Found problems: 1782
There are $n$ boys and $n$ girls in a school class, where $n$ is a positive integer. The heights of all the children in this class are distinct. Every girl determines the number of boys that are taller than her, subtracts the number of girls that are taller than her, and writes the result on a piece of paper. Every boy determines the number of girls that are shorter than him, subtracts the number of boys that are shorter than him, and writes the result on a piece of paper.
Prove that the numbers written down by the girls are the same as the numbers written down by the boys (up to a permutation).
[i]Proposed by Stephan Wagner, Austria[/i]
Let $A=\{1,2,\ldots,2012\}, \: B=\{1,2,\ldots,19\}$ and $S$ be the set of all subsets of $A.$ Find the number of functions $f : S\to B$ satisfying $f(A_1\cap A_2)=\min\{f(A_1),f(A_2)\}$ for all $A_1, A_2 \in S.$
Each rational number is painted either white or red. Call such a coloring of the rationals [i]sanferminera[/i] if for any distinct rationals numbers $x$ and $y$ satisfying one of the following three conditions: [list=1][*]$xy=1$,
[*]$x+y=0$,
[*]$x+y=1$,[/list]we have $x$ and $y$ painted different colors. How many sanferminera colorings are there?
Let $n$ be positive integer, set $M = \{ 1, 2, \ldots, 2n \}$. Find the minimum positive integer $k$ such that for any subset $A$ (with $k$ elements) of set $M$, there exist four pairwise distinct elements in $A$ whose sum is $4n + 1$.
(a) Prove that, for any positive integers $m\le \ell$ given, there is a positive integer $n$ and positive integers $x_1,\cdots,x_n,y_1,\cdots,y_n$ such that the equality \[ \sum_{i=1}^nx_i^k=\sum_{i=1}^ny_i^k\] holds for every $k=1,2,\cdots,m-1,m+1,\cdots,\ell$, but does not hold for $k=m$.
(b) Prove that there is a solution of the problem, where all numbers $x_1,\cdots,x_n,y_1,\cdots,y_n$ are distinct.
[i]Proposed by Ilya Bogdanov and Géza Kós.[/i]
Let $n$ be a positive integer. Answer the following questions.
(1) Find the maximum value of $f_n(x)=x^{n}e^{-x}$ for $x\geq 0$.
(2) Show that $\lim_{x\to\infty} f_n(x)=0$.
(3) Let $I_n=\int_0^x f_n(t)\ dt$. Find $\lim_{x\to\infty} I_n(x)$.
Let A be a symmetric matrix such that the sum of elements of any row is zero.
Show that all elements in the main diagonal of cofator matrix of A are equal.
A set $A$ of integers is called [i]sum-full[/i] if $A \subseteq A + A$, i.e. each element $a \in A$ is the sum of some pair of (not necessarily different) elements $b,c \in A$. A set $A$ of integers is said to be [i]zero-sum-free[/i] if $0$ is the only integer that cannot be expressed as the sum of the elements of a finite nonempty subset of $A$.
Does there exist a sum-full zero-sum-free set of integers?
[i]Romania (Dan Schwarz)[/i]
Given a set $ \mathcal{H}$ of points in the plane, $ P$ is called an "intersection point of $ \mathcal{H}$" if distinct points $ A,B,C,D$ exist in $ \mathcal{H}$ such that lines $ AB$ and $ CD$ are distinct and intersect in $ P$.
Given a finite set $ \mathcal{A}_{0}$ of points in the plane, a sequence of sets is defined as follows: for any $ j\geq0$, $ \mathcal{A}_{j+1}$ is the union of $ \mathcal{A}_{j}$ and the intersection points of $ \mathcal{A}_{j}$.
Prove that, if the union of all the sets in the sequence is finite, then $ \mathcal{A}_{i}=\mathcal{A}_{1}$ for any $ i\geq1$.
Consider a $ n\times n $ square grid which is divided into $ n^2 $ unit squares(think of a chess-board). The set of all unit squares intersecting the main diagonal of the square or lying under it is called an $n$-staircase. Find the number of ways in which an $n$-stair case can be partitioned into several rectangles, with sides along the grid lines, having mutually distinct areas.
An $n\times n$ chessboard is given, where $n$ is an even positive integer. On every line, the unit squares are to be permuted, subject to the condition that the resulting table has to be symmetric with respect to its main diagonal (the diagonal from the top-left corner to the bottom-right corner). We say that a board is [i]alternative[/i] if it has at least one pair of complementary lines (two lines are complementary if the unit squares on them which lie on the same column have distinct colours). Otherwise, we call the board [i]nonalternative[/i]. For what values of $n$ do we always get from the $n\times n$ chessboard an alternative board?\\ \\
[i](Alexandru Petrescu and Andra Elena Mircea)[/i]
There are 2000 cities in Graphland; some of them are connected by roads.
For every city the number of roads going from it is counted. It is known that there are exactly two equal numbers among all the numbers obtained. What can be these numbers?
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]
Let $ f(x) \equal{} x^2 \plus{} 2007x \plus{} 1$. Prove that for every positive integer $ n$, the equation $ \underbrace{f(f(\ldots(f}_{n\ {\rm times}}(x))\ldots)) \equal{} 0$ has at least one real solution.
Cards numbered from 1 to $2^n$ are distributed among $k$ children, $1\leq k\leq 2^n$, so that each child gets at least one card. Prove that the number of ways to do that is divisible by $2^{k-1}$ but not by $2^k$.
[i] M. Ivanov [/i]
Show that, for any fixed integer $\,n \geq 1,\,$ the sequence \[ 2, \; 2^2, \; 2^{2^2}, \; 2^{2^{2^2}}, \ldots (\mbox{mod} \; n) \] is eventually constant.
[The tower of exponents is defined by $a_1 = 2, \; a_{i+1} = 2^{a_i}$. Also $a_i \; (\mbox{mod} \; n)$ means the remainder which results from dividing $a_i$ by $n$.]
For all natural $n$, an $n$-staircase is a figure consisting of unit squares, with one square in the first row, two squares in the second row, and so on, up to $n$ squares in the $n^{th}$ row, such that all the left-most squares in each row are aligned vertically.
Let $f(n)$ denote the minimum number of square tiles requires to tile the $n$-staircase, where the side lengths of the square tiles can be any natural number. e.g. $f(2)=3$ and $f(4)=7$.
(a) Find all $n$ such that $f(n)=n$.
(b) Find all $n$ such that $f(n) = n+1$.
Suppose that $n$ people each know exactly one piece of information, and all $n$ pieces are different. Every time person $A$ phones person $B$, $A$ tells $B$ everything that $A$ knows, while $B$ tells $A$ nothing. What is the minimum number of phone calls between pairs of people needed for everyone to know everything? Prove your answer is a minimum.
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!)
Determine all natural numbers $n$ such that for each natural number $a$ relatively prime with $n$ and $a \le 1 + \left\lfloor \sqrt{n} \right\rfloor$ there exists some integer $x$ with $a \equiv x^2 \mod n$.
Remark: "Natural numbers" is the set of positive integers.
Let $ \left( x_n\right)_{n\ge 1} $ be a sequence of integers defined recursively as $ x_{n+2}=5x_{n+1}-x_n. $
Prove that $ \left( x_n\right)_{n\ge 1} $ has a subsequence whose terms are multiples of $ 22 $ if $ \left( x_n\right)_{n\ge 1} $ has a term that is multiple of $ 22. $
We examine the following two sequences: The Fibonacci sequence: $F_{0}= 0, F_{1}= 1, F_{n}= F_{n-1}+F_{n-2 }$ for $n \geq 2$; The Lucas sequence: $L_{0}= 2, L_{1}= 1, L_{n}= L_{n-1}+L_{n-2}$ for $n \geq 2$. It is known that for all $n \geq 0$ \[F_{n}=\frac{\alpha^{n}-\beta^{n}}{\sqrt{5}},L_{n}=\alpha^{n}+\beta^{n},\] where $\alpha=\frac{1+\sqrt{5}}{2},\beta=\frac{1-\sqrt{5}}{2}$. These formulae can be used without proof.
Prove that $F_{n-1}F_{n}F_{n+1}L_{n-1}L_{n}L_{n+1}(n \geq 2)$ is not a perfect square.
Let $ G$ be a simple graph with $ 2 \cdot n$ vertices and $ n^{2}+1$ edges. Show that this graph $ G$ contains a $ K_{4}-\text{one edge}$, that is, two triangles with a common edge.
for any positive integer $n$ greater than $1$, show that
\[2^n<\binom{2n}{n}<\frac{2^n}{\prod\limits_{i=0}^{n-1} \left(1-\frac{i}{n}\right)}\]
Let $ n$ be a positive integer and $ a_{1}, \ldots, a_{n}$ be arbitrary integers. Suppose that a function $ f: \mathbb{Z}\to \mathbb{R}$ satisfies $ \sum_{i=1}^{n}f(k+a_{i}l) = 0$ whenever $ k$ and $ l$ are integers and $ l \ne 0$. Prove that $ f = 0$.