Found problems: 1782
2015 Belarus Team Selection Test, 1
Let $n \ge 2$ be an integer, and let $A_n$ be the set \[A_n = \{2^n - 2^k\mid k \in \mathbb{Z},\, 0 \le k < n\}.\] Determine the largest positive integer that cannot be written as the sum of one or more (not necessarily distinct) elements of $A_n$ .
[i]Proposed by Serbia[/i]
2014 Vietnam National Olympiad, 1
Let $({{x}_{n}}),({{y}_{n}})$ be two positive sequences defined by ${{x}_{1}}=1,{{y}_{1}}=\sqrt{3}$ and
\[ \begin{cases} {{x}_{n+1}}{{y}_{n+1}}-{{x}_{n}}=0 \\ x_{n+1}^{2}+{{y}_{n}}=2 \end{cases} \] for all $n=1,2,3,\ldots$.
Prove that they are converges and find their limits.
2004 Junior Balkan MO, 4
Consider a convex polygon having $n$ vertices, $n\geq 4$. We arbitrarily decompose the polygon into triangles having all the vertices among the vertices of the polygon, such that no two of the triangles have interior points in common. We paint in black the triangles that have two sides that are also sides of the polygon, in red if only one side of the triangle is also a side of the polygon and in white those triangles that have no sides that are sides of the polygon.
Prove that there are two more black triangles that white ones.
2007 Germany Team Selection Test, 2
Let $ n, k \in \mathbb{N}$ with $ 1 \leq k \leq \frac {n}{2} - 1.$ There are $ n$ points given on a circle. Arbitrarily we select $ nk + 1$ chords among the points on the circle. Prove that of these chords there are at least $ k + 1$ chords which pairwise do not have a point in common.
2020 Germany Team Selection Test, 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 $r + 2$.
2015 Taiwan TST Round 3, 1
Let $\mathbb{Q}^+$ be the set of all positive rational numbers. Find all functions $f:\mathbb{Q}^+\rightarrow \mathbb{Q}^+$ satisfying $f(1)=1$ and
\[ f(x+n)=f(x)+nf(\frac{1}{x}) \forall n\in\mathbb{N},x\in\mathbb{Q}^+\]
2024 Indonesia TST, 3
Let $n$ be a positive integer and let $a_1, a_2, \ldots, a_n$ be positive reals. Show that $$\sum_{i=1}^{n} \frac{1}{2^i}(\frac{2}{1+a_i})^{2^i} \geq \frac{2}{1+a_1a_2\ldots a_n}-\frac{1}{2^n}.$$
2011 China Western Mathematical Olympiad, 3
Let $n \geq 2$ be a given integer
$a)$ Prove that one can arrange all the subsets of the set $\{1,2... ,n\}$ as a sequence of subsets $A_{1}, A_{2},\cdots , A_{2^{n}}$, such that $|A_{i+1}| = |A_{i}| + 1$ or $|A_{i}| - 1$ where $i = 1,2,3,\cdots , 2^{n}$ and $A_{2^{n} + 1} = A_{1}$
$b)$ Determine all possible values of the sum $\sum \limits_{i = 1}^{2^n} (-1)^{i}S(A_{i})$ where $S(A_{i})$ denotes the sum of all elements in $A_{i}$ and $S(\emptyset) = 0$, for any subset sequence $A_{1},A_{2},\cdots ,A_{2^n}$ satisfying the condition in $a)$
2012 Online Math Open Problems, 44
Given a set of points in space, a [i]jump[/i] consists of taking two points, $P$ and $Q,$ and replacing $P$ with the reflection of $P$ over $Q$. Find the smallest number $n$ such that for any set of $n$ lattice points in $10$-dimensional-space, it is possible to perform a finite number of jumps so that some two points coincide.
[i]Author: Anderson Wang[/i]
2005 IberoAmerican Olympiad For University Students, 1
Let $P(x,y)=(x^2y^3,x^3y^5)$, $P^1=P$ and $P^{n+1}=P\circ P^n$. Also, let $p_n(x)$ be the first coordinate of $P^n(x,x)$, and $f(n)$ be the degree of $p_n(x)$. Find
\[\lim_{n\to\infty}f(n)^{1/n}\]
2012 IMO Shortlist, A6
Let $f: \mathbb{N} \rightarrow \mathbb{N}$ be a function, and let $f^m$ be $f$ applied $m$ times. Suppose that for every $n \in \mathbb{N}$ there exists a $k \in \mathbb{N}$ such that $f^{2k}(n)=n+k$, and let $k_n$ be the smallest such $k$. Prove that the sequence $k_1,k_2,\ldots $ is unbounded.
[i]Proposed by Palmer Mebane, United States[/i]
2007 Romania Team Selection Test, 1
In a circle with center $O$ is inscribed a polygon, which is triangulated. Show that the sum of the squares of the distances from $O$ to the incenters of the formed triangles is independent of the triangulation.
2014 USA Team Selection Test, 3
Let $n$ be an even positive integer, and let $G$ be an $n$-vertex graph with exactly $\tfrac{n^2}{4}$ edges, where there are no loops or multiple edges (each unordered pair of distinct vertices is joined by either 0 or 1 edge). An unordered pair of distinct vertices $\{x,y\}$ is said to be [i]amicable[/i] if they have a common neighbor (there is a vertex $z$ such that $xz$ and $yz$ are both edges). Prove that $G$ has at least $2\textstyle\binom{n/2}{2}$ pairs of vertices which are amicable.
[i]Zoltán Füredi (suggested by Po-Shen Loh)[/i]
1999 Hungary-Israel Binational, 3
Find all functions $ f:\mathbb{Q}\to\mathbb{R}$ that satisfy $ f(x\plus{}y)\equal{}f(x)f(y)\minus{}f(xy)\plus{}1$ for every $x,y\in\mathbb{Q}$.
PEN A Problems, 70
Suppose that $m=nq$, where $n$ and $q$ are positive integers. Prove that the sum of binomial coefficients \[\sum_{k=0}^{n-1}{ \gcd(n, k)q \choose \gcd(n, k)}\] is divisible by $m$.
2013 Kosovo National Mathematical Olympiad, 4
Let be $n$ positive integer than calculate:
$1\cdot 1!+2\cdot2!+...+n\cdot n!$
2004 Harvard-MIT Mathematics Tournament, 4
Let $f(x)=\cos(\cos(\cos(\cos(\cos(\cos(\cos(\cos(x))))))))$, and suppose that the number $a$ satisfies the equation $a=\cos a$. Express $f'(a)$ as a polynomial in $a$.
2014 Contests, 1
Let $({{x}_{n}}),({{y}_{n}})$ be two positive sequences defined by ${{x}_{1}}=1,{{y}_{1}}=\sqrt{3}$ and
\[ \begin{cases} {{x}_{n+1}}{{y}_{n+1}}-{{x}_{n}}=0 \\ x_{n+1}^{2}+{{y}_{n}}=2 \end{cases} \] for all $n=1,2,3,\ldots$.
Prove that they are converges and find their limits.
2006 Hong kong National Olympiad, 4
Let $(a_n)_{n\ge 1}$ be a sequence of positive numbers. If there is a constant $M > 0$ such that $a_2^2 + a_2^2 +\ldots + a_n^2 < Ma_{n+1}^2$ for all $n$, then prove that there is a constant $M ' > 0$ such that $a_1 + a_2 +\ldots + a_n < M ' a_{n+1}$ .
2006 Italy TST, 3
Find all functions $f : \mathbb{Z} \rightarrow \mathbb{Z}$ such that for all integers $m,n$,
\[f(m - n + f(n)) = f(m) + f(n).\]
2014 All-Russian Olympiad, 3
There are $n$ cells with indices from $1$ to $n$. Originally, in each cell, there is a card with the corresponding index on it. Vasya shifts the card such that in the $i$-th cell is now a card with the number $a_i$. Petya can swap any two cards with the numbers $x$ and $y$, but he must pay $2|x-y|$ coins. Show that Petya can return all the cards to their original position, not paying more than $|a_1-1|+|a_2-2|+\ldots +|a_n-n|$ coins.
2012 IMC, 5
Let $c \ge 1$ be a real number. Let $G$ be an Abelian group and let $A \subset G$ be a finite set satisfying $|A+A| \le c|A|$, where $X+Y:= \{x+y| x \in X, y \in Y\}$ and $|Z|$ denotes the cardinality of $Z$. Prove that
\[|\underbrace{A+A+\dots+A}_k| \le c^k |A|\]
for every positive integer $k$.
[i]Proposed by Przemyslaw Mazur, Jagiellonian University.[/i]
2019 239 Open Mathematical Olympiad, 5
We call an ordered set of distinct natural numbers good if for any two numbers in it, the larger one is divided by the smaller one. Prove that the number $(n + 1)! – 1$ can be represented as $x_1 + 2x_2 + \ldots + nx_n$, where $\{ x_1, x_2, \ldots , x_n \}$ is a good set, by at least $n!$ ways.
2000 USAMO, 4
Find the smallest positive integer $n$ such that if $n$ squares of a $1000 \times 1000$ chessboard are colored, then there will exist three colored squares whose centers form a right triangle with sides parallel to the edges of the board.
2000 Junior Balkan MO, 2
Find all positive integers $n\geq 1$ such that $n^2+3^n$ is the square of an integer.
[i]Bulgaria[/i]