Found problems: 1782
1998 Balkan MO, 2
Let $n\geq 2$ be an integer, and let $0 < a_1 < a_2 < \cdots < a_{2n+1}$ be real numbers. Prove the inequality
\[ \sqrt[n]{a_1} - \sqrt[n]{a_2} + \sqrt[n]{a_3} - \cdots + \sqrt[n]{a_{2n+1}} < \sqrt[n]{a_1 - a_2 + a_3 - \cdots + a_{2n+1}}. \]
[i]Bogdan Enescu, Romania[/i]
2009 China National Olympiad, 3
Given an integer $ n > 3.$ Prove that there exists a set $ S$ consisting of $ n$ pairwisely distinct positive integers such that for any two different non-empty subset of $ S$:$ A,B, \frac {\sum_{x\in A}x}{|A|}$ and $ \frac {\sum_{x\in B}x}{|B|}$ are two composites which share no common divisors.
2020 Thailand TST, 1
You are given a set of $n$ blocks, each weighing at least $1$; their total weight is $2n$. Prove that for every real number $r$ with $0 \leq r \leq 2n-2$ you can choose a subset of the blocks whose total weight is at least $r$ but at most $r + 2$.
1992 China Team Selection Test, 2
A $(3n + 1) \times (3n + 1)$ table $(n \in \mathbb{N})$ is given. Prove that deleting any one of its squares yields a shape cuttable into pieces of the following form and its rotations: ''L" shape formed by cutting one square from a $2 \times 2$ squares.
PEN A Problems, 88
Find all positive integers $n$ such that $9^{n}-1$ is divisible by $7^n$.
2005 MOP Homework, 3
Squares of an $n \times n$ table ($n \ge 3$) are painted black and white as in a chessboard. A move allows one to choose any $2 \times 2$ square and change all of its squares to the opposite color. Find all such n that there is a finite number of the moves described after which all squares are the same color.
2008 China Team Selection Test, 3
Find all positive integers $ n$ having the following properties:in two-dimensional Cartesian coordinates, there exists a convex $ n$ lattice polygon whose lengths of all sides are odd numbers, and unequal to each other. (where lattice polygon is defined as polygon whose coordinates of all vertices are integers in Cartesian coordinates.)
2004 Bulgaria Team Selection Test, 1
Let $n$ be a positive integer. Find all positive integers $m$ for which there exists a polynomial $f(x) = a_{0} + \cdots + a_{n}x^{n} \in \mathbb{Z}[X]$ ($a_{n} \not= 0$) such that $\gcd(a_{0},a_{1},\cdots,a_{n},m)=1$ and $m|f(k)$ for each $k \in \mathbb{Z}$.
2011 Bogdan Stan, 4
Let be a natural number $ n, $ two $ \text{n-tuplets} $ of real numbers $ a:=\left( a_1,a_2,\ldots, a_n \right) , b:=\left( b_1,b_2,\ldots, b_n \right) , $ and the function $ f:\mathbb{R}\longrightarrow\mathbb{R}, f(x)=\sum_{i=1}^na_i\cos \left( b_ix \right) $. Prove that if the numbers of $ b $ are all positive and pairwise distinct,
[b]a)[/b] then, $ f\ge 0 $ implies that the numbers of $ a $ are all equal.
[b]b)[/b] if the numbers of $ a $ are all nonzero and $ f $ is periodic, then the ratio of any two numbers of $ b $ is rational.
[i]Marin Tolosi[/i]
2013 USAMTS Problems, 4
An infinite sequence of real numbers $a_1,a_2,a_3,\dots$ is called $\emph{spooky}$ if $a_1=1$ and for all integers $n>1$,
\[\begin{array}{c@{\;\,}c@{\;\,}c@{\;\,}c@{\;\,}c@{\;\,}c@{\;\,}c@{\;\,}c@{\;\,}c@{\;\,}c@{\;\,}c@{\;\,}c@{\;\,}c}
na_1&+&(n-1)a_2&+&(n-2)a_3&+&\dots&+&2a_{n-1}&+&a_n&<&0,\\
n^2a_1&+&(n-1)^2a_2&+&(n-2)^2a_3&+&\dots&+&2^2a_{n-1}&+&a_n&>&0.
\end{array}\]Given any spooky sequence $a_1,a_2,a_3,\dots$, prove that
\[2013^3a_1+2012^3a_2+2011^3a_3+\cdots+2^3a_{2012}+a_{2013}<12345.\]
2007 Tournament Of Towns, 6
The audience arranges $n$ coins in a row. The sequence of heads and tails is chosen arbitrarily. The audience also chooses a number between $1$ and $n$ inclusive. Then the assistant turns one of the coins over, and the magician is brought in to examine the resulting sequence. By an agreement with the assistant beforehand, the magician tries to determine the number chosen by the audience.
[list][b](a)[/b] Prove that if this is possible for some $n$, then it is also possible for $2n$.
[b](b)[/b] Determine all $n$ for which this is possible.[/list]
1997 Taiwan National Olympiad, 3
Let $n>2$ be an integer. Suppose that $a_{1},a_{2},...,a_{n}$ are real numbers such that $k_{i}=\frac{a_{i-1}+a_{i+1}}{a_{i}}$ is a positive integer for all $i$(Here $a_{0}=a_{n},a_{n+1}=a_{1}$). Prove that $2n\leq a_{1}+a_{2}+...+a_{n}\leq 3n$.
2022 USAMTS Problems, 3
Prove that there is a unique $1000$-digit number $N$ in base $2022$ with the following properties:
[list=1]
[*] All of the digits of $N$ (in base $2022$) are $1$’s or $2$’s, and [/*]
[*] $N$ is a multiple of the base-$10$ number $2^{1000}$. [/*]
[/list]
(Note that you must prove both that such a number exists and that there is not more than one such number. You do not have to write down the number! In fact, please don’t!)
2004 USA Team Selection Test, 6
Define the function $f: \mathbb N \cup \{0\} \to \mathbb{Q}$ as follows: $f(0) = 0$ and \[ f(3n+k) = -\frac{3f(n)}{2} + k , \] for $k = 0, 1, 2$. Show that $f$ is one-to-one and determine the range of $f$.
2009 IMO Shortlist, 7
Let $ a_1, a_2, \ldots , a_n$ be distinct positive integers and let $ M$ be a set of $ n \minus{} 1$ positive integers not containing $ s \equal{} a_1 \plus{} a_2 \plus{} \ldots \plus{} a_n.$ A grasshopper is to jump along the real axis, starting at the point $ 0$ and making $ n$ jumps to the right with lengths $ a_1, a_2, \ldots , a_n$ in some order. Prove that the order can be chosen in such a way that the grasshopper never lands on any point in $ M.$
[i]Proposed by Dmitry Khramtsov, Russia[/i]
2013 NIMO Problems, 6
A strictly increasing sequence $\{x_i\}_{i=1}^{\infty}$ of positive integers is said to be [i]large[/i] if, for every real number $L$, there exists an integer $n$ such that $\frac{1}{x_1} + \frac{1}{x_2} + \cdots + \frac{1}{x_n} > L$. Do there exist large sequences $\{a_i\}_{i=1}^\infty$ and $\{b_i\}_{i=1}^{\infty}$ such that the sequence $\{a_i+b_i\}_{i=1}^{\infty}$ is not large?
[i]Proposed by Lewis Chen[/i]
2003 USA Team Selection Test, 4
Let $\mathbb{N}$ denote the set of positive integers. Find all functions $f: \mathbb{N} \to \mathbb{N}$ such that \[ f(m+n)f(m-n) = f(m^2) \] for $m,n \in \mathbb{N}$.
2006 IMC, 1
Let $V$ be a convex polygon.
(a) Show that if $V$ has $3k$ vertices, then $V$ can be triangulated such that each vertex is in an odd number of triangles.
(b) Show that if the number of vertices is not divisible with 3, then $V$ can be triangulated such that exactly 2 vertices have an even number of triangles.
2010 Pan African, 2
How many ways are there to line up $19$ girls (all of different heights) in a row so that no girl has a shorter girl both in front of and behind her?
2008 Balkan MO Shortlist, A2
Is there a sequence $ a_1,a_2,\ldots$ of positive reals satisfying simoultaneously the following inequalities for all positive integers $ n$:
a) $ a_1\plus{}a_2\plus{}\ldots\plus{}a_n\le n^2$
b) $ \frac1{a_1}\plus{}\frac1{a_2}\plus{}\ldots\plus{}\frac1{a_n}\le2008$?
2014 Canada National Olympiad, 1
Let $a_1,a_2,\dots,a_n$ be positive real numbers whose product is $1$. Show that the sum \[\textstyle\frac{a_1}{1+a_1}+\frac{a_2}{(1+a_1)(1+a_2)}+\frac{a_3}{(1+a_1)(1+a_2)(1+a_3)}+\cdots+\frac{a_n}{(1+a_1)(1+a_2)\cdots(1+a_n)}\] is greater than or equal to $\frac{2^n-1}{2^n}$.
2003 IberoAmerican, 3
Pablo copied from the blackboard the problem:
[list]Consider all the sequences of $2004$ real numbers $(x_0,x_1,x_2,\dots, x_{2003})$ such that: $x_0=1, 0\le x_1\le 2x_0,0\le x_2\le 2x_1\ldots ,0\le x_{2003}\le 2x_{2002}$. From all these sequences, determine the sequence which minimizes $S=\cdots$[/list]
As Pablo was copying the expression, it was erased from the board. The only thing that he could remember was that $S$ was of the form $S=\pm x_1\pm x_2\pm\cdots\pm x_{2002}+x_{2003}$. Show that, even when Pablo does not have the complete statement, he can determine the solution of the problem.
2005 QEDMO 1st, 3 (C2)
At a turnament between $n$ persons, everyone playes exactly one time against everyone else, and at one game there is everytime a winner and a looser.
Prove that one can arrange the participants in a chain$P_1 \to P_2 \to ... \to P_n$ such that the $i$-th person has won against the $(i+1)$-th person.
2011 Math Prize for Girls Olympiad, 1
Let $A_0$, $A_1$, $A_2$, ..., $A_n$ be nonnegative numbers such that
\[
A_0 \le A_1 \le A_2 \le \dots \le A_n.
\]
Prove that
\[
\left| \sum_{i = 0}^{\lfloor n/2 \rfloor} A_{2i}
- \frac{1}{2} \sum_{i = 0}^n A_i \right| \le \frac{A_n}{2} \, .
\]
(Note: $\lfloor x \rfloor$ means the greatest integer that is less than or equal to $x$.)
1995 Moldova Team Selection Test, 4
Find all functions $f:\mathbb{Z}\rightarrow \mathbb{Z}$ satisfying the following:
$i)$ $f(1)=1$;
$ii)$ $f(m+n)(f(m)-f(n))=f(m-n)(f(m)+f(n))$ for all $m,n \in \mathbb{Z}$.