2010 Korea National Olympiad, 2
Let $ ABCD$ be a cyclic convex quadrilateral. Let $ E $ be the intersection of lines $ AB, CD $. $ P $ is the intersection of line passing $ B $ and perpendicular to $ AC $, and line passing $ C $ and perpendicular to $ BD$. $ Q $ is the intersection of line passing $ D $ and perpendicular to $ AC $, and line passing $ A $ and perpendicular to $ BD $. Prove that three points $ E, P, Q $ are collinear.
2019 Junior Balkan Team Selection Tests - Romania, 2
Determine all positive integers $n$ such that $4k^2 +n$ is a prime number for all non-negative integer $k$ smaller than $n$.
2012 Regional Competition For Advanced Students, 2
Determine all integer solutions $(x, y)$ of the equation \[(x - 1)x(x + 1) + (y - 1)y(y + 1) = 24 - 9xy\mbox{.}\]