Olympiad problems

2005 China Team Selection Test, 1

Let $ b, m, n$ be positive integers such that $ b > 1$ and $ m \neq n.$ Prove that if $ b^m \minus{} 1$ and $ b^n \minus{} 1$ have the same prime divisors, then $ b \plus{} 1$ is a power of 2.

1967 Polish MO Finals, 1

Tags: number theory , power of 2

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.

1993 Tournament Of Towns, (369) 1

Tags: number theory , power of 2 , digit

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 $2$.

1998 Rioplatense Mathematical Olympiad, Level 3, 4

Tags: number theory , power of 2

Let $M$ be a subset of $\{1,2,..., 1998\}$ with $1000$ elements. Prove that it is always possible to find two elements $a$ and $b$ in $M$, not necessarily distinct, such that $a + b$ is a power of $2$.

1969 IMO Shortlist, 61

Tags: function , algebra , binomial theorem , number theory , power of 2

$(SWE 4)$ Let $a_0, a_1, a_2, \cdots$ be determined with $a_0 = 0, a_{n+1} = 2a_n + 2^n$. Prove that if $n$ is power of $2$, then so is $a_n$

2008 Postal Coaching, 2

Tags: floor function , power of 2 , number theory

Show that if $n \ge 4, n \in N$ and $\big [ \frac{2^n}{n} ]$ is a power of $2$, then $n$ is a power of $2$.

2016 Thailand Mathematical Olympiad, 6

Tags: prime , power of 2 , number theory

Let $m$ and $n$ be positive integers. Prove that if $m^{4^n+1} - 1$ is a prime number, then there exists an integer $t \ge 0$ such that $n = 2^t$.

2015 Balkan MO Shortlist, N5

Tags: number theory , maximum , power of 2 , divide , product

For a positive integer $s$, denote with $v_2(s)$ the maximum power of $2$ that divides $s$. Prove that for any positive integer $m$ that: $$v_2\left(\prod_{n=1}^{2^m}\binom{2n}{n}\right)=m2^{m-1}+1.$$ (FYROM)

2010 Contests, 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.

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.

STEMS 2021 Math Cat B, Q2

Tags: polynomial , power of 2 , prime number , divisibility , kobayashi

Determine all non-constant monic polynomials $P(x)$ with integer coefficients such that no prime $p>10^{100}$ divides any number of the form $P(2^n)$

1976 IMO Shortlist, 4

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

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)$.$

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?

1981 Brazil National Olympiad, 2

Tags: digit , power of 2 , number theory

Show that there are at least $3$ and at most $4$ powers of $2$ with $m$ digits. For which $m$ are there $4$?

1992 IMO Longlists, 46

Tags: number theory , power of 2 , arithmetic sequence

Prove that the sequence $5, 12, 19, 26, 33,\cdots $ contains no term of the form $2^n -1.$

2001 Switzerland Team Selection Test, 10

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

Prove that every $1000$-element subset $M$ of the set $\{0,1,...,2001\}$ contains either a power of two or two distinct numbers whose sum is a power of two.

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$.

2023 Germany Team Selection Test, 1

Tags: power of 2 , number theory , prime

Does there exist a positive odd integer $n$ so that there are primes $p_1$, $p_2$ dividing $2^n-1$ with $p_1-p_2=2$?

1971 All Soviet Union Mathematical Olympiad, 144

Tags: number theory , power of 2 , digit

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 ).

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.

1994 All-Russian Olympiad, 5

Tags: recursive , sequence , power of 2 , number theory

Let $a_1$ be a natural number not divisible by $5$. The sequence $a_1,a_2,a_3, . . .$ is defined by $a_{n+1} =a_n+b_n$, where $b_n$ is the last digit of $a_n$. Prove that the sequence contains infinitely many powers of two. (N. Agakhanov)

1988 Tournament Of Towns, (194) 1

Tags: number theory , power of 2 , digit

Is there a power of $2$ such that it is possible to rearrange the digits, giving another power of $2$?

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 $1 \le j \le i \le 2006$ ?

1997 IMO Shortlist, 14

Tags: number theory , prime number , divisibility , power of 2 , prime divisor

Let $ b, m, n$ be positive integers such that $ b > 1$ and $ m \neq n.$ Prove that if $ b^m \minus{} 1$ and $ b^n \minus{} 1$ have the same prime divisors, then $ b \plus{} 1$ is a power of 2.

1968 Spain Mathematical Olympiad, 7

Tags: number theory , digit , power of 2

In the sequence of powers of $2$ (written in the decimal system, beginning with $2^1 = 2$) there are three terms of one digit, another three of two digits, another three of $3$, four out of $4$, three out of $5$, etc. Clearly reason the answers to the following questions: a) Can there be only two terms with a certain number of digits? b) Can there be five consecutive terms with the same number of digits? c) Can there be four terms of n digits, followed by four with $n + 1$ digits? d) What is the maximum number of consecutive powers of $2$ that can be found without there being four among them with the same number of digits?