Found problems: 1782
For a set $S$ we denote its cardinality by $|S|$. Let $e_1,e_2,\ldots,e_k$ be non-negative integers. Let $A_k$ (respectively $B_k$) be the set of all $k$-tuples $(f_1,f_2,\ldots,f_k)$ of integers such that $0\leq f_i\leq e_i$ for all $i$ and $\sum_{i=1}^k f_i$ is even (respectively odd). Show that $|A_k|-|B_k|=0 \textrm{ or } 1$.
Let $X$ be a non empty subset of $\mathbb{N} = \{1,2,\ldots \}$. Suppose that for all $x \in X$, $4x \in X$ and $\lfloor \sqrt{x} \rfloor \in X$. Prove that $X=\mathbb{N}$.
Find all functions $f : \mathbb{Q} \to \mathbb{R}$ such that $f(x)f(y)f(x+y) = f(xy)(f(x) + f(y))$ for all $x,y\in\mathbb{Q}$.
[i]Sammy Luo and Alex Zhu.[/i]
There is a $2012\times 2012$ grid with rows numbered $1,2,\dots 2012$ and columns numbered $1,2,\dots, 2012$, and we place some rectangular napkins on it such that the sides of the napkins all lie on grid lines. Each napkin has a positive integer thickness. (in micrometers!)
(a) Show that there exist $2012^2$ unique integers $a_{i,j}$ where $i,j \in [1,2012]$ such that for all $x,y\in [1,2012]$, the sum \[ \sum _{i=1}^{x} \sum_{j=1}^{y} a_{i,j} \] is equal to the sum of the thicknesses of all the napkins that cover the grid square in row $x$ and column $y$.
(b) Show that if we use at most $500,000$ napkins, at least half of the $a_{i,j}$ will be $0$.
[i]Proposed by Ray Li[/i]
Let $M$ be the set of the integer numbers from the range $[-n, n]$. The subset $P$ of $M$ is called a [i]base subset[/i] if every number from $M$ can be expressed as a sum of some different numbers from $P$. Find the smallest natural number $k$ such that every $k$ numbers that belongs to $M$ form a base subset.
In the number arrangement
\[\begin{array}{ccccc}
\texttt{1}&&&&\\
\texttt{2}&\texttt{3}&&&\\
\texttt{4}&\texttt{5}&\texttt{6}&&\\
\texttt{7}&\texttt{8}&\texttt{9}&\texttt{10}&\\
\texttt{11}&\texttt{12}&\texttt{13}&\texttt{14}&\texttt{15}\\
\vdots&&&&
\end{array}\]
what is the number that will appear directly below the number $2010$?
A convex $n$-gon $A_1A_2\dots A_n$ $(n>3)$ is divided into triangles by non-intersecting diagonals.
For every vertex the number of sides issuing from it is even, except for the vertices
$A_{i_1},A_{i_2},\dots,A_{i_k}$, where $1\leq i_1<\dots<i_k\leq n$. Prove that $k$ is even and
\[n\equiv i_1-i_2+\dots+i_{k-1}-i_k\pmod3\]
if $k>0$ and
\[n\equiv0\pmod3\mbox{ for }k=0.\]
Note that this leads to generalization of one recent Tournament of towns problem about triangulating of square.
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]
Given a sequence $\{a_n\}$ of real numbers such that $|a_{k+m} - a_k - a_m| \leq 1$ for all positive integers $k$ and $m$, prove that, for all positive integers $p$ and $q$, \[|\frac{a_p}{p} - \frac{a_q}{q}| < \frac{1}{p} + \frac{1}{q}.\]
Let $C_1, C_2$ be circles of radius $1/2$ tangent to each other and both tangent internally to a circle $C$ of radius $1$. The circles $C_1$ and $C_2$ are the first two terms of an infinite sequence of distinct circles $C_n$ defined as follows:
$C_{n+2}$ is tangent externally to $C_n$ and $C_{n+1}$ and internally to $C$. Show that the radius of each $C_n$ is the reciprocal of an integer.
[b]Problem 1. [/b]In the cells of square table are written the numbers $1$, $0$ or $-1$ so that in every line there is exactly one $1$, amd exactly one $-1$. Each turn we change the places of two columns or two rows. Is it possible, from any such table, after finite number of turns to obtain its opposite table (two tables are opposite if the sum of the numbers written in any two corresponding squares is zero)?
[i] Emil Kolev[/i]
Find all pairs of polynomials $p(x)$ and $q(x)$ with real coefficients for which
\[p(x)q(x+1)-p(x+1)q(x)=1.\]
Prove that, for any integer $a_{1}>1$, there exist an increasing sequence of positive integers $a_{1}, a_{2}, a_{3}, \cdots$ such that \[a_{1}+a_{2}+\cdots+a_{n}\; \vert \; a_{1}^{2}+a_{2}^{2}+\cdots+a_{n}^{2}\] for all $n \in \mathbb{N}$.
A tournament on $2k$ vertices contains no $7$-cycles. Show that its vertices can be partitioned into two sets, each with size $k$, such that the edges between vertices of the same set do not determine any $3$-cycles.
[i]Calvin Deng.[/i]
Prove that the number $\underbrace{111\ldots 11}_{1997}\underbrace{22\ldots 22}_{1998}5$ (which has 1997 of 1-s and 1998 of 2-s) is a perfect square.
The $100$ vertices of a prism, whose base is a $50$-gon, are labeled with numbers $1, 2, 3, \ldots, 100$ in any order. Prove that there are two vertices, which are connected by an edge of the prism, with labels differing by not more than $48$.
Note: In all the triangles the three vertices do not lie on a straight line.
Let $f: (0,+\infty)\rightarrow (0,+\infty)$ be a function satisfying the following condition: for arbitrary positive real numbers $x$ and $y$, we have $f(xy)\le f(x)f(y)$. Show that for arbitrary positive real number $x$ and natural number $n$, inequality $f(x^n)\le f(x)f(x^2)^{\dfrac{1}{2}}\dots f(x^n)^{\dfrac{1}{n}}$ holds.
Some $n>2$ lamps are cyclically connected: lamp $1$ with lamp $2$, ..., lamp $k$ with lamp $k+1$,..., lamp $n-1$ with lamp $n$, lamp $n$ with lamp $1$. At the beginning all lamps are off. When one pushes the switch of a lamp, that lamp and the two ones connected to it change status (from off to on, or vice versa). Determine the number of configurations of lamps reachable from the initial one, through some set of switches being pushed.
Let $a$ and $b$ be positive integers with $a>1$ and $b>2$. Prove that $a^b+1\ge b(a+1)$ and determine when there is inequality.
A positive integer is written on a blackboard. Players $A$ and $B$ play the following game: in each move one has to choose a proper divisor $m$ of the number $n$ written on the blackboard ($1<m<n$) and replaces $n$ with $n-m$. Player $A$ makes the first move, then players move alternately. The player who can't make a move loses the game. For which starting numbers is there a winning strategy for player $B$?
Let $0 < a < 1$ be a real number. Prove that for all finite, strictly increasing sequences $k_1, k_2, \ldots , k_n$ of non-negative integers we have the inequality
\[\biggl( \sum_{i=1}^n a^{k_i} \biggr)^2 < \frac{1+a}{1-a} \sum_{i=1}^n a^{2k_i}.\]
How many one-to-one functions $f : \{1, 2, \cdots, 9\} \rightarrow \{1, 2, \cdots, 9\}$ satisfy (i) and (ii)?
(i) $f(1)>f(2)$, $f(9)<9$.
(ii) For each $i=3, 4, \cdots, 8$, if $f(1), \cdots, f(i-1)$ are smaller than $f(i)$, then $f(i+1)$ is also smaller than $f(i)$.
Let $P$ be the set of all $2012$ tuples $(x_1, x_2, \dots, x_{2012})$, where $x_i \in \{1,2,\dots 20\}$ for each $1\leq i \leq 2012$. The set $A \subset P$ is said to be decreasing if for each $(x_1,x_2,\dots ,x_{2012} ) \in A$ any $(y_1,y_2,\dots, y_{2012})$ satisfying $y_i \leq x_i (1\leq i \leq 2012)$ also belongs to $A$. The set $B \subset P$ is said to be increasing if for each $(x_1,x_2,\dots ,x_{2012} ) \in B$ any $(y_1,y_2,\dots, y_{2012})$ satisfying $y_i \geq x_i (1\leq i \leq 2012)$ also belongs to $B$. Find the maximum possible value of $f(A,B)= \dfrac {|A\cap B|}{|A|\cdot |B|}$, where $A$ and $B$ are nonempty decreasing and increasing sets ($\mid \cdot \mid$ denotes the number of elements of the set).
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.
Let the integer $n \ge 2$, and the real numbers $x_1,x_2,\cdots,x_n\in \left[0,1\right] $.Prove that\[\sum_{1\le k<j\le n} kx_kx_j\le \frac{n-1}{3}\sum_{k=1}^n kx_k.\]