Olympiad problems

2006 IMO, 2

Tags: combinatorics , polygon , Extremal combinatorics , IMO , IMO 2006 , IMO Shortlist , Dusan Djukic

Let $P$ be a regular $2006$-gon. A diagonal is called [i]good[/i] if its endpoints divide the boundary of $P$ into two parts, each composed of an odd number of sides of $P$. The sides of $P$ are also called [i]good[/i]. Suppose $P$ has been dissected into triangles by $2003$ diagonals, no two of which have a common point in the interior of $P$. Find the maximum number of isosceles triangles having two good sides that could appear in such a configuration.

1967 IMO, 3

Tags: number theory , Divisibility , binomial coefficients , IMO , IMO 1967

Let $k,m,n$ be natural numbers such that $m+k+1$ is a prime greater than $n+1$. Let $c_s=s(s+1)$. Prove that \[(c_{m+1}-c_k)(c_{m+2}-c_k)\ldots(c_{m+n}-c_k)\] is divisible by the product $c_1c_2\ldots c_n$.

1966 IMO, 1

Tags: combinatorics , system of equations , algebra , IMO , IMO 1966

In a mathematical contest, three problems, $A,B,C$ were posed. Among the participants ther were 25 students who solved at least one problem each. Of all the contestants who did not solve problem $A$, the number who solved $B$ was twice the number who solved $C$. The number of students who solved only problem $A$ was one more than the number of students who solved $A$ and at least one other problem. Of all students who solved just one problem, half did not solve problem $A$. How many students solved only problem $B$?

1997 IMO, 4

Tags: matrix , combinatorics , Coloring , IMO , IMO 1997 , induction

An $ n \times n$ matrix whose entries come from the set $ S \equal{} \{1, 2, \ldots , 2n \minus{} 1\}$ is called a [i]silver matrix[/i] if, for each $ i \equal{} 1, 2, \ldots , n$, the $ i$-th row and the $ i$-th column together contain all elements of $ S$. Show that: (a) there is no silver matrix for $ n \equal{} 1997$; (b) silver matrices exist for infinitely many values of $ n$.

2017 IMO, 1

Tags: number theory , IMO , IMO 2017 , IMO Shortlist , Sequence

For each integer $a_0 > 1$, define the sequence $a_0, a_1, a_2, \ldots$ for $n \geq 0$ as $$a_{n+1} = \begin{cases} \sqrt{a_n} & \text{if } \sqrt{a_n} \text{ is an integer,} \\ a_n + 3 & \text{otherwise.} \end{cases} $$ Determine all values of $a_0$ such that there exists a number $A$ such that $a_n = A$ for infinitely many values of $n$. [i]Proposed by Stephan Wagner, South Africa[/i]

1989 IMO Longlists, 2

Tags: geometry , circumcircle , inradius , area , geometric inequality , IMO , IMO 1989

$ ABC$ is a triangle, the bisector of angle $ A$ meets the circumcircle of triangle $ ABC$ in $ A_1$, points $ B_1$ and $ C_1$ are defined similarly. Let $ AA_1$ meet the lines that bisect the two external angles at $ B$ and $ C$ in $ A_0$. Define $ B_0$ and $ C_0$ similarly. Prove that the area of triangle $ A_0B_0C_0 \equal{} 2 \cdot$ area of hexagon $ AC_1BA_1CB_1 \geq 4 \cdot$ area of triangle $ ABC$.

1960 IMO Shortlist, 3

Tags: trigonometry , geometry , circumcircle , Trigonometric Identities , IMO , IMO 1960

In a given right triangle $ABC$, the hypotenuse $BC$, of length $a$, is divided into $n$ equal parts ($n$ and odd integer). Let $\alpha$ be the acute angel subtending, from $A$, that segment which contains the mdipoint of the hypotenuse. Let $h$ be the length of the altitude to the hypotenuse fo the triangle. Prove that: \[ \tan{\alpha}=\dfrac{4nh}{(n^2-1)a}. \]

1985 IMO Longlists, 94

Tags: geometry , circumcircle , IMO , IMO 1985 , angle , Triangle

A circle with center $O$ passes through the vertices $A$ and $C$ of the triangle $ABC$ and intersects the segments $AB$ and $BC$ again at distinct points $K$ and $N$ respectively. Let $M$ be the point of intersection of the circumcircles of triangles $ABC$ and $KBN$ (apart from $B$). Prove that $\angle OMB=90^{\circ}$.

2021 IMO, 6

Tags: IMO 2021 , algebra , linear algebra , IMO , Number system , other bases , combinatorics

Let $m\ge 2$ be an integer, $A$ a finite set of integers (not necessarily positive) and $B_1,B_2,...,B_m$ subsets of $A$. Suppose that, for every $k=1,2,...,m$, the sum of the elements of $B_k$ is $m^k$. Prove that $A$ contains at least $\dfrac{m}{2}$ elements.

2019 IMO Shortlist, C5

Tags: combinatorics , IMO 2019 , IMO , graph theory , reachability , termination , IMO Shortlist 2019

A social network has $2019$ users, some pairs of whom are friends. Whenever user $A$ is friends with user $B$, user $B$ is also friends with user $A$. Events of the following kind may happen repeatedly, one at a time: [list] [*] Three users $A$, $B$, and $C$ such that $A$ is friends with both $B$ and $C$, but $B$ and $C$ are not friends, change their friendship statuses such that $B$ and $C$ are now friends, but $A$ is no longer friends with $B$, and no longer friends with $C$. All other friendship statuses are unchanged. [/list] Initially, $1010$ users have $1009$ friends each, and $1009$ users have $1010$ friends each. Prove that there exists a sequence of such events after which each user is friends with at most one other user. [i]Proposed by Adrian Beker, Croatia[/i]

1994 IMO, 5

Tags: function , algebra , functional equation , IMO , IMO 1994 , increasing functions , david monk

Let $ S$ be the set of all real numbers strictly greater than −1. Find all functions $ f: S \to S$ satisfying the two conditions: (a) $ f(x \plus{} f(y) \plus{} xf(y)) \equal{} y \plus{} f(x) \plus{} yf(x)$ for all $ x, y$ in $ S$; (b) $ \frac {f(x)}{x}$ is strictly increasing on each of the two intervals $ \minus{} 1 < x < 0$ and $ 0 < x$.

1997 IMO Shortlist, 21

Tags: inequalities , permutation , combinatorics , IMO , IMO 1997

Let $ x_1$, $ x_2$, $ \ldots$, $ x_n$ be real numbers satisfying the conditions: \[ \left\{\begin{array}{cccc} |x_1 \plus{} x_2 \plus{} \cdots \plus{} x_n | & \equal{} & 1 & \ \\ |x_i| & \leq & \displaystyle \frac {n \plus{} 1}{2} & \ \textrm{ for }i \equal{} 1, 2, \ldots , n. \end{array} \right. \] Show that there exists a permutation $ y_1$, $ y_2$, $ \ldots$, $ y_n$ of $ x_1$, $ x_2$, $ \ldots$, $ x_n$ such that \[ | y_1 \plus{} 2 y_2 \plus{} \cdots \plus{} n y_n | \leq \frac {n \plus{} 1}{2}. \]

1986 IMO Longlists, 14

Tags: geometry , rotation , Triangles , Fixed point , Equilateral Triangle , IMO , IMO 1986

Given a point $P_0$ in the plane of the triangle $A_1A_2A_3$. Define $A_s=A_{s-3}$ for all $s\ge4$. Construct a set of points $P_1,P_2,P_3,\ldots$ such that $P_{k+1}$ is the image of $P_k$ under a rotation center $A_{k+1}$ through an angle $120^o$ clockwise for $k=0,1,2,\ldots$. Prove that if $P_{1986}=P_0$, then the triangle $A_1A_2A_3$ is equilateral.

2010 Contests, 2

Tags: geometry , IMO Shortlist , Inversion , IMO 2010 , P2 , IMO

Given a triangle $ABC$, with $I$ as its incenter and $\Gamma$ as its circumcircle, $AI$ intersects $\Gamma$ again at $D$. Let $E$ be a point on the arc $BDC$, and $F$ a point on the segment $BC$, such that $\angle BAF=\angle CAE < \dfrac12\angle BAC$. If $G$ is the midpoint of $IF$, prove that the meeting point of the lines $EI$ and $DG$ lies on $\Gamma$. [i]Proposed by Tai Wai Ming and Wang Chongli, Hong Kong[/i]

2013 IMO, 5

Tags: algebra , function , IMO , IMO 2013 , functional equation , IMO Shortlist , Hi

Let $\mathbb Q_{>0}$ be the set of all positive rational numbers. Let $f:\mathbb Q_{>0}\to\mathbb R$ be a function satisfying the following three conditions: (i) for all $x,y\in\mathbb Q_{>0}$, we have $f(x)f(y)\geq f(xy)$; (ii) for all $x,y\in\mathbb Q_{>0}$, we have $f(x+y)\geq f(x)+f(y)$; (iii) there exists a rational number $a>1$ such that $f(a)=a$. Prove that $f(x)=x$ for all $x\in\mathbb Q_{>0}$. [i]Proposed by Bulgaria[/i]

2023 CMI B.Sc. Entrance Exam, 6

Tags: function , number theory , strong induction , CMI , IMO

Consider a positive integer $a > 1$. If $a$ is not a perfect square then at the next move we add $3$ to it and if it is a perfect square we take the square root of it. Define the trajectory of a number $a$ as the set obtained by performing this operation on $a$. For example the cardinality of $3$ is $\{3, 6, 9\}$. Find all $n$ such that the cardinality of $n$ is finite. The following part problems may attract partial credit. $\textbf{(a)}$Show that the cardinality of the trajectory of a number cannot be $1$ or $2$. $\textbf{(b)}$Show that $\{3, 6, 9\}$ is the only trajectory with cardinality $3$. $\textbf{(c)}$ Show that there for all $k \geq 3$, there exists a number such that the cardinality of its trajectory is $k$. $\textbf{(d)}$ Give an example of a number with cardinality of trajectory as infinity.

1987 IMO Shortlist, 22

Tags: function , algebra , functional equation , IMO Shortlist , IMO , IMO 1987

Does there exist a function $f : \mathbb N \to \mathbb N$, such that $f(f(n)) =n + 1987$ for every natural number $n$? [i](IMO Problem 4)[/i] [i]Proposed by Vietnam.[/i]

1968 IMO Shortlist, 26

Tags: function , algebra , periodic function , functional equation , IMO , IMO 1968

Let $f$ be a real-valued function defined for all real numbers, such that for some $a>0$ we have \[ f(x+a)={1\over2}+\sqrt{f(x)-f(x)^2} \] for all $x$. Prove that $f$ is periodic, and give an example of such a non-constant $f$ for $a=1$.

2012 IMO Shortlist, A3

Tags: inequalities , IMO , IMO 2012 , IMO Shortlist , n-variable inequality , Hi , ineqstd

Let $n\ge 3$ be an integer, and let $a_2,a_3,\ldots ,a_n$ be positive real numbers such that $a_{2}a_{3}\cdots a_{n}=1$. Prove that \[(1 + a_2)^2 (1 + a_3)^3 \dotsm (1 + a_n)^n > n^n.\] [i]Proposed by Angelo Di Pasquale, Australia[/i]

1987 IMO Longlists, 29

Tags: geometry , IMO , IMO 1987 , IMO Shortlist , coordinate geometry , area

Is it possible to put $1987$ points in the Euclidean plane such that the distance between each pair of points is irrational and each three points determine a non-degenerate triangle with rational area? [i](IMO Problem 5)[/i] [i]Proposed by Germany, DR[/i]

1970 IMO Longlists, 17

Tags: geometry , 3D geometry , tetrahedron , geometric inequality , IMO , IMO 1970

In the tetrahedron $ABCD,\angle BDC=90^o$ and the foot of the perpendicular from $D$ to $ABC$ is the intersection of the altitudes of $ABC$. Prove that: \[ (AB+BC+CA)^2\le6(AD^2+BD^2+CD^2). \] When do we have equality?

2004 IMO Shortlist, 5

Tags: induction , modular arithmetic , number theory , IMO , IMO 2004 , IMO Shortlist , decimal representation

We call a positive integer [i]alternating[/i] if every two consecutive digits in its decimal representation are of different parity. Find all positive integers $n$ such that $n$ has a multiple which is alternating.

2009 IMO Shortlist, 1

Tags: induction , modular arithmetic , number theory , IMO 2009 , IMO Shortlist , IMO

Let $ n$ be a positive integer and let $ a_1,a_2,a_3,\ldots,a_k$ $ ( k\ge 2)$ be distinct integers in the set $ { 1,2,\ldots,n}$ such that $ n$ divides $ a_i(a_{i + 1} - 1)$ for $ i = 1,2,\ldots,k - 1$. Prove that $ n$ does not divide $ a_k(a_1 - 1).$ [i]Proposed by Ross Atkins, Australia [/i]

2001 IMO, 2

Tags: IMO Shortlist , IMO 2001 , inequalities , hojoo lee , IMO , Hi

Prove that for all positive real numbers $a,b,c$, \[ \frac{a}{\sqrt{a^2 + 8bc}} + \frac{b}{\sqrt{b^2 + 8ca}} + \frac{c}{\sqrt{c^2 + 8ab}} \geq 1. \]

1962 IMO, 5

Tags: geometry , circumcircle , incenter , angle bisector , IMO , IMO 1962

On the circle $K$ there are given three distinct points $A,B,C$. Construct (using only a straightedge and a compass) a fourth point $D$ on $K$ such that a circle can be inscribed in the quadrilateral thus obtained.