Found problems: 27
Each pair $(x, y)$ of nonnegative integers is assigned number $f(x, y)$ according the conditions:
$f(0, 0) = 0$;
$f(2x, 2y) = f(2x + 1, 2y + 1) = f(x, y)$,
$f(2x + 1, 2y) = f(2x, 2y + 1) = f(x ,y) + 1$ for $x, y \ge 0$.
Let $n$ be a fixed nonnegative integer and let $a$, $b$ be nonnegative integers such that $f(a, b) = n$. Decide how many numbers satisfy the equation $f(a, x) + f(b, x) = n$.
For all rational $x$ satisfying $0 \leq x < 1$, the functions $f$ is defined by
\[f(x)=\begin{cases}\frac{f(2x)}{4},&\mbox{for }0 \leq x < \frac 12,\\ \frac 34+ \frac{f(2x - 1)}{4}, & \mbox{for } \frac 12 \leq x < 1.\end{cases}\]
Given that $x = 0.b_1b_2b_3 \cdots $ is the binary representation of $x$, find, with proof, $f(x)$.
The sequence $a_i$ is defined as $a_1 = 2, a_2 = 3$, and
$a_{n+1} = 2a_{n-1}$ or $a_{n+1} = 3a_n - 2a_{n-1}$ for all integers $n \ge 2$.
Prove that no term in $a_i$ is in the range $[1612, 2012]$.
Let $ n$ be a composite natural number and $ p$ a proper divisor of $ n.$ Find the binary representation of the smallest natural number $ N$ such that
\[ \frac{(1 \plus{} 2^p \plus{} 2^{n\minus{}p})N \minus{} 1}{2^n}\]
is an integer.
A function $ f$ defined on the positive integers (and taking positive integers values) is given by:
$ \begin{matrix} f(1) \equal{} 1, f(3) \equal{} 3 \\
f(2 \cdot n) \equal{} f(n) \\
f(4 \cdot n \plus{} 1) \equal{} 2 \cdot f(2 \cdot n \plus{} 1) \minus{} f(n) \\
f(4 \cdot n \plus{} 3) \equal{} 3 \cdot f(2 \cdot n \plus{} 1) \minus{} 2 \cdot f(n), \end{matrix}$
for all positive integers $ n.$ Determine with proof the number of positive integers $ \leq 1988$ for which $ f(n) \equal{} n.$
Let the base $2$ representation of $x\in[0;1)$ be $x=\sum_{i=0}^\infty \frac{x_i}{2^{i+1}}$. (If $x$ is dyadically rational, i.e. $x\in\left\{\frac{k}{2^n}\,:\, k,n\in\mathbb{Z}\right\}$, then we choose the finite representation.) Define function $f_n:[0;1)\to\mathbb{Z}$ by
$$f_n(x)=\sum_{j=0}^{n-1}(-1)^{\sum_{i=0}^j x_i}.$$Does there exist a function $\varphi:[0;\infty)\to[0;\infty)$ such that $\lim_{x\to\infty} \varphi(x)=\infty$ and
$$\sup_{n\in\mathbb{N}}\int_0^1 \varphi(|f_n(x)|)\mathrm{d}x<\infty\, ?$$
A function $ f$ defined on the positive integers (and taking positive integers values) is given by:
$ \begin{matrix} f(1) \equal{} 1, f(3) \equal{} 3 \\
f(2 \cdot n) \equal{} f(n) \\
f(4 \cdot n \plus{} 1) \equal{} 2 \cdot f(2 \cdot n \plus{} 1) \minus{} f(n) \\
f(4 \cdot n \plus{} 3) \equal{} 3 \cdot f(2 \cdot n \plus{} 1) \minus{} 2 \cdot f(n), \end{matrix}$
for all positive integers $ n.$ Determine with proof the number of positive integers $ \leq 1988$ for which $ f(n) \equal{} n.$
A function $ f$ defined on the positive integers (and taking positive integers values) is given by:
$ \begin{matrix} f(1) \equal{} 1, f(3) \equal{} 3 \\
f(2 \cdot n) \equal{} f(n) \\
f(4 \cdot n \plus{} 1) \equal{} 2 \cdot f(2 \cdot n \plus{} 1) \minus{} f(n) \\
f(4 \cdot n \plus{} 3) \equal{} 3 \cdot f(2 \cdot n \plus{} 1) \minus{} 2 \cdot f(n), \end{matrix}$
for all positive integers $ n.$ Determine with proof the number of positive integers $ \leq 1988$ for which $ f(n) \equal{} n.$
Consider the function $ f: \mathbb{N}_0\to\mathbb{N}_0$, where $ \mathbb{N}_0$ is the set of all non-negative
integers, defined by the following conditions :
$ (i)$ $ f(0) \equal{} 0$; $ (ii)$ $ f(2n) \equal{} 2f(n)$ and $ (iii)$ $ f(2n \plus{} 1) \equal{} n \plus{} 2f(n)$ for all $ n\geq 0$.
$ (a)$ Determine the three sets $ L \equal{} \{ n | f(n) < f(n \plus{} 1) \}$, $ E \equal{} \{n | f(n) \equal{} f(n \plus{} 1) \}$, and $ G \equal{} \{n | f(n) > f(n \plus{} 1) \}$.
$ (b)$ For each $ k \geq 0$, find a formula for $ a_k \equal{} \max\{f(n) : 0 \leq n \leq 2^k\}$ in terms of $ k$.
Let $P(n)$ be the number of ways to split a natural number $n$ to the sum of powers of two, when the order does not matter. For example $P(5) = 4$, as $5=4+1=2+2+1=2+1+1+1=1+1+1+1+1$. Prove that for any natural the identity $P(n) + (-1)^{a_1} P(n-1) + (-1)^{a_2} P(n-2) + \ldots + (-1)^{a_{n-1}}
P(1) + (-1)^{a_n} = 0,$ is true, where $a_k$ is the number of units in the binary number record $k$ .
[url=http://matol.kz/comments/2720/show]source[/url]
Let $ \alpha(n)$ be the number of digits equal to one in the binary representation of a positive integer $ n.$ Prove that:
(a) the inequality $ \alpha(n) (n^2 ) \leq \frac{1}{2} \alpha(n)(\alpha(n) + 1)$ holds;
(b) the above inequality is an equality for infinitely many positive integers, and
(c) there exists a sequence $ (n_i )^{\infty}_1$ such that $ \frac{\alpha ( n^2_i )}{\alpha (n_i }$
goes to zero as $ i$ goes to $ \infty.$
[i]Alternative problem:[/i] Prove that there exists a sequence a sequence $ (n_i )^{\infty}_1$ such that $ \frac{\alpha ( n^2_i )}{\alpha (n_i )}$
(d) $ \infty;$
(e) an arbitrary real number $ \gamma \in (0,1)$;
(f) an arbitrary real number $ \gamma \geq 0$;
as $ i$ goes to $ \infty.$
Let $x, y$ and $k$ be three positive integers. Prove that there exist a positive integer $N$ and a set of $k + 1$ positive integers $\{b_0,b_1, b_2, ... ,b_k\}$, such that, for every $i = 0, 1, ... , k$ , the $b_i$-ary expansion of $N$ is a $3$-digit palindrome, and the $b_0$-ary expansion is exactly $\overline{\mbox{xyx}}$.
proposed by Bojan Basic, Serbia
Prova that any integer $ Z $ has a unique representation
$$ a_0+a_12+a_22^2+a_32^3+\cdots +a_n2^n, $$
where $ n $ is natural, $ a_i\in\{ -1,0,+1\} $ for $ i=\overline{0,n} $ and $ a_ka_{k-1}=0 $ for $ k=\overline{1,n} . $
The sequence $a_i$ is defined as $a_1 = 2, a_2 = 3$, and
$a_{n+1} = 2a_{n-1}$ or $a_{n+1} = 3a_n - 2a_{n-1}$ for all integers $n \ge 2$.
Prove that no term in $a_i$ is in the range $[1612, 2012]$.
Let $ n$ be a composite natural number and $ p$ a proper divisor of $ n.$ Find the binary representation of the smallest natural number $ N$ such that
\[ \frac{(1 \plus{} 2^p \plus{} 2^{n\minus{}p})N \minus{} 1}{2^n}\]
is an integer.
Find, with proof, the least positive integer $n$ having the following property: in the binary representation of $\frac 1n$, all the binary representations of $1, 2, \ldots, 1990$ (each consist of consecutive digits) are appeared after the decimal point.
Let $\mathcal{F}$ be the family of all nonempty finite subsets of $\mathbb{N} \cup \{0\}.$ Find all real numbers $a$ for which the series
$$\sum_{A \in \mathcal{F}} \frac{1}{\sum_{k \in A}a^k}$$
is convergent.
For each positive integer $n$, let $a_n = 0$ (or $1$) if the number of $1$’s in the binary representation of $n$ is even (or
odd), respectively. Show that there do not exist positive integers $k$ and $m$ such that
$$a_{k+j}=a_{k+m+j} =a_{k+2m+j}$$
for $0 \leq j \leq m-1.$
For all rational $x$ satisfying $0 \leq x < 1$, the functions $f$ is defined by
\[f(x)=\begin{cases}\frac{f(2x)}{4},&\mbox{for }0 \leq x < \frac 12,\\ \frac 34+ \frac{f(2x - 1)}{4}, & \mbox{for } \frac 12 \leq x < 1.\end{cases}\]
Given that $x = 0.b_1b_2b_3 \cdots $ is the binary representation of $x$, find, with proof, $f(x)$.
Player Zero and Player One play a game on a $n \times n$ board ($n \ge 1$). The columns of this $n \times n$ board are numbered $1,2,4,\dots,2^{n-1}$. Turn my turn, the players put their own number in one of the free cells (thus Player Zero puts a $0$ and Player One puts a $1$). Player Zero begins. When the board is filled, the game ends and each row yields a (reverse binary) number obtained by adding the values of the columns with a $1$ in that row. For instance, when $n=4$, a row with $0101$ yields the number $0 \cdot1+1 \cdot 2+0 \cdot 4+1 \cdot 8=10$.
a) For which natural numbers $n$ can Player One always ensure that at least one of the row numbers is divisible by $4$?
b) For which natural numbers $n$ can Player One always ensure that at least one of the row numbers is divisible by $3$?
For any positive integer $ k$, let $ f_k$ be the number of elements in the set $ \{ k \plus{} 1, k \plus{} 2, \ldots, 2k\}$ whose base 2 representation contains exactly three 1s.
(a) Prove that for any positive integer $ m$, there exists at least one positive integer $ k$ such that $ f(k) \equal{} m$.
(b) Determine all positive integers $ m$ for which there exists [i]exactly one[/i] $ k$ with $ f(k) \equal{} m$.
Let $\sqrt 3 = 1.b_1b_2b_3 \dots _{(2)}$ be the binary representation of $\sqrt 3$. Prove that for any positive integer $n$, at least one of the digits $b_n$, $b_{n+1}$, $\dots$, $b_{2n}$ equals $1$.
Given $n$ numbers $a_1,a_2,...,a_n$ ($n\geq 3$) where $a_i\in\{0,1\}$ for all $i=1,2.,,,.n$. Consider $n$ following $n$-tuples \[ \begin{aligned} S_1 & =(a_1,a_2,...,a_{n-1},a_n)\\ S_2 & =(a_2,a_3,...,a_n,a_1)\\ & \vdots\\ S_n & =(a_n,a_1,...,a_{n-2},a_{n-1}).\end{aligned}\] For each tuple $r=(b_1,b_2,...,b_n)$, let \[ \omega (r)=b_1\cdot 2^{n-1}+b_2\cdot 2^{n-2}+\cdots+b_n. \] Assume that the numbers $\omega (S_1),\omega (S_2),...,\omega (S_n)$ receive exactly $k$ different values.
a) Prove that $k|n$ and $\frac{2^n-1}{2^k-1}|\omega (S_i)\quad\forall i=1,2,...,n.$
b) Let \[ \begin{aligned} M & =\max _{i=\overline{1,n}}\omega (S_i)\\ m & =\min _{i=\overline{1,n}}\omega (S_i). \end{aligned} \] Prove that \[ M-m\geq\frac{(2^n-1)(2^{k-1}-1)}{2^k-1}. \]
For any positive integer $ k$, let $ f_k$ be the number of elements in the set $ \{ k \plus{} 1, k \plus{} 2, \ldots, 2k\}$ whose base 2 representation contains exactly three 1s.
(a) Prove that for any positive integer $ m$, there exists at least one positive integer $ k$ such that $ f(k) \equal{} m$.
(b) Determine all positive integers $ m$ for which there exists [i]exactly one[/i] $ k$ with $ f(k) \equal{} m$.
We call the [i]tribi [/i] of a positive integer $k$ (denoted $T(k)$) the number of all pairs $11$ in the binary representation of $k$. e.g $$T(1)=T(2)=0,\, T(3)=1, \,T(4)=T(5)=0,\,T(6)=1,\,T(7)=2.$$
Calculate $S_n=\sum_{k=1}^{2^n}T(K)$.