Olympiad problems

2015 Argentina National Olympiad, 2

Find all pairs of natural numbers $a,b$ , with $a\ne b$ , such that $a+b$ and $ab+1$ are powers of $2$.

2012 Tournament of Towns, 1

Tags: digit , power of 2 , number theory

The decimal representation of an integer uses only two different digits. The number is at least $10$ digits long, and any two neighbouring digits are distinct. What is the greatest power of two that can divide this number?

2013 Tournament of Towns, 1

Tags: number theory , power of 2 , sum

Several positive integers are written on a blackboard. The sum of any two of them is some power of two (for example, $2, 4, 8,...$). What is the maximal possible number of different integers on the blackboard?

2016 Portugal MO, 5

Tags: power of 2 , number theory

Determine all natural numbers $x, y$ and $z$ such that the number $2^x +4^y +8^z +16^2$ is a power of $2$.

1978 Bundeswettbewerb Mathematik, 3

Tags: division , sum , power of 2 , number theory

For every positive integer $n$, define the remainder sum $r(n)$ as the sum of the remainders upon division of $n$ by each of the numbers $1$ through $n$. Prove that $r(2^{k}-1) =r(2^{k})$ for every $k\geq 1.$

2003 Belarusian National Olympiad, 6

Tags: power of 2 , number theory

a) A positive integer is called [i]nice [/i] if it can be represented as an arithmetic mean of some (not necessarily distinct) positive integers each being a nonnegative power of $2$. Prove that all positive integers are nice. b) A positive integer is called [i]ugly [/i] if it can not be represented as an arithmetic mean of some pairwise distinct positive integers each being a nonnegative power of $2$. Prove that there exist infinitely many ugly positive integers. (A. Romanenko, D. Zmeikov)

1984 IMO, 3

Tags: equation , divisibility , power of 2 , number theory

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 $a=1$.

1939 Eotvos Mathematical Competition, 2

Tags: factorial , power of 2 , divide , number theory

Determine the highest power of $2$ that divides $2^n!$.

Mathley 2014-15, 8

Tags: divisible , power of 2 , divide , number theory

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

2005 China Team Selection Test, 3

Tags: modular arithmetic , number theory , binomial expansion , fermat number , power of 2

$n$ 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 $2^{n+2}(n+1)$.

Kvant 2019, M2568

Tags: power of 2 , combinatorics

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.

2016 India IMO Training Camp, 1

Tags: number theory , recurrence relation , power of 2 , geometry , bir tinga qimmat ekan boru

Let $n$ be a natural number. We define sequences $\langle a_i\rangle$ and $\langle b_i\rangle$ of integers as follows. We let $a_0=1$ and $b_0=n$. For $i>0$, we let $$\left( a_i,b_i\right)=\begin{cases} \left(2a_{i-1}+1,b_{i-1}-a_{i-1}-1\right) & \text{if } a_{i-1}<b_{i-1},\\ \left( a_{i-1}-b_{i-1}-1,2b_{i-1}+1\right) & \text{if } a_{i-1}>b_{i-1},\\ \left(a_{i-1},b_{i-1}\right) & \text{if } a_{i-1}=b_{i-1}.\end{cases}$$ Given that $a_k=b_k$ for some natural number $k$, prove that $n+3$ is a power of two.

2007 Regional Olympiad of Mexico Center Zone, 4

Tags: number theory , power of 2

Is there a power of $2$ that when written in the decimal system has all its digits different from zero and it is possible to reorder them to form another power of $2$?

2018 Saudi Arabia BMO TST, 1

Tags: number theory , power of 2

Find the smallest positive integer $n$ which can not be expressed as $n =\frac{2^a - 2^b}{2^c - 2^d}$ for some positive integers $a, b, c, d$

2000 Czech and Slovak Match, 3

Tags: number theory , divisor , power of 2

Let $n$ be a positive integer. Prove that $n$ is a power of two if and only if there exists an integer $m$ such that $2^n-1$ is a divisor of $m^2 +9$.

2005 Germany Team Selection Test, 2

Tags: algebra , polynomial , function , power of 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$.

1976 IMO Longlists, 13

Tags: algebra , sequence , recurrence relation , floor function , power of 2

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 $\leq x)$

1976 IMO, 3

Tags: algebra , sequence , recurrence relation , floor function , power of 2

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 $\leq x)$

2019 Caucasus Mathematical Olympiad, 7

Tags: power of 2 , combinatorics

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.

2004 Tournament Of Towns, 1

Tags: arithmetic progression , sum , power of 2 , number theory

The sum of all terms of a finite arithmetical progression of integers is a power of two. Prove that the number of terms is also a power of two.

2010 Mathcenter Contest, 2

Tags: number theory , digit , 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]

1998 All-Russian Olympiad Regional Round, 10.4

Tags: combinatorics , number theory , power of 2

In the first $1999$ cells of the computer are written numbers in the specified order:: $1$, $2$, $4$,$... $, $2^{1998}$. Two programmers take turns reducing in one move per unit number in five different cells. If a negative number appears in one of the cells, then the computer breaks down and the broken repairs are paid for. Which programmer can protect himself from financial losses, regardless of his partner’s moves, and how should he do this act?

2007 Cuba MO, 5

Tags: number theory , digit , divide , power of 2

Prove that there is a unique positive integer formed only by the digits $2$ and $5$, which has $ 2007$ digits and is divisible by $2^{2007}$.

2023 Euler Olympiad, Round 2, 1

Tags: algebra , euler , 100 , power of 2 , residue , number theory

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]

2010 Dutch Mathematical Olympiad, 2

Tags: number theory , power of 2 , consecutive

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.