This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 85335

KoMaL A Problems 2023/2024, A. 864
Tags: geometry
Let $ABC$ be a triangle and $O$ be its circumcenter. Let $D$, $E$ and $F$ be the respective tangent points of the incircle of $\triangle ABC$, and sides $BC$, $CA$ and $AB$. Let $M$ and $N$ be the respective midpoints of sides $AB$ and $AC$. Let $M'$ and $N'$ be the respective reflections of points $M$ and $N$ across lines $DE$ and $DF$. Let lines $CM'$ and $BN'$ intersect lines $DE$ and $DF$ at points $H$ and $J$, respectively. Prove that the points $H$, $J$ and $O$ are collinear. [i]Proposed by Luu Dong, Vietnam[/i]

1977 Chisinau City MO, 153
Tags: irrational , trigonometry , algebra
Prove that the number $\tan \frac{\pi}{3^n}$ is irrational for any natural $n$.

2014 Serbia National Math Olympiad, 4
Tags: number theory
We call natural number $n$ [i]$crazy$[/i] iff there exist natural numbers $a$, $b >1$ such that $n=a^b+b$. Whether there exist $2014$ consecutive natural numbers among which are $2012$ [i]$crazy$[/i] numbers? [i]Proposed by Milos Milosavljevic[/i]

1994 Korea National Olympiad, Problem 3
Tags: trigonometric identities , trigonometry , geometry
Let $\alpha,\beta ,\gamma$ be the angles of $\triangle ABC$. a) Show that $cos^2\alpha +cos^2\beta +cos^2 \gamma =1-2cos\alpha cos\beta cos\gamma$ . b) Given that $cos\alpha : cos\beta : cos\gamma = 39 : 33 : 25$, find $sin\alpha : sin\beta : sin\gamma$ .

1967 IMO Shortlist, 5
Tags: trigonometry , algebra , system of equations , trigonometric equation
If $x,y,z$ are real numbers satisfying relations \[x+y+z = 1 \quad \textrm{and} \quad \arctan x + \arctan y + \arctan z = \frac{\pi}{4},\] prove that $x^{2n+1} + y^{2n+1} + z^{2n+1} = 1$ holds for all positive integers $n$.

2019 IFYM, Sozopol, 3
Tags: geometry , orthocenter
The perpendicular bisector of $AB$ of an acute $\Delta ABC$ intersects $BC$ and the continuation of $AC$ in points $P$ and $Q$ respectively. $M$ and $N$ are the middle points of side $AB$ and segment $PQ$ respectively. If the lines $AB$ and $CN$ intersect in point $D$, prove that $\Delta ABC$ and $\Delta DCM$ have a common orthocenter.

2018 PUMaC Geometry A, 1
Tags: geometry
Frist Campus Center is located $1$ mile north and $1$ mile west of Fine Hall. The area within $5$ miles of Fine Hall that is located north and east of Frist can be expressed in the form $\frac{a}{b} \pi - c$, where $a, b, c$ are positive integers and $a$ and $b$ are relatively prime. Find $a + b + c$.

2009 Brazil National Olympiad, 3
Tags: function , limit , cauchy inequality , inequalities
Let $ n > 3$ be a fixed integer and $ x_1,x_2,\ldots, x_n$ be positive real numbers. Find, in terms of $ n$, all possible real values of \[ {x_1\over x_n\plus{}x_1\plus{}x_2} \plus{} {x_2\over x_1\plus{}x_2\plus{}x_3} \plus{} {x_3\over x_2\plus{}x_3\plus{}x_4} \plus{} \cdots \plus{} {x_{n\minus{}1}\over x_{n\minus{}2}\plus{}x_{n\minus{}1}\plus{}x_n} \plus{} {x_n\over x_{n\minus{}1}\plus{}x_n\plus{}x_1}\]

2015 India Regional MathematicaI Olympiad, 2
Tags: integer , polynomial , quadratic trinomial , algebra
Let \(P(x)=x^{2}+ax+b\) be a quadratic polynomial where \(a\) is real and \(b \neq 2\), is rational. Suppose \(P(0)^{2},P(1)^{2},P(2)^{2}\) are integers, prove that \(a\) and \(b\) are integers.

2011 Purple Comet Problems, 16
Tags:
Let $a$ and $b$ be nonzero real numbers such that $\tfrac{1}{3a}+\tfrac{1}{b}=2011$ and $\tfrac{1}{a}+\tfrac{1}{3b}=1$. What is the quotient when $a+b$ is divided by $ab$?

2024 Mathematical Talent Reward Programme, 7
Tags: geometry
$\bigtriangleup ABC$ triangle such that $AB = AC, \angle BAC = 20 \textdegree$. $P$ is on $AB$ such that $AP = BC$, find $\frac{1}{2}\angle APC$ in degrees.

1984 AIME Problems, 15
Tags: algebra , polynomial , limit , algorithm
Determine $w^2+x^2+y^2+z^2$ if \[ \begin{array}{l} \displaystyle \frac{x^2}{2^2-1}+\frac{y^2}{2^2-3^2}+\frac{z^2}{2^2-5^2}+\frac{w^2}{2^2-7^2}=1 \\ \displaystyle \frac{x^2}{4^2-1}+\frac{y^2}{4^2-3^2}+\frac{z^2}{4^2-5^2}+\frac{w^2}{4^2-7^2}=1 \\ \displaystyle \frac{x^2}{6^2-1}+\frac{y^2}{6^2-3^2}+\frac{z^2}{6^2-5^2}+\frac{w^2}{6^2-7^2}=1 \\ \displaystyle \frac{x^2}{8^2-1}+\frac{y^2}{8^2-3^2}+\frac{z^2}{8^2-5^2}+\frac{w^2}{8^2-7^2}=1 \\ \end{array} \]

2005 Taiwan National Olympiad, 2
Tags: trigonometry , geometry
Given a line segment $AB=7$, $C$ is constructed on $AB$ so that $AC=5$. Two equilateral triangles are constructed on the same side of $AB$ with $AC$ and $BC$ as a side. Find the length of the segment connecting their two circumcenters.

2002 China Team Selection Test, 1
Tags: algebra
Given that $ a_1\equal{}1$, $ a_2\equal{}5$, $ \displaystyle a_{n\plus{}1} \equal{} \frac{a_n \cdot a_{n\minus{}1}}{\sqrt{a_n^2 \plus{} a_{n\minus{}1}^2 \plus{} 1}}$. Find a expression of the general term of $ \{ a_n \}$.

2016 Iran MO (3rd Round), 3
Tags: combinatorics , graph theory , directed graph
There are $24$ robots on the plane. Each robot has a $70^{\circ}$ field of view. What is the maximum number of observing relations? (Observing is a one-sided relation)

2019 Serbia National Math Olympiad, 1
Tags: number theory , relatively prime
Find all positive integers $n, n>1$ for wich holds : If $a_1, a_2 ,\dots ,a_k$ are all numbers less than $n$ and relatively prime to $n$ , and holds $a_1<a_2<\dots <a_k $, then none of sums $a_i+a_{i+1}$ for $i=1,2,3,\dots k-1 $ are divisible by $3$.

2003 Romania National Olympiad, 4
Tags: geometry , 3d geometry , tetrahedron
In tetrahedron $ ABCD$, $ G_1,G_2$ and $ G_3$ are barycenters of the faces $ ACD,ABD$ and $ BCD$ respectively. (a) Prove that the straight lines $ BG_1,CG_2$ and $ AG_3$ are concurrent. (b) Knowing that $ AG_3\equal{}8,BG_1\equal{}12$ and $ CG_2\equal{}20$ compute the maximum possible value of the volume of $ ABCD$.

2005 MOP Homework, 3
Tags: relatively prime , number theory
For any positive integer $n$, the sum $1+\frac{1}{2}+\frac{1}{3}+\cdots +\frac{1}{n}$ is written in the lowest form $\frac{p_n}{q_n}$; that is, $p_n$ and $q_n$ are relatively prime positive integers. Find all $n$ such that $p_n$ is divisible by $3$.

BIMO 2022, 2
Tags: combinatorial geometry , combinatorics
Let $\mathcal{S}$ be a set of $2023$ points in a plane, and it is known that the distances of any two different points in $S$ are all distinct. Ivan colors the points with $k$ colors such that for every point $P \in \mathcal{S}$, the closest and the furthest point from $P$ in $\mathcal{S}$ also have the same color as $P$. What is the maximum possible value of $k$? [i]Proposed by Ivan Chan Kai Chin[/i]

1997 Moldova Team Selection Test, 1
Tags: number theory
Let $a$ and $b$ be two odd positive integers. Define the sequence $(x_n)_{n\in\mathbb{N}}$ as such: $x_1=a, x_2=b,$ for every $n\geq3$ the term $x_n{}$ is the greatest odd integer of $x_{n-1}+x_{n-2}$. Show that starting with a term, all the following terms are constant.

2008 Junior Balkan Team Selection Tests - Romania, 4
Tags: floor function , irrational number , number theory
Let $ a,b$ be real nonzero numbers, such that number $ \lfloor an \plus{} b \rfloor$ is an even integer for every $ n \in \mathbb{N}$. Prove that $ a$ is an even integer.

1987 Spain Mathematical Olympiad, 2
Tags: binomial coefficient , positive integer , inequalities
Show that for each natural number $n > 1$ $1 \cdot \sqrt{{n \choose 1}}+ 2 \cdot \sqrt{{n \choose 2}}+...+n \cdot \sqrt{{n \choose n}} <\sqrt{2^{n-1}n^3}$

2018 Stanford Mathematics Tournament, 3
Tags: geometry
A triangle has side lengths of $7$, $8$, and $9$. Find the radius of the largest possible semicircle inscribed in the triangle.

III Soros Olympiad 1996 - 97 (Russia), 10.3
Tags: radical , algebra
Solve the equation $$\sqrt{x(x+7)}+\sqrt{(x+7)(x+17)}+\sqrt{(x+17)(x+24)}=12+17\sqrt2$$

2015 India PRMO, 6
Tags: perfect square , number theory , digit , sum of digits
$6.$ How many two digit positive integers $N$ have the property that the sum of $N$ and the number obtained by reversing the order of the digits of $N$ is a perfect square $?$