Found problems: 1782
Find the number of ten-digit natural numbers (which do not start with zero) containing no block $ 1991$.
Determine all functions $f:\mathbf{R} \rightarrow \mathbf{R}^+$ such that for all real numbers $x,y$ the following conditions hold:
$\begin{array}{rl}
i. & f(x^2) = f(x)^2 -2xf(x) \\
ii. & f(-x) = f(x-1)\\
iii. & 1<x<y \Longrightarrow f(x) < f(y).
\end{array}$
If $x$ is a positive rational number show that $x$ can be uniquely expressed in the form $x = \sum^n_{k=1} \frac{a_k}{k!}$ where $a_1, a_2, \ldots$ are integers, $0 \leq a_n \leq n - 1$, for $n > 1,$ and the series terminates. Show that $x$ can be expressed as the sum of reciprocals of different integers, each of which is greater than $10^6.$
$f$ is a function whose domain is the set of nonnegative integers and whose range is contained in the set of nonnegative integers. $f$ satisfies the condition that $f(f(n))+f(n)=2n+3$ for all nonnegative integers $n$. Find $f(2014)$.
Suppose the function $\psi$ satisfies $\psi(1)=\sqrt{2+\sqrt{2+\sqrt2}}$ and $\psi(3x)+3\psi(x)=\psi(x)^3$ for all real $x$. Determine the greatest integer less than $\textstyle\prod_{n=1}^{100}\psi(3^n)$.
Prove that for all positive integers $n$ there exists a single positive integer divisible with $5^n$ which in decimal base is written using $n$ digits from the set $\{1,2,3,4,5\}$.
Let $ k>1$ be an integer. Prove that there exists infinitely many natural numbers such as $ n$ such that: \[ n|1^n\plus{}2^n\plus{}\dots\plus{}k^n\]
A square is divided into congruent rectangles with sides of integer lengths. A rectangle is important if it has at least one point in common with a given diagonal of the square. Prove that this diagonal bisects the total area of the important rectangles
Given a finite sequence $ S \equal{} (a_1,a_2,\ldots,a_n)$ of $ n$ real numbers, let $ A(S)$ be the sequence
\[ \left(\frac {a_1 \plus{} a_2}2,\frac {a_2 \plus{} a_3}2,\ldots,\frac {a_{n \minus{} 1} \plus{} a_n}2\right)
\]of $ n \minus{} 1$ real numbers. Define $ A^1(S) \equal{} A(S)$ and, for each integer $ m$, $ 2\le m\le n \minus{} 1$, define $ A^m(S) \equal{} A(A^{m \minus{} 1}(S)).$ Suppose $ x > 0$, and let $ S \equal{} (1,x,x^2,\ldots,x^{100})$. If $ A^{100}(S) \equal{} (1/2^{50})$, then what is $ x$?
$ \textbf{(A) } 1 \minus{} \frac {\sqrt {2}}2\qquad \textbf{(B) } \sqrt {2} \minus{} 1\qquad \textbf{(C) } \frac 12\qquad \textbf{(D) } 2 \minus{} \sqrt {2}\qquad \textbf{(E) } \frac {\sqrt {2}}2$
Let $S$ be a set of integers such that [list][*] there exist $a, b \in S$ with $\gcd(a, b)=\gcd(a-2,b-2)=1$, [*] if $x,y\in S$, then $x^2 -y\in S$.[/list] Prove that $S=\mathbb{Z}$.
Let $A_{1}, ..., A_{n}$ be different subsets of an $n$-element set $X$. Show that there exists $x\in X$ such that the sets
$A_{1}-\{x\}, A_{2}-\{x\}, ..., A_{n}-\{x\}$ are all different.
Let us consider a simle graph with vertex set $ V$. All ordered pair $ (a,b)$ of integers with $ gcd(a,b) \equal{} 1$, are elements of V.
$ (a,b)$ is connected to $ (a,b \plus{} kab)$ by an edge and to $ (a \plus{} kab,b)$ by another edge for all integer k.
Prove that for all $ (a,b)\in V$, there exists a path fromm $ (1,1)$ to $ (a,b)$.
What is the least number of moves it takes a knight to get from one corner of an $n\times n$ chessboard, where $n\ge 4$, to the diagonally opposite corner?
The numbers $ 1, 2,\ldots, 50 $ are written on a blackboard. Each minute any two numbers are erased and their positive difference is written instead. At the end one number remains. Which values can take this number?
In each cell of an $n\times n$ board is a lightbulb. Initially, all of the lights are off. Each move consists of changing the state of all of the lights in a row or of all of the lights in a column (off lights are turned on and on lights are turned off).
Show that if after a certain number of moves, at least one light is on, then at this moment at least $n$ lights are on.
For $n$ an odd positive integer, the unit squares of an $n\times n$ chessboard are coloured alternately black and white, with the four corners coloured black. A [i]tromino[/i] is an $L$-shape formed by three connected unit squares.
$(a)$ For which values of $n$ is it possible to cover all the black squares with non-overlapping trominoes lying entirely on the chessboard?
$(b)$ When it is possible, find the minimum number of trominoes needed.
For any real numbers sequence $\{x_n\}$ ,suppose that $\{y_n\}$ is a sequence such that:
$y_1=x_1, y_{n+1}=x_{n+1}-(\sum\limits_{i = 1}^{n} {x^2_i})^{ \frac{1}{2}}$ ${(n \ge 1})$ .
Find the smallest positive number $\lambda$ such that for any real numbers sequence $\{x_n\}$ and all positive integers $m$ , have $\frac{1}{m}\sum\limits_{i = 1}^{m} {x^2_i}\le\sum\limits_{i = 1}^{m} {\lambda^{m-i}y^2_i} .$
(High School Affiliated to Nanjing Normal University )
Find the greatest real number $ \alpha$ for which there exists a sequence of infinitive integers $ (a_n)$, ($ n \equal{} 1, 2, 3, \ldots$) satisfying the following conditions:
1) $ a_n > 1997n$ for every $ n \in\mathbb{N}^{*}$;
2) For every $ n\ge 2$, $ U_n\ge a^{\alpha}_n$, where $ U_n \equal{} \gcd\{a_i \plus{} a_k | i \plus{} k \equal{} n\}$.
Three nonnegative real numbers $ r_1$, $ r_2$, $ r_3$ are written on a blackboard. These numbers have the property that there exist integers $ a_1$, $ a_2$, $ a_3$, not all zero, satisfying $ a_1r_1 \plus{} a_2r_2 \plus{} a_3r_3 \equal{} 0$. We are permitted to perform the following operation: find two numbers $ x$, $ y$ on the blackboard with $ x \le y$, then erase $ y$ and write $ y \minus{} x$ in its place. Prove that after a finite number of such operations, we can end up with at least one $ 0$ on the blackboard.
Prove that $ \prod_{i\equal{}1}^{n}(1\plus{}x_{1}\plus{}x_{2}\plus{}...\plus{}x_{i})\geq\sqrt{(n\plus{}1)^{n\plus{}1}x_{1}x_{2}...x_{n}}\forall x_{1},...,x_{n}> 0$.
Define a [i]beautiful number[/i] to be an integer of the form $a^n$, where $a\in\{3,4,5,6\}$ and $n$ is a positive integer.
Prove that each integer greater than $2$ can be expressed as the sum of pairwise distinct beautiful numbers.
[i]Proposed by Matthew Babbitt[/i]
At an international conference there are four official languages. Any two participants can speak in one of these languages. Show that at least $60\%$ of the participants can speak the same language.
[i]Mihai Baluna[/i]
Find all the functions $f(x),$ continuous on the whole real axis, such that for every real $x$ \[f(3x-2)\leq f(x)\leq f(2x-1).\]
[i]Proposed by A. Golovanov[/i]
Let $\alpha$ and $\beta$ be the roots of the equation $x^2 + mx -1 = 0$ where $m$ is an odd integer. Let $\lambda _n = \alpha ^n + \beta ^n , n \geq 0$
Prove that
(A) $\lambda _n$ is an integer
(B) gcd ( $\lambda _n , \lambda_{n+1}$) = 1 .
Let $n \ge 2$ be an integer. There are $n$ houses in a town. All distances between pairs of houses are different. Every house sends a visitor to the house closest to it. Find all possible values of $n$ (with full justification) for which we can design a town with $n$ houses where every house is visited.