Olympiad problems

2009 India IMO Training Camp, 6

Tags: function , combinatorics

Prove The Following identity: $ \sum_{j \equal{} 0}^n \left ({3n \plus{} 2 \minus{} j \choose j}2^j \minus{} {3n \plus{} 1 \minus{} j \choose j \minus{} 1}2^{j \minus{} 1}\right ) \equal{} 2^{3n}$. The Second term on left hand side is to be regarded zero for j=0.

2009 Bosnia And Herzegovina - Regional Olympiad, 3

Tags: board , minimal , combinatorics

There are $n$ positive integers on the board. We can add only positive integers $c=\frac{a+b}{a-b}$, where $a$ and $b$ are numbers already writted on the board. $a)$ Find minimal value of $n$, such that with adding numbers with described method, we can get any positive integer number written on the board $b)$ For such $n$, find numbers written on the board at the beginning

2015 AMC 8, 4

Tags:

The Centerville Middle School chess team consists of two boys and three girls. A photographer wants to take a picture of the team to appear in the local newspaper. She decides to have them sit in a row with a boy at each end and the three girls in the middle. How many such arrangements are possible? $\textbf{(A) }2\qquad\textbf{(B) }4\qquad\textbf{(C) }5\qquad\textbf{(D) }6\qquad \textbf{(E) }12$

2007 Purple Comet Problems, 6

Tags:

Find the sum of all the positive integers that are divisors of either $96$ or $180$.

2024 Cono Sur Olympiad, 3

Tags: number theory

Find all positive integers $n$ such that $3^n - 2^n - 1$ is a perfect square.

2013 Turkey MO (2nd round), 3

Tags: rectangle , geometry

Let $n$ be a positive integer and $P_1, P_2, \ldots, P_n$ be different points on the plane such that distances between them are all integers. Furthermore, we know that the distances $P_iP_1, P_iP_2, \ldots, P_iP_n$ forms the same sequence for all $i=1,2, \ldots, n$ when these numbers are arranged in a non-decreasing order. Find all possible values of $n$.

1972 Miklós Schweitzer, 4

Tags: superior algebra , abstract algebra , group theory , search

Let $ G$ be a solvable torsion group in which every Abelian subgroup is finitely generated. Prove that $ G$ is finite. [i]J. Pelikan[/i]

2007 Harvard-MIT Mathematics Tournament, 10

Tags: calculus , integration

Compute \[\int_0^\infty \dfrac{e^{-x}\sin(x)}{x}dx\]

2022 JBMO Shortlist, C6

Tags: combinatorics , board , shortlist

Let $n \ge 2$ be an integer. In each cell of a $4n \times 4n$ table we write the sum of the cell row index and the cell column index. Initially, no cell is colored. A move consists of choosing two cells which are not colored and coloring one of them in red and one of them in blue. Show that, however Alex perfors $n^2$ moves, Jane can afterwards perform a number of moves (eventually none) after which the sum of the numbers written in the red cells is the same as the sum of the numbers written in the blue ones.

2014 PUMaC Geometry A, 3

Tags: geometry , circumcircle , trigonometry , perpendicular bisector

Let $O$ be the circumcenter of triangle $ABC$ with circumradius $15$. Let $G$ be the centroid of $ABC$ and let $M$ be the midpoint of $BC$. If $BC=18$ and $\angle MOA=150^\circ$, find the area of $OMG$.

2000 Bundeswettbewerb Mathematik, 2

Tags: number theory

Prove that for every integer $n \geq 2$ there exist $n$ different positive integers such that for any two of these integers $a$ and $b$ their sum $a+b$ is divisible by their difference $a - b.$

2005 Taiwan TST Round 1, 2

Tags: 3d geometry , tetrahedron , geometry

Show that for any tetrahedron, the condition that opposite edges have the same length is equivalent to the condition that the segment drawn between the midpoints of any two opposite edges is perpendicular to the two edges.

2017 Peru Iberoamerican Team Selection Test, P6

Tags: number theory

For each positive integer $k$, let $S(k)$ be the sum of the digits of $k$ in the decimal system. Prove that there exists a positive integer $k$, which does not have the digit $9$ in its decimal representation, such that: $$S(2^{24^{2017}}k)=S(k)$$

2017 Miklós Schweitzer, 9

Tags: normed vector space , functional analysis , analysis , advanced fields

Let $N$ be a normed linear space with a dense linear subspace $M$. Prove that if $L_1,\ldots,L_m$ are continuous linear functionals on $N$, then for all $x\in N$ there exists a sequence $(y_n)$ in $M$ converging to $x$ satisfying $L_j(y_n)=L_j(x)$ for all $j=1,\ldots,m$ and $n\in \mathbb{N}$.

2024 Argentina Cono Sur TST, 4

Tags: algebra

Determine the least possible value of $\dfrac{(x^2+1)(4y^2+1)(9z^2+1)}{6xyz}$ if $x$, $y$ and $z$ are positive real numbers.

2000 Portugal MO, 1

Tags: algebra , combinatorics

Consider the following table where initially all squares contain zeros: $ \begin{tabular}{ | l | c | r| } \hline 0 & 0 & 0 \\ \hline 0 & 0 & 0 \\ \hline 0 & 0 & 0 \\ \hline \end{tabular} $ To change the table, the following operation is allowed: a $2 \times 2$ square formed by adjacent squares is chosen, and a unit is added to all its numbers. Complete the following table, knowing that it was obtained by a sequence of permitted operations $ \begin{tabular}{ | l | c | r| } \hline 14 & & \\ \hline 19 & 36 & \\ \hline & 16 & \\ \hline \end{tabular} $

1984 Swedish Mathematical Competition, 6

Tags: sum , number theory , combinatorics

Assume $a_1,a_2,...,a_{14}$ are positive integers such that $\sum_{i=1}^{14}3^{a_i} = 6558$. Prove that the numbers $a_1,a_2,...,a_{14}$ consist of the numbers $1,...,7$, each taken twice.

1973 Kurschak Competition, 3

Tags: combinatorics , combinatorial geometry

$n > 4$ planes are in general position (so every $3$ planes have just one common point, and no point belongs to more than $3$ planes). Show that there are at least $\frac{2n-3}{ 4}$ tetrahedra among the regions formed by the planes.

2001 ITAMO, 1

Tags: geometry

A hexagon has all its angles equal, and the lengths of four consecutive sides are $5$, $3$, $6$ and $7$, respectively. Find the lengths of the remaining two edges.

2009 Germany Team Selection Test, 1

Tags: prime number , number theory

For which $ n \geq 2, n \in \mathbb{N}$ are there positive integers $ A_1, A_2, \ldots, A_n$ which are not the same pairwise and have the property that the product $ \prod^n_{i \equal{} 1} (A_i \plus{} k)$ is a power for each natural number $ k.$

2024 USAJMO, 3

Tags: awesomeness_in_a_bun

Let $a(n)$ be the sequence defined by $a(1)=2$ and $a(n+1)=(a(n))^{n+1}-1$ for each integer $n\geq 1$. Suppose that $p>2$ is a prime and $k$ is a positive integer. Prove that some term of the sequence $a(n)$ is divisible by $p^k$. [i]Proposed by John Berman[/i]

2016 Middle European Mathematical Olympiad, 5

Tags: circumcircle , law of cosines , geometry

Let $ABC$ be an acute triangle for which $AB \neq AC$, and let $O$ be its circumcenter. Line $AO$ meets the circumcircle of $ABC$ again in $D$, and the line $BC$ in $E$. The circumcircle of $CDE$ meets the line $CA$ again in $P$. The lines $PE$ and $AB$ intersect in $Q$. Line passing through $O$ parallel to the line $PE$ intersects the $A$-altitude of $ABC$ in $F$. Prove that $FP = FQ$.

2011 NIMO Problems, 11

Tags: number theory , least common multiple , greatest common divisor

How many ordered pairs of positive integers $(m, n)$ satisfy the system \begin{align*} \gcd (m^3, n^2) & = 2^2 \cdot 3^2, \\ \text{LCM} [m^2, n^3] & = 2^4 \cdot 3^4 \cdot 5^6, \end{align*} where $\gcd(a, b)$ and $\text{LCM}[a, b]$ denote the greatest common divisor and least common multiple of $a$ and $b$, respectively?

1990 Romania Team Selection Test, 3

Tags: polynomial , algebra , functional , functional equation

Find all polynomials $P(x)$ such that $2P(2x^2 -1) = P(x)^2 -1$ for all $x$.

2004 Purple Comet Problems, 3

Tags: algebra , polynomial

How many real numbers are roots of the polynomial \[x^9 - 37x^8 - 2x^7 + 74x^6 + x^4 - 37x^3 - 2x^2 + 74x?\]