Found problems: 1782
Find all integers $n$, $n \ge 1$, such that $n \cdot 2^{n+1}+1$ is a perfect square.
A sequence of real numbers $ \{a_{n}\}$ is defineds by $ a_{0}\neq 0,1$, $ a_1\equal{}1\minus{}a_0$,$ a_{n\plus{}1}\equal{}1\minus{}a_n(1\minus{}a_n)$, $ n\equal{}1,2,...$.
Prove that for any positive integer $ n$, we have
$ a_{0}a_{1}...a_{n}(\frac{1}{a_0}\plus{}\frac{1}{a_1}\plus{}...\plus{}\frac{1}{a_n})\equal{}1$
Determine all pairs $ (x,y)$ of positive integers satisfying the equation
\[ x!\plus{}y!\equal{}x^{y}.\]
The sequence $ \{a_n\}$ of integers is defined by
\[ a_1 \equal{} 2, a_2 \equal{} 7
\]
and
\[ \minus{} \frac {1}{2} < a_{n \plus{} 1} \minus{} \frac {a^2_n}{a_{n \minus{} 1}} \leq \frac {}{}, n \geq 2.
\]
Prove that $ a_n$ is odd for all $ n > 1.$
We are given one red and $k>1$ blue cells, and a pack of $2n$ cards, enumerated by the numbers from $1$ to $2n$. Initially, the pack is situated on the red cell and arranged in an arbitrary order. In each move, we are allowed to take the top card from one of the cells and place it either onto the top of another cell on which the number on the top card is greater by $1$, or onto an empty cell. Given $k$, what is the maximal $n$ for which it is always possible to move all the cards onto a blue cell?
We wish to find the sum of $40$ given numbers utilizing $40$ processors. Initially, we have the number $0$ on the screen of each processor. Each processor adds the number on its screen with a number entered directly (only the given numbers could be entered directly to the processors) or transferred from another processor in a unit time. Whenever a number is transferred from a processor to another, the former processor resets. Find the least time needed to find the desired sum.
Let $n$ be a positive integer not divisible by $2$ or $3$. Prove that for all integers $k$, the number $(k+1)^n-k^n-1$ is divisible by $k^2+k+1$.
Let $n$ be a positive integer, and $a_j$, for $j=1,2,\ldots,n$ are complex numbers. Suppose $I$ is an arbitrary nonempty subset of $\{1,2,\ldots,n\}$, the inequality $\left|-1+ \prod_{j\in I} (1+a_j) \right| \leq \frac 12$ always holds.
Prove that $\sum_{j=1}^n |a_j| \leq 3$.
The audience arranges $n$ coins in a row. The sequence of heads and tails is chosen arbitrarily. The audience also chooses a number between $1$ and $n$ inclusive. Then the assistant turns one of the coins over, and the magician is brought in to examine the resulting sequence. By an agreement with the assistant beforehand, the magician tries to determine the number chosen by the audience.
[list][b](a)[/b] Prove that if this is possible for some $n$, then it is also possible for $2n$.
[b](b)[/b] Determine all $n$ for which this is possible.[/list]
Let $F$ be a family of subsets of $S = \left \{ 1,2,...,n \right \}$ ($n \geq 2$). A valid play is to choose two disjoint sets $A$ and $B$ from $F$ and add $A \cup B$ to $F$ (without removing $A$ and $B$).
Initially, $F$ has all the subsets that contain only one element of $S$. The goal is to have all subsets of $n - 1$ elements of $S$ in $F$ using valid plays.
Determine the lowest number of plays required in order to achieve the goal.
A family $L$ of 2006 lines on the plane is given in such a way that it doesn't contain
parallel lines and it doesn't contain three lines with a common point.We say that
the line $l_1\in L$ is [i]bounding[/i] the line $l_2\in L$,if all intersection points
of the line $l_2$ with other lines from $L$ lie on the one side of the line $l_1$.
Prove that in the family $L$ there are two lines $l$ and $l'$ such that the following
2 conditions are satisfied simultaneously:
[b]1)[/b] The line $l$ is bounding the line $l'$;
[b]2)[/b] the line $l'$ is not bounding the line $l$.
Prove that for all odd $ n > 1$, we have $ 8n \plus{} 4|C^{4n}_{2n}$.
Find all functions $f$ from the set of real numbers to itself satisfying
\[ f(x(1+y)) = f(x)(1 + f(y)) \]
for all real numbers $x, y$.
The infinite sequence $P_0(x),P_1(x),P_2(x),\ldots,P_n(x),\ldots$ is defined as
\[P_0(x)=x,\quad P_n(x) = P_{n-1}(x-1)\cdot P_{n-1}(x+1),\quad n\ge 1.\]
Find the largest $k$ such that $P_{2014}(x)$ is divisible by $x^k$.
Find all functions $f: Q^{+}\rightarrow R$ such that
$f(x)+f(y)+2xyf(xy)=\frac{f(xy)}{f(x+y)}$ for all $x,y\in Q^{+}$
A directed graph has each vertex with outdegree 2. Prove that it is possible to split the vertices into 3 sets so that for each vertex $v$, $v$ is not simultaneously in the same set with both of the vertices that it points to.
[i]David Yang.[/i]
[hide="Stronger Version"]See [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?f=42&t=492100]here[/url].[/hide]
Define $a_1=1$ and $a_{n+1}=\frac{a_n}{n}+\frac{n}{a_n}$ for $n\in\mathbb{N}$. Find the greatest integer not exceeding $a_{2000}$ and prove your claim.
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.
Let $ R$ be a finite commutative ring. Prove that $ R$ has a multiplicative identity element $ (1)$ if and only if the annihilator of $ R$ is $ 0$ (that is, $ aR\equal{}0, \;a\in R $ imply $ a\equal{}0$).
Define a sequence $\{u_n\}_{n=0}^{\infty}$ by $u_0=u_1=u_2=1,$ and thereafter by the condition that
$\det\begin{vmatrix} u_n & u_{n+1} \\ u_{n+2} & u_{n+3} \end{vmatrix}=n!$
for all $n\ge 0.$ Show that $u_n$ is an integer for all $n.$ (By convention, $0!=1$.)
Let $ f(x) \equal{} \frac{x}{1 \minus{} x}$ and let $ a$ be a real number. If $ x_0 \equal{} a, x_1 \equal{} f(x_0), x_2 \equal{} f(x_1), ...., x_{1996} \equal{} f(x_{1995})$ and $ x_{1996} \equal{} 1,$ what is $ a$?
A. 0
B. 1/1997
C. 1995
D. 1995/1996
E. None of these
Prove that for any positive integer $n$, the number
\[ S_n = {2n+1\choose 0}\cdot 2^{2n}+{2n+1\choose 2}\cdot 2^{2n-2}\cdot 3 +\cdots + {2n+1 \choose 2n}\cdot 3^n \] is the sum of two consecutive perfect squares.
[i]Dorin Andrica[/i]
Let $n$ be a positive integer. Show that \begin{align*}&\quad\,\,\frac{1}{\binom{n}{1}}+\frac{1}{2\binom{n}{2}}+\frac{1}{3\binom{n}{3}}+\cdots+\frac{1}{n\binom{n}{n}}\\&=\frac{1}{2^{n-1}}+\frac{1}{2\cdot2^{n-2}}+\frac{1}{3\cdot2^{n-3}}+\cdots+\frac{1}{n\cdot2^0}.\end{align*}
Evaluate
\[\int_0^1 (1-x^2)^n dx\ (n=0,1,2,\cdots)\]
i) If $x = \left(1+\frac{1}{n}\right)^{n}$ and $y=\left(1+\frac{1}{n}\right)^{n+1}$, show that $y^{x}= x^{y}$.
ii) Show that, for all positive integers $n$, \[1^{2}-2^{2}+3^{2}-4^{2}+\cdots+(-1)^{n}(n-1)^{2}+(-1)^{n+1}n^{2}= (-1)^{n+1}(1+2+\cdots+n).\]