Found problems: 5923
[color=blue][size=150]PERU TST IMO - 2006[/size]
Saturday, may 20.[/color]
[b]Question 03[/b]
In each square of a board drawn into squares of $2^n$ rows and
$n$ columns $(n\geq 1)$ are written a 1 or a -1, in such a way
that the rows of the board constitute all the possible sequences
of length $n$ that they are possible to be formed with numbers 1
and -1.
Next, some of the numbers are replaced by zeros.
Prove that it is possible to choose some of the rows of the board
(It could be a row only) so that in the chosen rows, is fulfilled that the
sum of the numbers in each column is zero.
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[url=http://www.mathlinks.ro/Forum/viewtopic.php?t=88511]Spanish version[/url]
$\text{\LaTeX}{}$ed by carlosbr
On a blackboard are written all the integers from $1$ to $2008$ inclusive. Two numbers are deleted and their difference is written. For example, if you erase $5$ and $241$, you write $236$. This continues, erasing two numbers and writing their difference, until only one number remains. Determine if the number left at the end can be $2008$. What about $2007$? In each case, if the answer is affirmative, indicate a sequence with that final number, and if it is negative, explain why.
Each term of the sequence $5, 12, 19, 26, \cdots$ is $7$ more than the term that precedes it. What is the first term of the sequence that is greater than $2017$?
$\text{(A) }2018\qquad\text{(B) }2019\qquad\text{(C) }2020\qquad\text{(D) }2021\qquad\text{(E) }2022$
Define the sequence $(a_n)$ as $a_1 = 1$, $a_{2n} = a_n$ and $a_{2n+1} = a_n + 1$ for all $n\geq 1$.
a) Find all positive integers $n$ such that $a_{kn} = a_n$ for all integers $1 \leq k \leq n$.
b) Prove that there exist infinitely many positive integers $m$ such that $a_{km} \geq a_m$ for all positive integers $k$.
A chessboard has 64 cells painted black and white in the usual way.
A bishop path is a sequence of distinct cells such that two consecutive cells have
exactly one common point. At least how many bishop paths can the set of all white
cells be divided into?
Consider the sequence $(a_n)_{n\ge 1}$ such that $a_1=1$ and $a_{n+1}=\sqrt{a_n+n^2}$, $\forall n\ge 1$.
$\textbf{(a)}$ Prove that there is exactly one rational number among the numbers $a_1,a_2,a_3,\dots$.
$\textbf{(b)}$ Consider the sequence $(S_n)_{n\ge 1}$ such that
$$S_n=\sum_{i=1}^n\frac{4}{\left (\left \lfloor a_{i+1}^2\right \rfloor-\left \lfloor a_i^2\right \rfloor\right)\left(\left \lfloor a_{i+2}^2\right \rfloor-\left \lfloor a_{i+1}^2\right \rfloor\right)}.$$
Prove that there exists an integer $N$ such that $S_n>0.9$, $\forall n>N$.
[i] (Stefan Obadă)[/i]
In the sequence $1, 0, 1, 0, 1, 0, 3, 5, \cdots$, each member after the sixth one is equal to the last digit of the sum of the six members just preceeding it. Prove that in this sequence one cannot find the following group of six consecutive members: \[0, 1, 0, 1, 0, 1\]
The first term of a sequence is $2014$. Each succeeding term is the sum of the cubes of the digits of the previous term. What is the $2014$ th term of the sequence?
An integer sequence $\{a_{n}\}_{n \ge 1}$ is defined by \[a_{0}=0, \; a_{1}=1, \; a_{n+2}=2a_{n+1}+a_{n}\] Show that $2^{k}$ divides $a_{n}$ if and only if $2^{k}$ divides $n$.
The infinite sequence $a_0,a_1,\ldots$ is given by $a_1=\frac12$, $a_{n+1} = \sqrt{\frac{1+a_n}{2}}$. Determine the infinite product $a_1a_2a_3\cdots$.
Define the sequence $a_0,a_1,a_2,\hdots$ by $a_n=2^n+2^{\lfloor n/2\rfloor}$. Prove that there are infinitely many terms of the sequence which can be expressed as a sum of (two or more) distinct terms of the sequence, as well as infinitely many of those which cannot be expressed in such a way.
A sequence $a_1,a_2,...,a_{2007}$ where $a_i \in\{2,3\}$ for $i = 1,2,...,2007$ and an integer sequence $x_1,x_2,...,x_{2007}$ satis fies the following: $a_ix_i + x_{i+2 }\equiv 0$ ($mod 5$) , where the indices are taken modulo $2007$. Prove that $x_1,x_2,...,x_{2007}$ are all multiples of $5$.
Fibonacci sequences is defined as $f_1=1$,$f_2=2$, $f_{n+1}=f_{n}+f_{n-1}$ for $n \ge 2$.
a) Prove that every positive integer can be represented as sum of several distinct Fibonacci number.
b) A positive integer is called [i]Fib-unique[/i] if the way to represent it as sum of several distinct Fibonacci number is unique. Example: $13$ is not Fib-unique because $13 = 13 = 8 + 5 = 8 + 3 + 2$. Find all Fib-unique.
Given a sequence of positive integers $a_1, a_2, a_3, \dots, a_{n}$, define the \emph{power tower function} \[f(a_1, a_2, a_3, \dots, a_{n})=a_1^{a_2^{a_3^{\mathstrut^{ .^{.^{.^{a_{n}}}}}}}}.\] Let $b_1, b_2, b_3, \dots, b_{2017}$ be positive integers such that for any $i$ between 1 and 2017 inclusive, \[f(a_1, a_2, a_3, \dots, a_i, \dots, a_{2017})\equiv f(a_1, a_2, a_3, \dots, a_i+b_i, \dots, a_{2017}) \pmod{2017}\] for all sequences $a_1, a_2, a_3, \dots, a_{2017}$ of positive integers greater than 2017. Find the smallest possible value of $b_1+b_2+b_3+\dots+b_{2017}$.
[i]Proposed by Yannick Yao
Let $ \left( a_i \right)_{1\le i\le n} $ and $ \left( b_i \right)_{1\le i\le n} $ be two sequences, the former being a decreasing sequence and the latter being an increasing sequence. All the terms of $ \left( a_i \right)_{1\le i\le n} $ and $ \left( b_i \right)_{1\le i\le n} $ form the set $ \{1,2,3,\ldots ,2n \} . $ Prove that:
$$ \left| a_1-b_1 \right| +\left| a_2-b_2 \right| +\cdots +\left| a_n-b_n \right|=n^2 $$
In a “Fibonacci” sequence of numbers, each term beginning with the third, is the sum of the previous two terms. The first number in such a sequence is 2 and the third is 9. What is the eighth term in the sequence?
$\textbf{(A)}\ 34 \qquad \textbf{(B)}\ 36 \qquad \textbf{(C)}\ 107 \qquad \textbf{(D)}\ 152 \qquad \textbf{(E)}\ 245$
The sequence $\{a_n\}_{n}$ satisfies the relations $a_1=a_2=1$ and for all positive integers $n$,
\[ a_{n+2} = \frac 1{a_{n+1}} + a_n . \]
Find $a_{2004}$.
In what case does the system of equations
$\begin{matrix} x + y + mz = a \\ x + my + z = b \\ mx + y + z = c \end{matrix}$
have a solution? Find conditions under which the unique solution of the above system is an arithmetic progression.
What relationship should be between the positive real numbers $ a $ and $ b $ such that the sequence $ \left(\left( a\sqrt[n]{n} +b \right)^{\frac{n}{\ln n}}\right)_{n\ge 1} $ has a nonzero and finite limit? For such $ a,b, $ calculate the limit of this sequence.
[i]Ion Cucurezeanu[/i]
The first five terms of a sequence are $1, 2, 3, 4$ and $5$. From the sixth term on, each term is $1$ less than the product of all the proceeding ones. Prove that the product of the first$ 70$ terms is equal to the sum of their squares.
(LD Kurliandchik)
Let $f$ be a real-valued function defined on $I = (0,+\infty)$ and having no zeros on $I$. Suppose that
\[\lim_{x \to +\infty} \frac{f'(x)}{f(x)}=+\infty.\]
For the sequence $u_n = \ln \left| \frac{f(n+1)}{f(n)} \right|$, prove that $u_n \to +\infty$ as $n \to +\infty.$
A laticial cycle of length $n$ is a sequence of lattice points $(x_k, y_k)$, $k = 0, 1,\cdots, n$, such that $(x_0, y_0) = (x_n, y_n) = (0, 0)$ and $|x_{k+1} -x_{k}|+|y_{k+1} - y_{k}| = 1$ for each $k$. Prove that for all $n$, the number of latticial cycles of length $n$ is a perfect square.
Hari is obsessed with cubics. He comes up with a cubic with leading coefficient 1, rational coefficients and real roots $0 < a < b < c < 1$. He knows the following three facts: $P(0) = -\frac{1}{8}$, the roots form a geometric progression in the order $a,b,c$, and \[ \sum_{k=1}^{\infty} (a^k + b^k + c^k) = \dfrac{9}{2} \] The value $a + b + c$ can be expressed as $\frac{m}{n}$, where $m,n$ are relatively prime positive integers. Find $m + n$.
[i]Proposed by Akshar Yeccherla (TopNotchMath)[/i]
Let $f$ be a bijection of the set of all natural numbers on to itself. Prove that there exists positive integers $a < a+d < a+ 2d$ such that $f(a) < f(a+d) <f(a+2d)$
Consider the sequence $(a_n)_{n\in \mathbb{N}}$ with $a_0=a_1=a_2=a_3=1$ and $a_na_{n-4}=a_{n-1}a_{n-3} + a^2_{n-2}$. Prove that all the terms of this sequence are integer numbers.