Found problems: 85335
1984 Polish MO Finals, 2
Let $n$ be a positive integer. For all $i, j \in \{1,2,...,n\}$ define $a_{j,i} = 1$ if $j = i$ and $a_{j,i} = 0$ otherwise. Also, for $i = n+1,...,2n$ and $j = 1,...,n$ define $a_{j,i} = -\frac{1}{n}$.
Prove that for any permutation $p$ of the set $\{1,2,...,2n\}$ the following inequality holds: $\sum_{j=1}^{n}\left|\sum_{k=1}^{n} a_{j,p}(k)\right| \ge \frac{n}{2}$
2021 All-Russian Olympiad, 1
On the side $BC$ of the parallelogram $ABCD$, points $E$ and $F$ are given ($E$ lies between $B$ and $F$) and the diagonals $AC, BD$ meet at $O$. If it's known that $AE, DF$ are tangent to the circumcircle of $\triangle AOD$, prove that they're tangent to the circumcircle of $\triangle EOF$ as well.
2015 Irish Math Olympiad, 5
Suppose a doubly infinite sequence of real numbers $. . . , a_{-2}, a_{-1}, a_0, a_1, a_2, . . .$ has the property that $$a_{n+3} =\frac{a_n + a_{n+1} + a_{n+2}}{3},$$ for all integers $n .$ Show that if this sequence is bounded (i.e., if there exists a number $R$ such that $|a_n| \leq R$ for all $n$), then $a_n$ has the same value for all $n.$
2018 HMNT, 6
Call a polygon [i]normal[/i] if it can be inscribed in a unit circle. How many non-congruent normal polygons are there such that the square of each side length is a positive integer?
2001 China Team Selection Test, 1
Let $p(x)$ be a polynomial with real coefficients such that $p(0)=p(n)$. Prove that there are at least $n$ pairs of real numbers $(x,y)$ where $p(x)=p(y)$ and $y-x$ is a positive integer
2012 Peru IMO TST, 3
Suppose that $1000$ students are standing in a circle. Prove that there exists an integer $k$ with $100 \leq k \leq 300$ such that in this circle there exists a contiguous group of $2k$ students, for which the first half contains the same number of girls as the second half.
[i]Proposed by Gerhard Wöginger, Austria[/i]
Brazil L2 Finals (OBM) - geometry, 2016.2
The inner bisections of the $ \angle ABC $ and $ \angle ACB $ angles of the $ ABC $ triangle are at $ I $. The $ BI $ parallel line that runs through the point $ A $ finds the $ CI $ line at the point $ D $. The $ CI $ parallel line for $ A $ finds the $ BI $ line at the point $ E $. The lines $ BD $ and $ CE $ are at the point $ F $. Show that $ F, A $, and $ I $ are collinear if and only if $ AB = AC. $
1962 AMC 12/AHSME, 13
$ R$ varies directly as $ S$ and inverse as $ T$. When $ R \equal{} \frac43$ and $ T \equal{} \frac {9}{14}$, $ S \equal{} \frac37.$ Find $ S$ when $ R \equal{} \sqrt {48}$ and $ T \equal{} \sqrt {75}.$
$ \textbf{(A)}\ 28 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 40 \qquad \textbf{(D)}\ 42 \qquad \textbf{(E)}\ 60$
2019 Switzerland Team Selection Test, 2
Find the largest prime $p$ such that there exist positive integers $a,b$ satisfying $$p=\frac{b}{2}\sqrt{\frac{a-b}{a+b}}.$$
2009 Stanford Mathematics Tournament, 15
What is the largest integer $n$ for which $\frac{2008!}{31^n}$ is an integer?
PEN K Problems, 13
Find all functions $f: \mathbb{Z}\to \mathbb{Z}$ such that for all $m\in \mathbb{Z}$: \[f(f(m))=m+1.\]
2003 IMO Shortlist, 5
Let $\mathbb{R}^+$ be the set of all positive real numbers. Find all functions $f: \mathbb{R}^+ \to \mathbb{R}^+$ that satisfy the following conditions:
- $f(xyz)+f(x)+f(y)+f(z)=f(\sqrt{xy})f(\sqrt{yz})f(\sqrt{zx})$ for all $x,y,z\in\mathbb{R}^+$;
- $f(x)<f(y)$ for all $1\le x<y$.
[i]Proposed by Hojoo Lee, Korea[/i]
2011 Vietnam Team Selection Test, 5
Find all positive integers $n$ such that $A=2^{n+2}(2^n-1)-8\cdot 3^n +1$ is a perfect square.
2006 Bundeswettbewerb Mathematik, 4
A piece of paper with the shape of a square lies on the desk. It gets dissected step by step into smaller pieces: in every step, one piece is taken from the desk and cut into two pieces by a straight cut; these pieces are put back on the desk then.
Find the smallest number of cuts needed to get $100$ $20$-gons.
2009 Ukraine National Mathematical Olympiad, 1
Pairwise distinct real numbers $a, b, c$ satisfies the equality
\[a +\frac 1b =b + \frac 1c =c+\frac 1a.\]
Find all possible values of $abc .$
Indonesia MO Shortlist - geometry, g12
In triangle $ABC$, the incircle is tangent to $BC$ at $D$, to $AC$ at $E$, and to $AB$ at $F$. Prove that:
$$\frac{CE-EA}{\sqrt{AB}}+\frac{AF-FB}{\sqrt{BC}} +\frac{BD-DC}{\sqrt{CA}} \ge \frac{BD-DC}{\sqrt{AB}}
+\frac{CE-EA}{\sqrt{BC}} +\frac{AF-FB}{\sqrt{CA}}$$
2018 Junior Balkan Team Selection Tests - Romania, 4
In $n$ transparent boxes there are red balls and blue balls. One needs to choose $50$ boxes such that, together, they contain at least half of the red balls and at least half of the blue balls. Is such a choice possible irrespective on the number of balls and on the way they are distributed in the boxes, if:
a) $n = 100$
b) $n = 99$?
2018 Regional Olympiad of Mexico Southeast, 5
Let $ABC$ an isosceles triangle with $CA=CB$ and $\Gamma$ it´s circumcircle. The perpendicular to $CB$ through $B$ intersect $\Gamma$ in points $B$ and $E$. The parallel to $BC$ through $A$ intersect $\Gamma$ in points $A$ and $D$. Let $F$ the intersection of $ED$ and $BC, I$ the intersection of $BD$ and $EC, \Omega$ the cricumcircle of the triangle $ADI$ and $\Phi$ the circumcircle of $BEF$.If $O$ and $P$ are the centers of $\Gamma$ and $\Phi$, respectively, prove that $OP$ is tangent to $\Omega$
2014 Saudi Arabia Pre-TST, 4.3
Fatima and Asma are playing the following game. First, Fatima chooses $2013$ pairwise different numbers, called $a_1, a_2, ..., a_{2013}$. Then, Asma tries to know the value of each number $a_1, a_2, ..., a_{2013}$.. At each time, Asma chooses $1 \le i < j \le 2013$ and asks Fatima ''[i]What is the set $\{a_i,a_j\}$?[/i]'' (For example, if Asma asks what is the set $\{a_i,a_j\}$, and $a_1 = 17$ and $a_2 = 13$, Fatima will answer $\{13. 17\}$). Find the least number of questions Asma needs to ask, to know the value of all the numbers $a_1, a_2, ..., a_{2013}$.
2023 ISI Entrance UGB, 2
Let $a_0 = \frac{1}{2}$ and $a_n$ be defined inductively by
\[a_n = \sqrt{\frac{1+a_{n-1}}{2}} \text{, $n \ge 1$.} \]
[list=a]
[*] Show that for $n = 0,1,2, \ldots,$
\[a_n = \cos(\theta_n) \text{ for some $0 < \theta_n < \frac{\pi}{2}$, }\]
and determine $\theta_n$.
[*] Using (a) or otherwise, calculate
\[ \lim_{n \to \infty} 4^n (1 - a_n).\]
[/list]
2013 India IMO Training Camp, 3
In a triangle $ABC$, with $AB \ne BC$, $E$ is a point on the line $AC$ such that $BE$ is perpendicular to $AC$. A circle passing through $A$ and touching the line $BE$ at a point $P \ne B$ intersects the line $AB$ for the second time at $X$. Let $Q$ be a point on the line $PB$ different from $P$ such that $BQ = BP$. Let $Y$ be the point of intersection of the lines $CP$ and $AQ$. Prove that the points $C, X, Y, A$ are concyclic if and only if $CX$ is perpendicular to $AB$.
2018 IMC, 9
Determine all pairs $P(x),Q(x)$ of complex polynomials with leading coefficient $1$ such that $P(x)$ divides $Q(x)^2+1$ and $Q(x)$ divides $P(x)^2+1$.
[i]Proposed by Rodrigo Angelo, Princeton University and Matheus Secco, PUC, Rio de Janeiro[/i]
2024 Kazakhstan National Olympiad, 5
In triangle $ABC$ ($AB\ne AC$), where all angles are greater than $45^\circ$, the altitude $AD$ is drawn. Let $\omega_1$ and $\omega_2$ be-- circles with diameters $AC$ and $AB$, respectively. The angle bisector of $\angle ADB$ secondarily intersects $\omega_1$ at point $P$, and the angle bisector of $\angle ADC$ secondarily intersects $\omega_2$ at point $Q$. The line $AP$ intersects $\omega_2$ at the point $R$. Prove that the circumcenter of triangle $PQR$ lies on line $BC$.
2016 Belarus Team Selection Test, 1
Find all functions $f:\mathbb{R}\to \mathbb{R},g:\mathbb{R}\to \mathbb{R}$ such that
$$f(x-2f(y))= xf(y)-yf(x)+g(x)$$ for all real $x,y$
2014 Greece Team Selection Test, 1
Let $(x_{n}) \ n\geq 1$ be a sequence of real numbers with $x_{1}=1$ satisfying $2x_{n+1}=3x_{n}+\sqrt{5x_{n}^{2}-4}$
a) Prove that the sequence consists only of natural numbers.
b) Check if there are terms of the sequence divisible by $2011$.