Found problems: 1782
Suppose $A\subset \{(a_1,a_2,\dots,a_n)\mid a_i\in \mathbb{R},i=1,2\dots,n\}$. For any $\alpha=(a_1,a_2,\dots,a_n)\in A$ and $\beta=(b_1,b_2,\dots,b_n)\in A$, we define
\[ \gamma(\alpha,\beta)=(|a_1-b_1|,|a_2-b_2|,\dots,|a_n-b_n|), \] \[ D(A)=\{\gamma(\alpha,\beta)\mid\alpha,\beta\in A\}. \] Please show that $|D(A)|\geq |A|$.
Let $a_1, b_1, a_2, b_2, \dots , a_n, b_n$ be nonnegative real numbers. Prove that
\[
\sum_{i, j = 1}^{n} \min\{a_ia_j, b_ib_j\} \le \sum_{i, j = 1}^{n} \min\{a_ib_j, a_jb_i\}.
\]
A sequence $ (x_n)$ is given by $ x_1\equal{}2$ and $ nx_n\equal{}2(2n\minus{}1)x_{n\minus{}1}$ for $ n>1$. Prove that $ x_n$ is an integer for every $ n \in \mathbb{N}$.
Prove that $N^2$ arbitrary distinct positive integers ($N>10$) can be arranged in a $N\times N$ table, so that all $2N$ sums in rows and columns are distinct.
[i]Proposed by S. Volchenkov[/i]
Let $n$ be a positive integer. On the table, we have $n^2$ ornaments in $n$ different colours, not necessarily $n$ of each colour. Prove that we can hang the ornaments on $n$ Christmas trees in such a way that there are exactly $n$ ornaments on each tree and the ornaments on every tree are of at most $2$ different colours.
Find all functions $f:\mathbb{N}\rightarrow\mathbb{N}$ such that \[f(n+1)>\frac{f(n)+f(f(n))}{2}\] for all $n\in\mathbb{N}$, where $\mathbb{N}$ is the set of strictly positive integers.
Function $f(x, y): \mathbb N \times \mathbb N \to \mathbb Q$ satisfies the conditions:
(i) $f(1, 1) =1$,
(ii) $f(p + 1, q) + f(p, q + 1) = f(p, q)$ for all $p, q \in \mathbb N$, and
(iii) $qf(p + 1, q) = pf(p, q + 1)$ for all $p, q \in \mathbb N$.
Find $f(1990, 31).$
An [i]animal[/i] with $n$ [i]cells[/i] is a connected figure consisting of $n$ equal-sized cells[1].
A [i]dinosaur[/i] is an animal with at least $2007$ cells. It is said to be [i]primitive[/i] it its cells cannot be partitioned into two or more dinosaurs. Find with proof the maximum number of cells in a primitive dinosaur.
(1) Animals are also called [i]polyominoes[/i]. They can be defined inductively. Two cells are [i]adjacent[/i] if they share a complete edge. A single cell is an animal, and given an animal with $n$ cells, one with $n+1$ cells is obtained by adjoining a new cell by making it adjacent to one or more existing cells.
For a positive integer $k\ge 2$ define $\mathcal{T}_k=\{(x,y)\mid x,y=0,1,\ldots, k-1\}$ to be a collection of $k^2$ lattice points on the cartesian coordinate plane. Let $d_1(k)>d_2(k)>\cdots$ be the decreasing sequence of the distinct distances between any two points in $T_k$. Suppose $S_i(k)$ be the number of distances equal to $d_i(k)$.
Prove that for any three positive integers $m>n>i$ we have $S_i(m)=S_i(n)$.
The sequence $<a_n>$ is defined as follows, $a_1=a_2=1$, $a_3=2$,
$$a_{n+3}=\frac{a_{n+2}a_{n+1}+n!}{a_n},$$
$n \ge 1$.
Prove that all the terms in the sequence are integers.
A given rectangle $ R$ is divided into $mn$ small rectangles by straight lines parallel to its sides. (The distances between the parallel lines may not be equal.) What is the minimum number of appropriately selected rectangles’ areas that should be known in order to determine the area of $ R$?
Suppose $\, q_{0}, \, q_{1}, \, q_{2}, \ldots \; \,$ is an infinite sequence of integers satisfying the following two conditions:
(i) $\, m-n \,$ divides $\, q_{m}-q_{n}\,$ for $\, m > n \geq 0,$
(ii) there is a polynomial $\, P \,$ such that $\, |q_{n}| < P(n) \,$ for all $\, n$
Prove that there is a polynomial $\, Q \,$ such that $\, q_{n}= Q(n) \,$ for all $\, n$.
Given two positive integers $m$ and $n$, find the smallest positive integer $k$ such that among any $k$ people, either there are $2m$ of them who form $m$ pairs of mutually acquainted people or there are $2n$ of them forming $n$ pairs of mutually unacquainted people.
A Pythagorean triple is a solution of the equation $x^2 + y^2 = z^2$ in positive integers such that $x < y$. Given any non-negative integer $n$ , show that some positive integer appears in precisely $n$ distinct Pythagorean triples.
The number $2013$ is expressed in the form \[2013=\frac{a_1!a_2!\cdots a_m!}{b_1!b_2!\cdots b_n!},\] where $a_1\ge a_2\ge\cdots\ge a_m$ and $b_1\ge b_2\ge\cdots\ge b_n$ are positive integers and $a_1+b_1$ is as small as possible. What is $|a_1-b_1|$?
${ \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D}}\ 4\qquad\textbf{(E)}\ 5 $
Find all functions $f: \mathbb{Q}\to \mathbb{Q}$ such that for all $x,y,z \in \mathbb{Q}$: \[f(x+y+z)+f(x-y)+f(y-z)+f(z-x)=3f(x)+3f(y)+3f(z).\]
Consider a function $f:\mathbb{Z}\to \mathbb{Z}$ such that:
\[f(m^2+f(n))=f^2(m)+n,\ \forall m,n\in \mathbb{Z}\]
Prove that:
a)$f(0)=0$;
b)$f(1)=1$;
c)$f(n)=n,\ \forall n\in \mathbb{Z}$
[i]Lucian Dragomir[/i]
Let $n$ be an integer greater than or equal to $2$. There are $n$ people in one line, each of which is either a [i]scoundrel[/i] (who always lie) or a [i]knight[/i] (who always tells the truth). Every person, except the first, indicates a person in front of him/her and says "This person is a scoundrel" or "This person is a knight." Knowing that there are strictly more scoundrel than knights, seeing the statements show that it is possible to determine each person whether he/she is a scoundrel or a knight.
Prove that for each positive integer $ n$, there are pairwise relatively prime integers $ k_0,k_1,\ldots,k_n$, all strictly greater than $ 1$, such that $ k_0k_1\ldots k_n\minus{}1$ is the product of two consecutive integers.
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 $x$.)
Let $(a_n)_{n=0}^{\infty}$ be a sequence of real numbers defined as follows:
[list]
[*] $a_0 = 3$, $a_1 = 2$, and $a_2 = 12$; and
[*] $2a_{n + 3} - a_{n + 2} - 8a_{n + 1} + 4a_n = 0$ for $n \geq 0$.
[/list]
Show that $a_n$ is always a strictly positive integer.
The infinite sequence of 2's and 3's \[\begin{array}{l}2,3,3,2,3,3,3,2,3,3,3,2,3,3,2,3,3, \\ 3,2,3,3,3,2,3,3,3,2,3,3,2,3,3,3,2,\cdots \end{array}\] has the property that, if one forms a second sequence that records the number of 3's between successive 2's, the result is identical to the given sequence. Show that there exists a real number $r$ such that, for any $n$, the $n$th term of the sequence is 2 if and only if $n = 1+\lfloor rm \rfloor$ for some nonnegative integer $m$.
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 $r + 2$.
If $a_{1}$, $a_{2}$, $\ldots$, $a_{n}\geq 0$ are such that \[a_{1}^{2}+\cdots+a_{n}^{2}=1,\]
then find the maximum value of the product $(1-a_{1})\cdots (1-a_{n})$.
Let $ X: \equal{} \{x_1,x_2,\ldots,x_{29}\}$ be a set of $ 29$ boys: they play with each other in a tournament of Pro Evolution Soccer 2009, in respect of the following rules:
[list]i) every boy play one and only one time against each other boy (so we can assume that every match has the form $ (x_i \text{ Vs } x_j)$ for some $ i \neq j$);
ii) if the match $ (x_i \text{ Vs } x_j)$, with $ i \neq j$, ends with the win of the boy $ x_i$, then $ x_i$ gains $ 1$ point, and $ x_j$ doesn’t gain any point;
iii) if the match $ (x_i \text{ Vs } x_j)$, with $ i \neq j$, ends with the parity of the two boys, then $ \frac {1}{2}$ point is assigned to both boys.
[/list]
(We assume for simplicity that in the imaginary match $ (x_i \text{ Vs } x_i)$ the boy $ x_i$ doesn’t gain any point).
Show that for some positive integer $ k \le 29$ there exist a set of boys $ \{x_{t_1},x_{t_2},\ldots,x_{t_k}\} \subseteq X$ such that, for all choice of the positive integer $ i \le 29$, the boy $ x_i$ gains always a integer number of points in the total of the matches $ \{(x_i \text{ Vs } x_{t_1}),(x_i \text{ Vs } x_{t_2}),\ldots, (x_i \text{ Vs } x_{t_k})\}$.
[i](Paolo Leonetti)[/i]