Found problems: 766
Find the magnitude of the product of all complex numbers $c$ such that the recurrence defined by $x_1 = 1$, $x_2 = c^2 - 4c + 7$, and $x_{n+1} = (c^2 - 2c)^2 x_n x_{n-1} + 2x_n - x_{n-1}$ also satisfies $x_{1006} = 2011$.
[i]Author: Alex Zhu[/i]
Find all functions $ f \colon \mathbb{Z} \mapsto \mathbb{R}$ which satisfy:
i) $ f(x)f(y) \equal{} f(x \plus{} y) \plus{} f(x \minus{} y)$ for all integers $ x$, $ y$
ii) $ f(0) \neq 0$
iii) $ f(1) \equal{} \frac {5}{2}$
Find all function(s) $f:\mathbb R\to\mathbb R$ satisfying the equation
\[f(x+y)+f(x)f(y)=(1+y)f(x)+(1+x)f(y)+f(xy);\]
For all $x,y\in\mathbb R.$
A sequence $a_n$ is defined by
$$a_0=0,\qquad a_1=3;$$$$a_n=8a_{n-1}+9a_{n-2}+16\text{ for }n\ge2.$$Find the least positive integer $h$ such that $a_{n+h}-a_n$ is divisible by $1999$ for all $n\ge0$.
A sequence of real numbers $ a_{0},\ a_{1},\ a_{2},\dots$ is defined by the formula
\[ a_{i \plus{} 1} \equal{} \left\lfloor a_{i}\right\rfloor\cdot \left\langle a_{i}\right\rangle\qquad\text{for}\quad i\geq 0;
\]here $a_0$ is an arbitrary real number, $\lfloor a_i\rfloor$ denotes the greatest integer not exceeding $a_i$, and $\left\langle a_i\right\rangle=a_i-\lfloor a_i\rfloor$. Prove that $a_i=a_{i+2}$ for $i$ sufficiently large.
[i]Proposed by Harmel Nestra, Estionia[/i]
Four sequences of real numbers $(a_n), (b_n), (c_n), (d_n)$ satisfy for all $n$,
$$a_{n+1} = a_n +b_n, b_{n+1} = b_n +c_n,$$
$$c_{n+1} = c_n +d_n, d_{n+1} = d_n +a_n.$$
Prove that if $a_{k+m} = a_m, b_{k+m} = b_m, c_{k+m} = c_m, d_{k+m} = d_m$ for some $k\ge 1,n \ge 1$, then $a_2 = b_2 = c_2 = d_2 = 0$.
Let $n \in \mathbb N$ and $A_n$ set of all permutations $(a_1, \ldots, a_n)$ of the set $\{1, 2, \ldots , n\}$ for which
\[k|2(a_1 + \cdots+ a_k), \text{ for all } 1 \leq k \leq n.\]
Find the number of elements of the set $A_n$.
[i]Proposed by Vidan Govedarica, Serbia[/i]
Let $a$ be a permutation on $\{0,1,\ldots ,2015\}$ and $b,c$ are also permutations on $\{1,2,\ldots ,2015\}$. For all $x\in \{1,2,\ldots ,2015\}$, the following conditions are satisfied:
(i) $a(x)-a(x-1)\neq 1$,\\
(ii) if $b(x)\neq x$, then $c(x)=x$,\\
Prove that the number of $a$'s is equal to the number of ordered pairs of $(b,c)$.
Define sequence of positive integers $(a_n)$ as $a_1 = a$ and $a_{n+1} = a^2_n + 1$ for $n \ge 1$. Prove that there is no index $n$ for which $$\prod_{k=1}^{n} \left(a^2_k + a_k + 1\right)$$ is a perfect square.
Let $(x_n)_{n\geq 1}$ be the sequence defined recursively as such: \[x_1=1, \ x_{n+1}=\frac{x_1}{n+1}+\frac{x_2}{n+2}+\cdots+\frac{x_n}{2n} \ \forall n\geq 1.\]Consider the sequence $(y_n)_{n\geq 1}$ such that $y_n=(x_1^2+x_2^2+\cdots x_n^2)/n$ for all $n\geq 1.$ Prove that
[list=a]
[*]$x_{n+1}^2<y_n/2$ and $y_{n+1}<(2n+1)/(2n+2)\cdot y_n$ for all $n\geq 1;$
[*]$\lim_{n\to\infty}x_n=0.$
[/list]
Define the sequence $x_1,x_2,\dots$ by the initial terms $x_1=2, x_2=4$, and the recurrence relation
\[x_{n+2}=3x_{n+1}-2x_n+\frac{2^n}{x_n} \quad \text{for} \quad n \ge 1.\]
Prove that $\lim_{n \to \infty} \frac{x_n}{2^n}$ exists and satisfies
\[\frac{1+\sqrt{3}}{2} \le \lim_{n \to \infty} \frac{x_n}{2^n} \le \frac{3}{2}.\]
$a_0 = a_1 = 1$ and ${a_{n+1} . a_{n-1}} = a_n . (a_n + 1)$ for all positive integers n.
prove that $a_n$ is one integer for all positive integers n.
(1) Determine if there exists an infinite sequence $\{a_n\}$ with positive integer terms, such that $a^2_{n+1}\ge 2a_na_{n+2}$ for any positive integer $n$.
(2) Determine if there exists an infinite sequence $\{a_n\}$ with positive irrational terms, such that $a^2_{n+1}\ge 2a_na_{n+2}$ for any positive integer $n$.
There are $n$ students sitting on a round table. You collect all of $ n $ name tags and give them back arbitrarily. Each student gets one of $n$ name tags. Now $n$ students repeat following operation:
The students who have their own name tags exit the table. The other students give their name tags to the student who is sitting right to him.
Find the number of ways giving name tags such that there exist a student who don't exit the table after 4 operations.
An eccentric mathematician has a ladder with $ n$ rungs that he always ascends and descends in the following way: When he ascends, each step he takes covers $ a$ rungs of the ladder, and when he descends, each step he takes covers $ b$ rungs of the ladder, where $ a$ and $ b$ are fixed positive integers. By a sequence of ascending and descending steps he can climb from ground level to the top rung of the ladder and come back down to ground level again. Find, with proof, the minimum value of $ n,$ expressed in terms of $ a$ and $ b.$
Show that the recursion $n=x_n(x_{n-1}+x_n+x_{n+1})$, $n=1,2,\ldots$, $x_0=0$ has exaclty one nonnegative solution.
(translated by L. Erdős)
Consider sequences that consist entirely of $ A$'s and $ B$'s and that have the property that every run of consecutive $ A$'s has even length, and every run of consecutive $ B$'s has odd length. Examples of such sequences are $ AA$, $ B$, and $ AABAA$, while $ BBAB$ is not such a sequence. How many such sequences have length 14?
Define a sequence of integers $a_0=m, a_1=n$ and $a_{k+1}=4a_k-5a_{k-1}$ for all $k \ge 1$. Suppose $p>5$ is a prime with $p \equiv 1 \pmod{4}$. Prove that it is possible to choose $m,n$ such that $p \nmid a_k$ for any $k \ge 0$.
A sequence $(u_{n})$ is defined by \[ u_{0}=2 \quad u_{1}=\frac{5}{2}, u_{n+1}=u_{n}(u_{n-1}^{2}-2)-u_{1} \quad \textnormal{for } n=1,\ldots \] Prove that for any positive integer $n$ we have \[ [u_{n}]=2^{\frac{(2^{n}-(-1)^{n})}{3}} \](where [x] denotes the smallest integer $\leq$ x)$.$
A sequence $\{a_n\}$ , $a_1=1,a_2=2,a_{n+1}=\dfrac{a_n^2+(-1)^n}{a_{n-1}}$. Show that $a_m^2+a_{m+1}^2\in\{a_n\},\forall m\in\Bbb N$
A sequence of integers $a_0, a_1 …$ is called [i]kawaii[/i] if $a_0 =0, a_1=1,$ and $$(a_{n+2}-3a_{n+1}+2a_n)(a_{n+2}-4a_{n+1}+3a_n)=0$$ for all integers $n \geq 0$. An integer is called [i]kawaii[/i] if it belongs to some kawaii sequence.
Suppose that two consecutive integers $m$ and $m+1$ are both kawaii (not necessarily belonging to the same kawaii sequence). Prove that $m$ is divisible by $3,$ and that $m/3$ is also kawaii.
Given a real number $\alpha$, a function $f$ is defined on pairs of nonnegative integers by
$f(0,0) = 1, f(m,0) = f(0,m) = 0$ for $m > 0$,
$f(m,n) = \alpha f(m,n-1)+(1- \alpha)f(m -1,n-1)$ for $m,n > 0$.
Find the values of $\alpha$ such that $| f(m,n)| < 1989$ holds for any integers $m,n \ge 0$.
Determine all functions $f: \mathbb{Z}\to\mathbb{Z}$ satisfying \[f\big(f(m)+n\big)+f(m)=f(n)+f(3m)+2014\] for all integers $m$ and $n$.
[i]Proposed by Netherlands[/i]
For any two positive real numbers $x_0 > 0$, $x_1 > 0$, a sequence of real numbers is defined recursively by $$x_{n+1} =\frac{4 \max\{x_n, 4\}}{x_{n-1}}$$ for $n \ge 1$. Find $x_{2010}$.
Let $n$ be positive integer and fix $2n$ distinct points on a circle. Determine the number of ways to connect the points with $n$ arrows (oriented line segments) such that all of the following conditions hold: [list] [*]each of the $2n$ points is a startpoint or endpoint of an arrow; [*]no two arrows intersect; and [*]there are no two arrows $\overrightarrow{AB}$ and $\overrightarrow{CD}$ such that $A$, $B$, $C$ and $D$ appear in clockwise order around the circle (not necessarily consecutively). [/list]