Found problems: 85335
2012 Mediterranean Mathematics Olympiad, 2
In an acute $\triangle ABC$, prove that
\begin{align*}\frac{1}{3}\left(\frac{\tan^2A}{\tan B\tan C}+\frac{\tan^2 B}{\tan C\tan A}+\frac{\tan^2 C}{\tan A\tan B}\right) \\ +3\left(\frac{1}{\tan A+\tan B+\tan C}\right)^{\frac{2}{3}}\ge 2.\end{align*}
2010 IberoAmerican, 2
Let $ABCD$ be a cyclic quadrilateral whose diagonals $AC$ and $BD$ are perpendicular. Let $O$ be the circumcenter of $ABCD$, $K$ the intersection of the diagonals, $ L\neq O $ the intersection of the circles circumscribed to $OAC$ and $OBD$, and $G$ the intersection of the diagonals of the quadrilateral whose vertices are the midpoints of the sides of $ABCD$. Prove that $O, K, L$ and $G$ are collinear
2004 All-Russian Olympiad Regional Round, 8.2
There is a set of weights with the following properties:
1) It contains 5 weights, pairs of different weights.
2) For any two weights, there are two other weights of the same total weight.
What is the smallest number of weights that can be in this set?
2003 Costa Rica - Final Round, 3
If $a>1$ and $b>2$ are positive integers, show that $a^{b}+1 \geq b(a+1)$, and determine when equality holds.
2005 Georgia Team Selection Test, 8
In a convex quadrilateral $ ABCD$ the points $ P$ and $ Q$ are chosen on the sides $ BC$ and $ CD$ respectively so that $ \angle{BAP}\equal{}\angle{DAQ}$. Prove that the line, passing through the orthocenters of triangles $ ABP$ and $ ADQ$, is perpendicular to $ AC$ if and only if the triangles $ ABP$ and $ ADQ$ have the same areas.
2024 Brazil Team Selection Test, 3
Let $n$ be a positive integer and let $a_1, a_2, \ldots, a_n$ be positive reals. Show that $$\sum_{i=1}^{n} \frac{1}{2^i}(\frac{2}{1+a_i})^{2^i} \geq \frac{2}{1+a_1a_2\ldots a_n}-\frac{1}{2^n}.$$
1976 IMO Longlists, 35
Let $P$ be a polynomial with real coefficients such that $P(x) > 0$ if $x > 0$. Prove that there exist polynomials $Q$ and $R$ with nonnegative coefficients such that $P(x) = \frac{Q(x)}{R(x)}$ if $x > 0.$
2019 PUMaC Combinatorics B, 2
Suppose Alan, Michael, Kevin, Igor, and Big Rahul are in a running race. It is given that exactly one pair of people tie (for example, two people both get second place), so that no other pair of people end in the same position. Each competitor has equal skill; this means that each outcome of the race, given that exactly two people tie, is equally likely. The probability that Big Rahul gets first place (either by himself or he ties for first) can be expressed in the form $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$.
2008 Brazil Team Selection Test, 4
The diagonals of a trapezoid $ ABCD$ intersect at point $ P$. Point $ Q$ lies between the parallel lines $ BC$ and $ AD$ such that $ \angle AQD \equal{} \angle CQB$, and line $ CD$ separates points $ P$ and $ Q$. Prove that $ \angle BQP \equal{} \angle DAQ$.
[i]Author: Vyacheslav Yasinskiy, Ukraine[/i]
1989 All Soviet Union Mathematical Olympiad, 504
$ABC$ is a triangle. Points $D, E, F$ are chosen on $BC, CA, AB$ such that $B$ is equidistant from $D$ and $F$, and $C$ is equidistant from $D$ and $E$. Show that the circumcenter of $AEF$ lies on the bisector of $EDF$.
2020 IMEO, Problem 6
Let $O$, $I$, and $\omega$ be the circumcenter, the incenter, and the incircle of nonequilateral $\triangle ABC$. Let $\omega_A$ be the unique circle tangent to $AB$ and $AC$, such that the common chord of $\omega_A$ and $\omega$ passes through the center of $\omega_A$ . Let $O_A$ be the center of $\omega_A$. Define $\omega_B, O_B, \omega_C, O_C$ similarly. If $\omega$ touches $BC$, $CA$, $AB$ at $D$, $E$, $F$ respectively, prove that the perpendiculars from $D$, $E$, $F$ to $O_BO_C , O_CO_A , O_AO_B$ are concurrent on the line $OI$.
[i]Pitchayut Saengrungkongka[/i]
2002 Iran MO (3rd Round), 2
$f: \mathbb R\longrightarrow\mathbb R^{+}$ is a non-decreasing function. Prove that there is a point $a\in\mathbb R$ that \[f(a+\frac1{f(a)})<2f(a)\]
2014 Harvard-MIT Mathematics Tournament, 5
Eli, Joy, Paul, and Sam want to form a company; the company will have 16 shares to split among the $4$ people. The following constraints are imposed:
$\bullet$ Every person must get a positive integer number of shares, and all $16$ shares must be given out.
$\bullet$ No one person can have more shares than the other three people combined.
Assuming that shares are indistinguishable, but people are distinguishable, in how many ways can the shares be given out?
2009 All-Russian Olympiad Regional Round, 11.8
11 integers are placed along the circle. It is known that any two neighbors differ by at least 20 and sum of any two neighbors is no more than 100. Find the minimal possible sum of all numbers.
2014 Contests, 1
Find all triples $(f,g,h)$ of injective functions from the set of real numbers to itself satisfying
\begin{align*}
f(x+f(y)) &= g(x) + h(y) \\
g(x+g(y)) &= h(x) + f(y) \\
h(x+h(y)) &= f(x) + g(y)
\end{align*}
for all real numbers $x$ and $y$. (We say a function $F$ is [i]injective[/i] if $F(a)\neq F(b)$ for any distinct real numbers $a$ and $b$.)
[i]Proposed by Evan Chen[/i]
2020 DMO Stage 1, 2.
[b]Q.[/b] Consider in the plane $n>3$ different points. These have the properties, that all $3$ points can be included in a triangle with maximum area $1$. Prove that all the $n>3$ points can be included in a triangle with maximum area $4$.
[i]Proposed by TuZo[/i]
2013 Princeton University Math Competition, 3
Find the smallest positive integer $n$ with the following property: for every sequence of positive integers $a_1,a_2,\ldots , a_n$ with $a_1+a_2+\ldots +a_n=2013$, there exist some (possibly one) consecutive term(s) in the sequence that add up to $70$.
2009 Hungary-Israel Binational, 2
Denote the three real roots of the cubic $ x^3 \minus{} 3x \minus{} 1 \equal{} 0$ by $ x_1$, $ x_2$, $ x_3$ in order of increasing magnitude. (You may assume that the equation in fact has three distinct real roots.) Prove that $ x_3^2 \minus{} x_2^2 \equal{} x_3 \minus{} x_1$.
2000 Kazakhstan National Olympiad, 3
In a country with $ n $ ($ n \geq 3 $) airports, the government only licenses air travel to those airlines whose airline system meets the following conditions:
a) Each airline must connect any two airports with one and only one one-way airline;
b) For each airline there is an airport from which the passenger could fly off and fly back, using the services of only this airline.
What is the maximum number of airlines with different airline systems?
2023 IFYM, Sozopol, 2
Find all functions $f: \mathbb{Z} \to \mathbb{Z}$ such that
\[
f(x) + f(y - 1) + f(f(y - f(x))) = 1
\]
for all integers $x$ and $y$.
2005 Cuba MO, 3
There are two piles of cards, one with $n$ cards and the other with $m$ cards.
$A$ and $B$ play alternately, performing one of the following actions in each turn. following operations:
a) Remove a card from a pile.
b) Remove one card from each pile.
c) Move a card from one pile to the other.
Player $A$ always starts the game and whoever takes the last one letter wins . Determine if there is a winning strategy based on $m$ and $n$, so that one of the players following her can win always.
2003 District Olympiad, 4
Let $ABC$ be a triangle. Let $B'$ be the symmetric of $B$ with respect to $C, C'$ the symmetry of $C$ with respect to $A$ and $A'$ the symmetry of $A$ with respect to $B$.
a) Prove that the area of triangle $AC'A'$ is twice the area of triangle $ABC$.
b) If we delete points $A, B, C$, how can they be reconstituted? Justify your reasoning.
2014 JBMO TST - Macedonia, 2
Point $M$ is an arbitrary point in the plane and let points $G$ and $H$ be the intersection points of the tangents from point M and the circle $k$. Let $O$ be the center of the circle $k$ and let $K$ be the orthocenter of the triangle $MGH$. Prove that ${\angle}GMH={\angle}OGK$.
2021 Purple Comet Problems, 24
Let $x$ be a real number such that $$4^{2x}+2^{-x}+1=(129+8\sqrt2)(4^{x}+2^{-x}-2^{x}).$$ Find $10x$.
2012 Today's Calculation Of Integral, 804
For $a>0$, find the minimum value of $I(a)=\int_1^e |\ln ax|\ dx.$