Olympiad problems

1967 IMO Shortlist, 1

In a sports meeting a total of $m$ medals were awarded over $n$ days. On the first day one medal and $\frac{1}{7}$ of the remaining medals were awarded. On the second day two medals and $\frac{1}{7}$ of the remaining medals were awarded, and so on. On the last day, the remaining $n$ medals were awarded. How many medals did the meeting last, and what was the total number of medals ?

1990 IMO Shortlist, 6

Tags: combinatorics , game , game strategy , algorithm , IMO , IMO 1990

Given an initial integer $ n_0 > 1$, two players, $ {\mathcal A}$ and $ {\mathcal B}$, choose integers $ n_1$, $ n_2$, $ n_3$, $ \ldots$ alternately according to the following rules : [b]I.)[/b] Knowing $ n_{2k}$, $ {\mathcal A}$ chooses any integer $ n_{2k \plus{} 1}$ such that \[ n_{2k} \leq n_{2k \plus{} 1} \leq n_{2k}^2. \] [b]II.)[/b] Knowing $ n_{2k \plus{} 1}$, $ {\mathcal B}$ chooses any integer $ n_{2k \plus{} 2}$ such that \[ \frac {n_{2k \plus{} 1}}{n_{2k \plus{} 2}} \] is a prime raised to a positive integer power. Player $ {\mathcal A}$ wins the game by choosing the number 1990; player $ {\mathcal B}$ wins by choosing the number 1. For which $ n_0$ does : [b]a.)[/b] $ {\mathcal A}$ have a winning strategy? [b]b.)[/b] $ {\mathcal B}$ have a winning strategy? [b]c.)[/b] Neither player have a winning strategy?

1971 IMO Longlists, 35

Tags: number theory , Euler s Theorem , Sequence , relatively prime , IMO , IMO 1970

Prove that we can find an infinite set of positive integers of the from $2^n-3$ (where $n$ is a positive integer) every pair of which are relatively prime.

1976 IMO Shortlist, 3

Tags: geometry , trigonometry , inequalities , area of a triangle , geometric inequality , IMO , imo 1976

In a convex quadrilateral (in the plane) with the area of $32 \text{ cm}^{2}$ the sum of two opposite sides and a diagonal is $16 \text{ cm}$. Determine all the possible values that the other diagonal can have.

1982 IMO Longlists, 55

Tags: geometry , combinatorial geometry , combinatorics , length , polygon , IMO , IMO 1982

Let $S$ be a square with sides length $100$. Let $L$ be a path within $S$ which does not meet itself and which is composed of line segments $A_0A_1,A_1A_2,A_2A_3,\ldots,A_{n-1}A_n$ with $A_0=A_n$. Suppose that for every point $P$ on the boundary of $S$ there is a point of $L$ at a distance from $P$ no greater than $\frac {1} {2}$. Prove that there are two points $X$ and $Y$ of $L$ such that the distance between $X$ and $Y$ is not greater than $1$ and the length of the part of $L$ which lies between $X$ and $Y$ is not smaller than $198$.

1964 IMO Shortlist, 2

Tags: inequalities , geometry , trigonometry , triangle inequality , IMO , IMO 1964 , algebra

Suppose $a,b,c$ are the sides of a triangle. Prove that \[ a^2(b+c-a)+b^2(a+c-b)+c^2(a+b-c) \leq 3abc \]

1997 IMO, 6

Tags: combinatorics , counting , Integer partitions , power of 2 , IMO , IMO 1997

For each positive integer $ n$, let $ f(n)$ denote the number of ways of representing $ n$ as a sum of powers of 2 with nonnegative integer exponents. Representations which differ only in the ordering of their summands are considered to be the same. For instance, $ f(4) \equal{} 4$, because the number 4 can be represented in the following four ways: 4; 2+2; 2+1+1; 1+1+1+1. Prove that, for any integer $ n \geq 3$ we have $ 2^{\frac {n^2}{4}} < f(2^n) < 2^{\frac {n^2}2}$.

1972 IMO, 1

Tags: combinatorics , pigeonhole principle , IMO , IMO 1972

Prove that from a set of ten distinct two-digit numbers, it is always possible to find two disjoint subsets whose members have the same sum.

2020 CHMMC Winter (2020-21), 15

Tags: combinatorics , AIME , USAMO , IMO

For an integer $n \ge 2$, let $G_n$ be an $n \times n$ grid of unit cells. A subset of cells $H \subseteq G_n$ is considered \textit{quasi-complete} if and only if each row of $G_n$ has at least one cell in $H$ and each column of $G_n$ has at least one cell in $H$. A subset of cells $K \subseteq G_n$ is considered \textit{quasi-perfect} if and only if there is a proper subset $L \subset K$ such that $|L| = n$ and no two elements in $L$ are in the same row or column. Let $\vartheta(n)$ be the smallest positive integer such that every quasi-complete subset $H \subseteq G_n$ with $|H| \ge \vartheta(n)$ is also quasi-perfect. Moreover, let $\varrho(n)$ be the number of quasi-complete subsets $H \subseteq G_n$ with $|H| = \vartheta(n) - 1$ such that $H$ is not quasi-perfect. Compute $\vartheta(20) + \varrho(20)$.

1973 IMO, 2

Tags: function , algebra , Fixed point , group theory , IMO , IMO 1973

$G$ is a set of non-constant functions $f$. Each $f$ is defined on the real line and has the form $f(x)=ax+b$ for some real $a,b$. If $f$ and $g$ are in $G$, then so is $fg$, where $fg$ is defined by $fg(x)=f(g(x))$. If $f$ is in $G$, then so is the inverse $f^{-1}$. If $f(x)=ax+b$, then $f^{-1}(x)= \frac{x-b}{a}$. Every $f$ in $G$ has a fixed point (in other words we can find $x_f$ such that $f(x_f)=x_f$. Prove that all the functions in $G$ have a common fixed point.

1968 IMO, 4

Tags: geometry , 3D geometry , tetrahedron , triangle inequality , IMO , IMO 1968

Prove that every tetrahedron has a vertex whose three edges have the right lengths to form a triangle.

1978 IMO Shortlist, 10

Tags: graph theory , Ramsey Theory , combinatorics , Extremal Graph Theory , Extremal combinatorics , IMO , IMO 1978

An international society has its members from six different countries. The list of members contain $1978$ names, numbered $1, 2, \dots, 1978$. Prove that there is at least one member whose number is the sum of the numbers of two members from his own country, or twice as large as the number of one member from his own country.

2010 IMO, 3

Tags: function , algebra , IMO Shortlist , IMO 2010 , Perfect Squares , number theory , IMO

Find all functions $g:\mathbb{N}\rightarrow\mathbb{N}$ such that \[\left(g(m)+n\right)\left(g(n)+m\right)\] is a perfect square for all $m,n\in\mathbb{N}.$ [i]Proposed by Gabriel Carroll, USA[/i]

1965 IMO, 1

Tags: inequalities , trigonometry , IMO , IMO 1965

Determine all values of $x$ in the interval $0 \leq x \leq 2\pi$ which satisfy the inequality \[ 2 \cos{x} \leq \sqrt{1+\sin{2x}}-\sqrt{1-\sin{2x}} \leq \sqrt{2}. \]

1975 IMO, 3

Tags: geometry , trigonometry , Triangle , angles , IMO , IMO 1975

In the plane of a triangle $ABC,$ in its exterior$,$ we draw the triangles $ABR, BCP, CAQ$ so that $\angle PBC = \angle CAQ = 45^{\circ}$, $\angle BCP = \angle QCA = 30^{\circ}$, $\angle ABR = \angle RAB = 15^{\circ}$. Prove that [b]a.)[/b] $\angle QRP = 90\,^{\circ},$ and [b]b.)[/b] $QR = RP.$

1962 IMO Shortlist, 3

Tags: geometry , 3D geometry , perimeter , vector , cube , IMO , IMO 1962

Consider the cube $ABCDA'B'C'D'$ ($ABCD$ and $A'B'C'D'$ are the upper and lower bases, repsectively, and edges $AA', BB', CC', DD'$ are parallel). The point $X$ moves at a constant speed along the perimeter of the square $ABCD$ in the direction $ABCDA$, and the point $Y$ moves at the same rate along the perimiter of the square $B'C'CB$ in the direction $B'C'CBB'$. Points $X$ and $Y$ begin their motion at the same instant from the starting positions $A$ and $B'$, respectively. Determine and draw the locus of the midpionts of the segments $XY$.

1986 IMO Longlists, 38

Tags: invariant , algorithm , combinatorics , pentagon , game strategy , IMO , IMO 1986

To each vertex of a regular pentagon an integer is assigned, so that the sum of all five numbers is positive. If three consecutive vertices are assigned the numbers $x,y,z$ respectively, and $y<0$, then the following operation is allowed: $x,y,z$ are replaced by $x+y,-y,z+y$ respectively. Such an operation is performed repeatedly as long as at least one of the five numbers is negative. Determine whether this procedure necessarily comes to an end after a finite number of steps.

2008 IMO, 3

Tags: quadratics , number theory , prime divisor , IMO 2008 , IMO , IMO Shortlist

Prove that there are infinitely many positive integers $ n$ such that $ n^{2} \plus{} 1$ has a prime divisor greater than $ 2n \plus{} \sqrt {2n}$. [i]Author: Kestutis Cesnavicius, Lithuania[/i]

1976 IMO Longlists, 8

Tags: geometry , trigonometry , inequalities , area of a triangle , geometric inequality , IMO , imo 1976

In a convex quadrilateral (in the plane) with the area of $32 \text{ cm}^{2}$ the sum of two opposite sides and a diagonal is $16 \text{ cm}$. Determine all the possible values that the other diagonal can have.

1999 IMO, 5

Tags: geometry , homothety , circumcircle , IMO , IMO 1999

Two circles $\Omega_{1}$ and $\Omega_{2}$ touch internally the circle $\Omega$ in M and N and the center of $\Omega_{2}$ is on $\Omega_{1}$. The common chord of the circles $\Omega_{1}$ and $\Omega_{2}$ intersects $\Omega$ in $A$ and $B$. $MA$ and $MB$ intersects $\Omega_{1}$ in $C$ and $D$. Prove that $\Omega_{2}$ is tangent to $CD$.

1987 IMO, 3

Tags: pigeonhole principle , n-variable inequality , IMO , IMO 1987 , algebra , combinatorics

Let $x_1,x_2,\ldots,x_n$ be real numbers satisfying $x_1^2+x_2^2+\ldots+x_n^2=1$. Prove that for every integer $k\ge2$ there are integers $a_1,a_2,\ldots,a_n$, not all zero, such that $|a_i|\le k-1$ for all $i$, and $|a_1x_1+a_2x_2+\ldots+a_nx_n|\le{(k-1)\sqrt n\over k^n-1}$.

1962 IMO, 6

Tags: geometry , circumcircle , trigonometry , circles , isosceles , IMO , IMO 1962

Consider an isosceles triangle. let $R$ be the radius of its circumscribed circle and $r$ be the radius of its inscribed circle. Prove that the distance $d$ between the centers of these two circle is \[ d=\sqrt{R(R-2r)} \]

2007 IMO Shortlist, 1

Tags: IMO , IMO 2007 , Sequence , algebra , IMO Shortlist

Real numbers $ a_{1}$, $ a_{2}$, $ \ldots$, $ a_{n}$ are given. For each $ i$, $ (1 \leq i \leq n )$, define \[ d_{i} \equal{} \max \{ a_{j}\mid 1 \leq j \leq i \} \minus{} \min \{ a_{j}\mid i \leq j \leq n \} \] and let $ d \equal{} \max \{d_{i}\mid 1 \leq i \leq n \}$. (a) Prove that, for any real numbers $ x_{1}\leq x_{2}\leq \cdots \leq x_{n}$, \[ \max \{ |x_{i} \minus{} a_{i}| \mid 1 \leq i \leq n \}\geq \frac {d}{2}. \quad \quad (*) \] (b) Show that there are real numbers $ x_{1}\leq x_{2}\leq \cdots \leq x_{n}$ such that the equality holds in (*). [i]Author: Michael Albert, New Zealand[/i]

1965 IMO, 6

Tags: geometry , diameter , point set , combinatorial geometry , IMO , IMO 1965

In a plane a set of $n\geq 3$ points is given. Each pair of points is connected by a segment. Let $d$ be the length of the longest of these segments. We define a diameter of the set to be any connecting segment of length $d$. Prove that the number of diameters of the given set is at most $n$.

1979 IMO Longlists, 25

Tags: number theory , relatively prime , series summation , Divisibility , prime , IMO , IMO 1979

If $p$ and $q$ are natural numbers so that \[ \frac{p}{q}=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+ \ldots -\frac{1}{1318}+\frac{1}{1319}, \] prove that $p$ is divisible with $1979$.