Olympiad problems

1976 Swedish Mathematical Competition, 1

Tags: combinatorics

In a tournament every team plays every other team just once. Each game is won by one of the teams (there are no draws). Each team loses at least once. Show that there must be three teams $A$, $B$, $C$ such that $A$ beat $B$, $B$ beat $C$ and $C$ beat $A$.

1999 IMO, 5

Tags: geometry , homothety , circumcircle

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 Longlists, 19

Tags: symmetry , combinatorics of words , counting , digit , recurrence relation , combinatorics

How many words with $n$ digits can be formed from the alphabet $\{0, 1, 2, 3, 4\}$, if neighboring digits must differ by exactly one? [i]Proposed by Germany, FR.[/i]

2021 Science ON all problems, 2

Tags: linear algebra

Consider $A,B\in\mathcal{M}_n(\mathbb{C})$ for which there exist $p,q\in\mathbb{C}$ such that $pAB-qBA=I_n$. Prove that either $(AB-BA)^n=O_n$ or the fraction $\frac{p}{q}$ is well-defined ($q \neq 0$) and it is a root of unity. [i](Sergiu Novac)[/i]

2006 AMC 12/AHSME, 6

Tags:

Francesca uses 100 grams of lemon juice, 100 grams of sugar, and 400 grams of water to make lemonade. There are 25 calories in 100 grams of lemon juice and 386 calories in 100 grams of sugar. Water contains no calories. How many calories are in 200 grams of her lemonade? $ \textbf{(A) } 129 \qquad \textbf{(B) } 137 \qquad \textbf{(C) } 174 \qquad \textbf{(D) } 223 \qquad \textbf{(E) } 411$

2007 Argentina National Olympiad, 5

Tags: number theory , sum of digits

We will say that a positive integer is [i]lucky [/i ]if the sum of its digits is divisible by $31$. What is the maximum possible difference between two consecutive [i]lucky [/i ] numbers?

2000 JBMO ShortLists, 7

Tags: number theory

Find all the pairs of positive integers $(m,n)$ such that the numbers $A=n^2+2mn+3m^2+3n$, $B=2n^2+3mn+m^2$, $C=3n^2+mn+2m^2$ are consecutive in some order.

2016 Greece Junior Math Olympiad, 2

Tags: algebra

Given is that $x, y, z$ are real numbers, different from 0, $x$ and $z$ are different, such that $(x+y) ^2+(2-xy)=9$ and $(y+z) ^2-(3+yz)=4$ Find the value of $A=(x/y+y^2/x^2+z^3/x^2y)(y/z+z^2/y^2+x^3/y^2z)(z/x+x^2/z^2+y^3/z^2x)=?$

2007 Thailand Mathematical Olympiad, 16

Tags: number theory , divisor

What is the smallest positive integer with $24$ positive divisors?

2022 Korea -Final Round, P4

Tags: geometry , concyclic , concentric

Let $ABC$ be a scalene triangle with incenter $I$ and let $AI$ meet the circumcircle of triangle $ABC$ again at $M$. The incircle $\omega$ of triangle $ABC$ is tangent to sides $AB, AC$ at $D, E$, respectively. Let $O$ be the circumcenter of triangle $BDE$ and let $L$ be the intersection of $\omega$ and the altitude from $A$ to $BC$ so that $A$ and $L$ lie on the same side with respect to $DE$. Denote by $\Omega$ a circle centered at $O$ and passing through $L$, and let $AL$ meet $\Omega$ again at $N$. Prove that the lines $LD$ and $MB$ meet on the circumcircle of triangle $LNE$.

2010 LMT, 29

Tags:

Let $S$ be the set of integers that represent the number of intersections of some four distinct lines in the plane. List the elements of $S$ in ascending order.

2020 Purple Comet Problems, 14

Tags: combinatorics

Six different small books and three different large books sit on a shelf. Three children may each take either two small books or one large book. Find the number of ways the three children can select their books.

1979 IMO, 1

Tags: number theory , relatively prime , series summation , divisibility , prime

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 $1979$.

2004 Silk Road, 2

Tags: number theory

find all primes $p$, for which exist natural numbers, such that $p=m^2+n^2$ and $p|(m^3+n^3-4)$.

1991 Arnold's Trivium, 73

Tags:

Find (to the first order in $\epsilon$) the influence of the imperfection of an almost spherical capacitor $R = 1 + \epsilon f(\varphi, \theta)$ on its capacity.

2005 AMC 8, 15

Tags: geometry , perimeter

How many different isosceles triangles have integer side lengths and perimeter 23? $ \textbf{(A)}\ 2\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 9\qquad\textbf{(E)}\ 11$

2021 Flanders Math Olympiad, 4

Tags: inequalities , functional , algebra , functional equation

(a) Prove that for every $x \in R$ holds that $$-1 \le \frac{x}{x^2 + x + 1} \le \frac 13$$ (b) Determine all functions $f : R \to R$ for which for every $x \in R$ holds that $$f \left( \frac{x}{x^2 + x + 1} \right) = \frac{x^2}{x^4 + x^2 + 1}$$

2005 Romania Team Selection Test, 1

Tags: geometry , rectangle , modular arithmetic , combinatorics

On a $2004 \times 2004$ chess table there are 2004 queens such that no two are attacking each other\footnote[1]{two queens attack each other if they lie on the same row, column or direction parallel with on of the main diagonals of the table}. Prove that there exist two queens such that in the rectangle in which the center of the squares on which the queens lie are two opposite corners, has a semiperimeter of 2004.

2012 Iran MO (3rd Round), 4

Tags: polynomial , algebra

Suppose $f(z)=z^n+a_1z^{n-1}+...+a_n$ for which $a_1,a_2,...,a_n\in \mathbb C$. Prove that the following polynomial has only one positive real root like $\alpha$ \[x^n+\Re(a_1)x^{n-1}-|a_2|x^{n-2}-...-|a_n|\] and the following polynomial has only one positive real root like $\beta$ \[x^n-\Re(a_1)x^{n-1}-|a_2|x^{n-2}-...-|a_n|.\] And roots of the polynomial $f(z)$ satisfy $-\beta \le \Re(z) \le \alpha$.

2008 Thailand Mathematical Olympiad, 4

Tags: number theory , binomial , divisible , divide

Let $n$ be a positive integer. Show that $${2n+1 \choose 1} -{2n+1 \choose 3}2008 + {2n+1 \choose 5}2008^2- ...+(-1)^{n}{2n+1 \choose 2n+1}2008^n $$ is not divisible by $19$.

1999 IMC, 5

Tags: combinatorics

Let $S$ be the set of words made from the letters $a,b$ and $c$. The equivalence relation $\sim$ on $S$ satisfies \[uu \sim u \] \[u \sim v \Rightarrow uw \sim vw \; \text{and} \; wu \sim wv\] for all words $u, v$ and $w$. Prove that every word in $S$ is equivalent to a word of length $\leq 8$.

2023 IRN-SGP-TWN Friendly Math Competition, 4

Tags: graph theory , combinatorics

On a connected graph $G$, one may perform the following operations: [list] [*]choose a vertice $v$, and add a vertice $v'$ such that $v'$ is connected to $v$ and all of its neighbours [*] choose a vertice $v$ with odd degree and delete it [/list] Show that for any connected graph $G$, we may perform a finite number of operations such that the resulting graph is a clique. Proposed by [i]idonthaveanaopsaccount[/i]

1951 AMC 12/AHSME, 6

Tags: geometry , 3d geometry

The bottom, side, and front areas of a rectangular box are known. The product of these areas is equal to: $ \textbf{(A)}\ \text{the volume of the box} \qquad\textbf{(B)}\ \text{the square root of the volume} \qquad\textbf{(C)}\ \text{twice the volume}$ $ \textbf{(D)}\ \text{the square of the volume} \qquad\textbf{(E)}\ \text{the cube of the volume}$

2013 Lusophon Mathematical Olympiad, 6

Tags: geometry , geometric transformation , homothety

Consider a triangle $ABC$. Let $S$ be a circumference in the interior of the triangle that is tangent to the sides $BC$, $CA$, $AB$ at the points $D$, $E$, $F$ respectively. In the exterior of the triangle we draw three circumferences $S_A$, $S_B$, $S_C$. The circumference $S_A$ is tangent to $BC$ at $L$ and to the prolongation of the lines $AB$, $AC$ at the points $M$, $N$ respectively. The circumference $S_B$ is tangent to $AC$ at $E$ and to the prolongation of the line $BC$ at $P$. The circumference $S_C$ is tangent to $AB$ at $F$ and to the prolongation of the line $BC$ at $Q$. Show that the lines $EP$, $FQ$ and $AL$ meet at a point of the circumference $S$.

PEN K Problems, 13

Tags: functional equation , function

Find all functions $f: \mathbb{Z}\to \mathbb{Z}$ such that for all $m\in \mathbb{Z}$: \[f(f(m))=m+1.\]