This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 86

1961 Poland - Second Round, 1
Tags: number theory , consecutive , power of 2
Prove that no number of the form $ 2^n $, where $ n $ is a natural number, is the sum of two or more consecutive natural numbers.

1984 IMO Longlists, 43
Tags: number theory , equation , divisibility , 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 $a=1$.

1997 IMO, 6
Tags: combinatorics , counting , integer partition , power of 2
For each positive integer $ n$, let $ f(n)$ denote the number of ways of representing $ n$ as a sum of powers of 2 with nonnegative integer exponents. Representations which differ only in the ordering of their summands are considered to be the same. For instance, $ f(4) \equal{} 4$, because the number 4 can be represented in the following four ways: 4; 2+2; 2+1+1; 1+1+1+1. Prove that, for any integer $ n \geq 3$ we have $ 2^{\frac {n^2}{4}} < f(2^n) < 2^{\frac {n^2}2}$.

2012 Tournament of Towns, 1
Tags: digit , number theory , power of 2
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?

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.

2010 Denmark MO - Mohr Contest, 4
Tags: number theory , power of 2
It is stated that $2^{2010}$ is a $606$-digit number that begins with $1$. How many of the numbers $1, 2,2^2,2^3, ..., 2^{2009}$ start with $4$?

1939 Eotvos Mathematical Competition, 2
Tags: number theory , factorial , power of 2 , divide
Determine the highest power of $2$ that divides $2^n!$.

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?

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 $\leq x)$

2004 Estonia Team Selection Test, 5
Tags: number theory , lcm , power of 2 , divisor
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 $k$.

2016 Silk Road, 4
Tags: binary representation , number theory , power of 2 , combinatorics
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]

1999 Estonia National Olympiad, 1
Tags: number theory , remainder , power of 2 , power of 3
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 $16$.

2000 Chile National Olympiad, 5
Tags: power of 2 , divide , divisible , number theory
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 $2^n$.

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.

1979 Austrian-Polish Competition, 9
Tags: floor function , power of 2 , number theory
Find the greatest power of $2$ that divides $a_n = [(3+\sqrt{11} )^{2n+1}]$, where $n$ is a given positive integer.

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 $k\geq 1.$

VMEO III 2006 Shortlist, N1
Tags: number theory , 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 $1 \le j \le i \le 2006$ ?

2009 Mathcenter Contest, 5
Tags: number theory , digit , power of 2
For $n\in\mathbb{N}$, prove that $2^n$ can begin with any sequence of digits. Hint: $\log 2$ is irrational number.

1997 IMO Shortlist, 24
Tags: combinatorics , counting , integer partition , power of 2
For each positive integer $ n$, let $ f(n)$ denote the number of ways of representing $ n$ as a sum of powers of 2 with nonnegative integer exponents. Representations which differ only in the ordering of their summands are considered to be the same. For instance, $ f(4) \equal{} 4$, because the number 4 can be represented in the following four ways: 4; 2+2; 2+1+1; 1+1+1+1. Prove that, for any integer $ n \geq 3$ we have $ 2^{\frac {n^2}{4}} < f(2^n) < 2^{\frac {n^2}2}$.

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 $m^2 +9$.

2002 Tuymaada Olympiad, 3
Tags: algebra , polynomial , integer polynomial , trinomial , quadratic trinomial , power of 2
Is there a quadratic trinomial with integer coefficients, such that all values which are natural to be natural powers of two?

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]

2024 Ukraine National Mathematical Olympiad, Problem 2
Tags: number theory , algebra , power of 2
You are given a positive integer $n$. Find the smallest positive integer $k$, for which there exist integers $a_1, a_2, \ldots, a_k$, for which the following equality holds: $$2^{a_1} + 2^{a_2} + \ldots + 2^{a_k} = 2^n - n + k$$ [i]Proposed by Mykhailo Shtandenko[/i]

1988 Tournament Of Towns, (171) 4
Tags: combinatorics , power of 2 , weighing
We have a set of weights with masses $1$ gm, $2$ gm, $4$ gm and so on, all values being powers of $2$ . Some of these weights may have equal mass. Some weights were put on both sides of a balance beam, resulting in equilibrium. It is known that on the left hand side all weights were distinct . Prove that on the right hand side there were no fewer weights than on the left hand side.

1981 Swedish Mathematical Competition, 6
Tags: number theory , perfect square , power of 2 , triangle inequality
Show that there are infinitely many triangles with side lengths $a$, $b$, $c$, where $a$ is a prime, $b$ is a power of $2$ and $c$ is the square of an odd integer.