Found problems: 86
For every $n$ positive integers we denote $$\frac{x_n}{y_n}=\sum_{k=1}^{n}{\frac{1}{k {n \choose k}}}$$
where $x_n, y_n$ are coprime positive integers. Prove that $y_n$ is not divisible by $2^n$ for any positive integers $n$.
Ha Duy Hung, high school specializing in the Ha University of Education, Hanoi, Xuan Thuy, Cau Giay, Hanoi
15 boxes are given. They all are initially empty. By one move it is allowed to choose some boxes and to put in them numbers of apricots which are pairwise distinct powers of 2. Find the least positive integer $k$ such that it is possible
to have equal numbers of apricots in all the boxes after $k$ moves.
Let $n$ be a positive number. Prove that there exists an integer $N =\overline{m_1m_2...m_n}$ with $m_i \in \{1, 2\}$ which is divisible by $2^n$.
Consider a sequence of 100 positive integers. Each member of the sequence, starting from the second one, is derived by either multiplying the previous number by 2 or dividing it by 16. Is it possible for the sum of these 100 numbers to be equal to $2^{2023}$?
[i]Proposed by Nika Glunchadze, Georgia[/i]
Let $n \ge 1$. Prove that it is possible to select some of the integers $1,2,...,2^n$ so that for each $p = 0,1,...,n - 1$ the sum of the $p$-th powers of the selected numbers is equal to the sum of the $p$-th powers of the remaining numbers.
15 boxes are given. They all are initially empty. By one move it is allowed to choose some boxes and to put in them numbers of apricots which are pairwise distinct powers of 2. Find the least positive integer $k$ such that it is possible
to have equal numbers of apricots in all the boxes after $k$ moves.
Find all integers of the form $2^n$ (where $n$ is a natural number) such that after deleting the first digit of its decimal representation we again get a power of $2$.
Let $k$ and $d$ be integers such that $k>1$ and $0\leq d<9$. Prove that there exists some integer $n$ such that the $k$th digit from the right of $2^n$ is $d$.
[i](tatari/nightmare)[/i]
A number is called polite if it can be written as $ m + (m+1)+...+ n$, for certain positive integers $ m <n$ . For example: $18$ is polite, since $18 =5 + 6 + 7$. A number is called a power of two if it can be written as $2^{\ell}$ for
some integer $\ell \ge 0$.
(a) Show that no number is both polite and a power of two.
(b) Show that every positive integer is polite or a power of two.
Find the greatest power of $2$ that divides $a_n = [(3+\sqrt{11} )^{2n+1}]$, where $n$ is a given positive integer.
Let $a, b, c$ and $d$ be non-negative integers. Prove that the numbers $2^a7^b$ and $2^c7^d$ give the same remainder when divided by $15$ iff the numbers $3^a5^b$ and $3^c5^d$ give the same remainder when divided by $16$.
Is there a power of $2$ such that it is possible to rearrange the digits, giving another power of $2$?
Let $a,b,c,d$ be odd integers such that $0<a<b<c<d$ and $ad=bc$. Prove that if $a+d=2^k$ and $b+c=2^m$ for some integers $k$ and $m$, then $a=1$.
Determine all natural numbers $x, y$ and $z$ such that the number $2^x +4^y +8^z +16^2$ is a power of $2$.
$f(n)$ denotes the largest integer $k$ such that that $2^k|n$.
$2006$ integers $a_i$ are such that $a_1<a_2<...<a_{2016}$.
Is it possible to find integers $k$ where $1 \le k\le 2006$ and $f(a_i-a_j)\ne k$ for every $1 \le j \le i \le 2006$ ?
Find the highest power of 2 that is a factor of the number $$ L_n = (n+1)(n+2)... 2n,$$ where $n$is a natural number.
Show that there are at least $3$ and at most $4$ powers of $2$ with $m$ digits. For which $m$ are there $4$?
1 - Prove that $55 < (1+\sqrt{3})^4 < 56$ .
2 - Find the largest power of $2$ that divides $\lceil(1+\sqrt{3})^{2n}\rceil$ for the positive integer $n$
Prove that for every natural $n$ there exists a number, containing only digits "$1$" and "$2$" in its decimal notation, that is divisible by $2^n$ ( $n$-th power of two ).
Find all natural numbers $n$ for which the number of all positive divisors of the number lcm $(1,2,..., n)$ is equal to $2^k$ for some non-negative integer $k$.
$n$ is a positive integer, $F_n=2^{2^{n}}+1$. Prove that for $n \geq 3$, there exists a prime factor of $F_n$ which is larger than $2^{n+2}(n+1)$.
Determine the smallest integer $k > 1$ such that there exist $k$ distinct primes whose squares sum to a power of $2$.
Does there exist a positive odd integer $n$ so that there are primes $p_1$, $p_2$ dividing $2^n-1$ with $p_1-p_2=2$?
For any positive integer $ n$, prove that there exists a polynomial $ P$ of degree $ n$ such that all coeffients of this polynomial $ P$ are integers, and such that the numbers $ P\left(0\right)$, $ P\left(1\right)$, $ P\left(2\right)$, ..., $ P\left(n\right)$ are pairwisely distinct powers of $ 2$.
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 $k$ .
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