Found problems: 766
We consider the two sequences $(a_n)_{n\ge 0}$ and $(b_n) _{n\ge 0}$ of integers, which are given by $a_0 = b_0 = 2$ and $a_1= b_1 = 14$ and for $n\ge 2$ they are defined as
$a_n = 14a_{n-1} + a_{n-2}$ ,
$b_n = 6b_{n-1}-b_{n-2}$.
Determine whether there are infinite numbers that occur in both sequences
Let $ n$ be a positive integer. Find the number of odd coefficients of the polynomial
\[ u_n(x) \equal{} (x^2 \plus{} x \plus{} 1)^n.
\]
The Fibonacci sequence $\{F_{n}\}$ is defined by \[F_{1}=1, \; F_{2}=1, \; F_{n+2}=F_{n+1}+F_{n}.\] Show that $F_{mn-1}-F_{n-1}^{m}$ is divisible by $F_{n}^{2}$ for all $m \ge 1$ and $n>1$.
For a complex number $z=1+2\sqrt{6}i$ and natural number $n=1,\ 2,\ 3,\ \cdots$, express the complex number $z^n$ in using real numbers $a_n,\ b_n$ as $z^n=a_n+b_ni$.
Answer the following questions.
(1) Show that $a_n^2+b_n^2=5^{2n}\ (n=1,\ 2,\ 3,\ \cdots).$
(2) Find the constants $p,\ q$ such that $a_{n+2}=pa_{n+1}+qa_n$ holds for all $n$.
(3) Show that $a_n$ is not a multiple of $5$ for any $n$.
(4) Show that $z^n\ (n=1,\ 2,\ 3,\ \cdots)$ is not a real number.
Numbers $u_{n,k} \ (1\leq k \leq n)$ are defined as follows
\[u_{1,1}=1, \quad u_{n,k}=\binom{n}{k} - \sum_{d \mid n, d \mid k, d>1} u_{n/d, k/d}.\]
(the empty sum is defined to be equal to zero). Prove that $n \mid u_{n,k}$ for every natural number $n$ and for every $k \ (1 \leq k \leq n).$
Define the sequence $(w_n)_{n\ge0}$ by the recurrence relation
$$w_{n+2}=2w_{n+1}+3w_n,\enspace\enspace w_0=1,w_1=i,\enspace n=0,1,\ldots$$
(1) Find the general formula for $w_n$ and compute the first $9$ terms.
(2) Show that $|\Re w_n-\Im w_n|=1$ for all $n\ge1$.
[i]Proposed by Ovidiu Bagdasar[/i]
In a sports meeting a total of $m$ medals were awarded over $n$ days. On the first day one medal and $\frac{1}{7}$ of the remaining medals were awarded. On the second day two medals and $\frac{1}{7}$ of the remaining medals were awarded, and so on. On the last day, the remaining $n$ medals were awarded. How many medals did the meeting last, and what was the total number of medals ?
The sequences $ (a_n),(b_n)$ are defined by $ a_1\equal{}1,b_1\equal{}2$ and \[a_{n \plus{} 1} \equal{} \frac {1 \plus{} a_n \plus{} a_nb_n}{b_n}, \quad b_{n \plus{} 1} \equal{} \frac {1 \plus{} b_n \plus{} a_nb_n}{a_n}.\]
Show that $ a_{2008} < 5$.
There are n cards on a table numbered from $1$ to $n$, where $n$ is an even number. Two people take turns taking away the cards. The first player will always take the card with the largest number on it, but the second player will take a random card. Prove: the probability that the first player takes the card with the number $i$ is $ \frac{i-1}{n-1} $
Find all functions $f:\mathbb{R}\to \mathbb{R}$ such that for all real numbers $a,b,$ and $c$:
(i) If $a+b+c\ge 0$ then $f(a^3)+f(b^3)+f(c^3)\ge 3f(abc).$
(ii) If $a+b+c\le 0$ then $f(a^3)+f(b^3)+f(c^3)\le 3f(abc).$
[i]Proposed by Ashwin Sah[/i]
A sequence of polynomials $P_m(x, y, z), m = 0, 1, 2, \cdots$, in $x, y$, and $z$ is defined by $P_0(x, y, z) = 1$ and by
\[P_m(x, y, z) = (x + z)(y + z)P_{m-1}(x, y, z + 1) - z^2P_{m-1}(x, y, z)\]
for $m > 0$. Prove that each $P_m(x, y, z)$ is symmetric, in other words, is unaltered by any permutation of $x, y, z.$
Let $ \left\{x_n\right\}$, with $ n \equal{} 1, 2, 3, \ldots$, be a sequence defined by $ x_1 \equal{} 603$, $ x_2 \equal{} 102$ and $ x_{n \plus{} 2} \equal{} x_{n \plus{} 1} \plus{} x_n \plus{} 2\sqrt {x_{n \plus{} 1} \cdot x_n \minus{} 2}$ $ \forall n \geq 1$. Show that:
[b](1)[/b] The number $ x_n$ is a positive integer for every $ n \geq 1$.
[b](2)[/b] There are infinitely many positive integers $ n$ for which the decimal representation of $ x_n$ ends with 2003.
[b](3)[/b] There exists no positive integer $ n$ for which the decimal representation of $ x_n$ ends with 2004.
In a sports meeting a total of $m$ medals were awarded over $n$ days. On the first day one medal and $\frac{1}{7}$ of the remaining medals were awarded. On the second day two medals and $\frac{1}{7}$ of the remaining medals were awarded, and so on. On the last day, the remaining $n$ medals were awarded. How many medals did the meeting last, and what was the total number of medals ?
$ k$ is a given natural number. Find all functions $ f: \mathbb{N}\rightarrow\mathbb{N}$ such that for each $ m,n\in\mathbb{N}$ the following holds: \[ f(m)\plus{}f(n)\mid (m\plus{}n)^k\]
Let $n$ be a positive integer, and define $f(n)=1!+2!+\ldots+n!$. Find polynomials $P$ and $Q$ such that
$$f(n+2)=P(n)f(n+1)+Q(n)f(n)$$for all $n\ge1$.
For $ n\geq 0$ define the matrices $ A_n$ and $ B_n$ as follows: $ A_0 \equal{} B_0 \equal{} (1)$, and for every $ n>0$ let
\[ A_n \equal{} \left( \begin{array}{cc} A_{n \minus{} 1} & A_{n \minus{} 1} \\
A_{n \minus{} 1} & B_{n \minus{} 1} \\
\end{array} \right) \ \textrm{and} \ B_n \equal{} \left( \begin{array}{cc} A_{n \minus{} 1} & A_{n \minus{} 1} \\
A_{n \minus{} 1} & 0 \\
\end{array} \right).
\]
Denote by $ S(M)$ the sum of all the elements of a matrix $ M$. Prove that $ S(A_n^{k \minus{} 1}) \equal{} S(A_k^{n \minus{} 1})$, for all $ n,k\geq 2$.
The harmonic table is a triangular array:
$1$
$\frac 12 \qquad \frac 12$
$\frac 13 \qquad \frac 16 \qquad \frac 13$
$\frac 14 \qquad \frac 1{12} \qquad \frac 1{12} \qquad \frac 14$
Where $a_{n,1} = \frac 1n$ and $a_{n,k+1} = a_{n-1,k} - a_{n,k}$ for $1 \leq k \leq n-1.$ Find the harmonic mean of the $1985^{th}$ row.
The sequence $ \{x_{n}\}$ is defined by $ x_{1} \equal{} 2,x_{2} \equal{} 12$, and $ x_{n \plus{} 2} \equal{} 6x_{n \plus{} 1} \minus{} x_{n}$, $ (n \equal{} 1,2,\ldots)$. Let $ p$ be an odd prime number, let $ q$ be a prime divisor of $ x_{p}$. Prove that if $ q\neq2,3,$ then $ q\geq 2p \minus{} 1$.
Find all functions $f:(0,\infty)\to(0,\infty)$ such that
$$f(f(f(x)))+4f(f(x))+f(x)=6x.$$
Let $ n > 1$ be an odd positive integer and $ A = (a_{ij})_{i, j = 1..n}$ be the $ n \times n$ matrix with
\[ a_{ij}= \begin{cases}2 & \text{if }i = j \\ 1 & \text{if }i-j \equiv \pm 2 \pmod n \\ 0 & \text{otherwise}\end{cases}.\]
Find $ \det A$.
Show that there is an infinite sequence $a_1,a_2,...$ of natural numbers such that $a^2_1+a^2_2+ ...+a^2_N$ is a perfect square for all $N$. Give a recurrent formula for one such sequence.
Let $a_1=1$, $a_2=2^{a_1}$, $a_3=3^{a_2}$, $a_4=4^{a_3}$, $\dots$, $a_9 = 9^{a_8}$. Find the last two digits of $a_9$.
There is no sequence $x_n$ strictly increasing with terms natural numbers such that : $$ x_n+x_{k}=x_{nk}, \ \ for \, any \,\,\, n, k \in \mathbb{N}^*$$
For each positive integer $ n$, evaluate the sum
\[ \sum_{k\equal{}0}^{2n}(\minus{}1)^{k}\frac{\binom{4n}{2k}}{\binom{2n}{k}}\]
Let $p$ be a prime, and let $a_1, a_2, a_3, . . .$ be a sequence of positive integers so that $a_na_{n+2} = a^2_{n+1} + p$ for all positive integers $n$. Show that $a_{n+1}$ divides $a_n + a_{n+2}$ for all positive integers $n$.