Olympiad problems

2018 Purple Comet Problems, 28

Tags: geometry

In $\vartriangle ABC$ points $D, E$, and $F$ lie on side $\overline{BC}$ such that $\overline{AD}$ is an angle bisector of $\angle BAC$, $\overline{AE}$ is a median, and $\overline{AF}$ is an altitude. Given that $AB = 154$ and $AC = 128$, and $9 \times DE = EF,$ find the side length $BC$.

2025 Thailand Mathematical Olympiad, 1

Tags: number theory

For each positive integer $m$, denote by $d(m)$ the number of positive divisors of $m$. We say that a positive integer $n$ is [i]Burapha[/i] integer if it satisfy the following condition [list] [*] $d(n)$ is an odd integer. [*] $d(k) \leqslant d(\ell)$ holds for every positive divisor $k, \ell$ of $n$, such that $k < \ell$ [/list] Find all Burapha integer.

2023 German National Olympiad, 6

Tags: cubic equation , number theory , algebra , ceiling function

The equation $x^3-3x^2+1=0$ has three real solutions $x_1<x_2<x_3$. Show that for any positive integer $n$, the number $\left\lceil x_3^n\right\rceil$ is a multiple of $3$.

1977 AMC 12/AHSME, 21

Tags: quadratic , special factorizations

For how many values of the coeﬃcient $a$ do the equations \begin{align*}x^2+ax+1=0 \\ x^2-x-a=0\end{align*} have a common real solution? $\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ \text{infinitely many}$

1994 All-Russian Olympiad Regional Round, 11.6

Tags: algebra , functional

Find all functions satisfying the equality $$(x-1)f \left(\dfrac{x+1}{x-1}\right)- f(x) = x$$ for all $x \ne 1$.

2010 Sharygin Geometry Olympiad, 23

Tags: geometry

A cyclic hexagon $ABCDEF$ is such that $AB \cdot CF= 2BC \cdot FA, CD \cdot EB = 2 DE \cdot BC$ and $EF \cdot AD = 2FA \cdot DE.$ Prove that the lines $AD, BE$ and $CF$ are concurrent.

2024 India IMOTC, 15

Tags: combinatorics

In a conference, mathematicians from $11$ different countries participate and they have integer-valued ages between $27$ and $33$ years (including $27$ and $33$). There is at least one mathematician from each country, and there is at least one mathematician of each possible age between $27$ and $33$. Show that we can find at least five mathematicians $m_1, \ldots, m_5$ such that for any $i \in \{1, \ldots, 5 \}$ there are more mathematicians in the conference having the same age as $m_i$ than those having the same nationality as $m_i$. [i]Proposed by S. Muralidharan[/i]

1997 Romania National Olympiad, 3

Tags: differentiable function , function , real analysis

Let $\mathcal{F}$ be the set of the differentiable functions $f: \mathbb{R} \to \mathbb{R}$ satisfying $f(x) \ge f(x+ \sin x)$ for any $x \in \mathbb{R}.$ a) Prove that there exist nonconstant functions in $\mathcal{F}.$ b) Prove that if $f \in \mathcal{F},$ then the set of solutions of the equation $f'(x)=0$ is infinite.

2020 Ecuador NMO (OMEC), 3

Tags: circumcircle , geometry

Let $ABC$ a triangle with circumcircle $\Gamma$ and circumcenter $O$. A point $X$, different from $A$, $B$, $C$, or their diametrically opposite points, on $\Gamma$, is chosen. Let $\omega$ the circumcircle of $COX$. Let $E$ the second intersection of $XA$ with $\omega$, $F$ the second intersection of $XB$ with $\omega$ and $D$ a point on line $AB$ such that $CD \perp EF$. Prove that $E$ is the circumcenter of $ADC$ and $F$ is the circumcenter of $BDC$.

2012 Turkmenistan National Math Olympiad, 3

Tags: logarithm , inequalities

Prove that : $\frac{1}{(\log_{bc} a)^n}+\frac{1}{(\log_{ac} b)^n}+\frac{1}{(\log_{bc} a)^n}\geq 3\cdot2^{n}$ where $a,b,c>1$ and $n$ is natural number.

2020 Regional Olympiad of Mexico Southeast, 6

Tags: prime number , sequence , number theory

Prove that for all $a, b$ and $x_0$ positive integers, in the sequence $x_1, x_2, x_3, \cdots$ defined by $$x_{n+1}=ax_n+b, n\geq 0$$ Exist an $x_i$ that is not prime for some $i\geq 1$

2014 All-Russian Olympiad, 4

Tags: geometry , perpendicular bisector , circumcircle , angle bisector , incenter , trigonometry

Given a triangle $ABC$ with $AB>BC$, let $ \Omega $ be the circumcircle. Let $M$, $N$ lie on the sides $AB$, $BC$ respectively, such that $AM=CN$. Let $K$ be the intersection of $MN$ and $AC$. Let $P$ be the incentre of the triangle $AMK$ and $Q$ be the $K$-excentre of the triangle $CNK$. If $R$ is midpoint of the arc $ABC$ of $ \Omega $ then prove that $RP=RQ$. [i]M. Kungodjin[/i]

2000 Manhattan Mathematical Olympiad, 1

Tags: algebra , polynomial

Prove there exists no polynomial $f(x)$, with integer coefficients, such that $f(7) = 11$ and $f(11) = 13$.

1992 Hungary-Israel Binational, 1

Tags: function , induction , inequalities

Prove that if $c$ is a positive number distinct from $1$ and $n$ a positive integer, then \[n^{2}\leq \frac{c^{n}+c^{-n}-2}{c+c^{-1}-2}. \]

1995 Spain Mathematical Olympiad, 4

Tags: diophantine equation , integer solutions , number theory

Given a prime number $p$, find all integer solutions of $p(x+y) = xy$.

1960 AMC 12/AHSME, 26

Tags: inequalities

Find the set of $x$-values satisfying the inequality $|\frac{5-x}{3}|<2$. [The symbol $|a|$ means $+a$ if $a$ is positive, $-a$ if $a$ is negative, 0 if $a$ is zero. The notation $1<a<2$ means that $a$ can have any value between $1$ and $2$, excluding $1$ and $2$. ] $ \textbf{(A)}\ 1 < x < 11\qquad\textbf{(B)}\ -1 < x < 11\qquad\textbf{(C)}\ x< 11\qquad$ $\textbf{(D)}\ x>11\qquad\textbf{(E)}\ |x| < 6 $

2025 CMIMC Team, 6

Tags: team

Suppose we have a regular $24$-gon labeled $A_1 \cdots A_{24}.$ We will draw $2$ similar $24$-gons within $A_1 \cdots A_{24}.$ For the sake of this problem, make $A_i=A_{i+24}.$ With our first configuration, we create $3$ stars by creating lines $\overline{A_iA_{i+9}}.$ A $24$-gon will be created in the center, which we denote as our first $24$-gon. With our second configuration, we create a start by creating lines $\overline{A_iA_{i+11}}.$ A $24$-gon will be created in the center, which we denote as our second $24$-gon. Find the ratio of the areas of the first $24$-gon to the second $24$-gon.

1997 Bundeswettbewerb Mathematik, 4

Tags: grid , combinatorics

There are $10000$ trees in a park, arranged in a square grid with $100$ rows and $100$ columns. Find the largest number of trees that can be cut down, so that sitting on any of the tree stumps one cannot see any other tree stump.

1990 AIME Problems, 10

Tags: greatest common divisor , least common multiple , number theory , complex number , relatively prime , modular arithmetic , trigonometry

The sets $A = \{z : z^{18} = 1\}$ and $B = \{w : w^{48} = 1\}$ are both sets of complex roots of unity. The set $C = \{zw : z \in A \ \text{and} \ w \in B\}$ is also a set of complex roots of unity. How many distinct elements are in $C$?

2022 Singapore MO Open, Q2

Tags: rectangle , number theory , geometry

Prove that if the length and breadth of a rectangle are both odd integers, then there does not exist a point $P$ inside the rectangle such that each of the distances from $P$ to the 4 corners of the rectangle is an integer.

2015 Balkan MO Shortlist, G3

Tags: combinatorial geometry , angle , geometry

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)

2016 Latvia National Olympiad, 4

Tags: number theory

Find the least prime factor of the number $\frac{2016^{2016}-3}{3}$.

1984 IMO Shortlist, 20

Tags: logarithm , inequalities , algebra

Determine all pairs $(a, b)$ of positive real numbers with $a \neq 1$ such that \[\log_a b < \log_{a+1} (b + 1).\]

2012 Grigore Moisil Intercounty, 2

Tags: real analysis

Let $ \left( x_n \right)_{n\ge 0} $ be a sequence of positive real numbers with $ x_0=1 $ and defined recursively: $$ x_{n+1}=x_n+\frac{x_0}{x_1+x_2+\cdots +x_n} $$ [b]a)[/b] Show that $ \lim_{n\to\infty } x_n=\infty . $ [b]b)[/b] Calculate $ \lim_{n\to\infty }\frac{x_n}{\sqrt{\ln n}} . $ [i]Ovidiu Furdui[/i]

2015 Brazil National Olympiad, 1

Tags: geometry , circumcircle

Let $\triangle ABC$ be an acute-scalene triangle, and let $N$ be the center of the circle wich pass trough the feet of altitudes. Let $D$ be the intersection of tangents to the circumcircle of $\triangle ABC$ at $B$ and $C$. Prove that $A$, $D$ and $N$ are collinear iff $\measuredangle BAC = 45º$.