Olympiad problems

1986 IMO Shortlist, 2

Let $f(x) = x^n$ where $n$ is a fixed positive integer and $x =1, 2, \cdots .$ Is the decimal expansion $a = 0.f (1)f(2)f(3) . . .$ rational for any value of $n$ ? The decimal expansion of a is defined as follows: If $f(x) = d_1(x)d_2(x) \cdots d_{r(x)}(x)$ is the decimal expansion of $f(x)$, then $a = 0.1d_1(2)d_2(2) \cdots d_{r(2)}(2)d_1(3) . . . d_{r(3)}(3)d_1(4) \cdots .$

2017 IMO Shortlist, N4

Tags: number theory , decimal representation , digit

Call a rational number [i]short[/i] if it has finitely many digits in its decimal expansion. For a positive integer $m$, we say that a positive integer $t$ is $m-$[i]tastic[/i] if there exists a number $c\in \{1,2,3,\ldots ,2017\}$ such that $\dfrac{10^t-1}{c\cdot m}$ is short, and such that $\dfrac{10^k-1}{c\cdot m}$ is not short for any $1\le k<t$. Let $S(m)$ be the set of $m-$tastic numbers. Consider $S(m)$ for $m=1,2,\ldots{}.$ What is the maximum number of elements in $S(m)$?

2004 Germany Team Selection Test, 1

Tags: number theory , combinatorics , decimal representation , algorithm

Each positive integer $a$ undergoes the following procedure in order to obtain the number $d = d\left(a\right)$: (i) move the last digit of $a$ to the first position to obtain the numb er $b$; (ii) square $b$ to obtain the number $c$; (iii) move the first digit of $c$ to the end to obtain the number $d$. (All the numbers in the problem are considered to be represented in base $10$.) For example, for $a=2003$, we get $b=3200$, $c=10240000$, and $d = 02400001 = 2400001 = d(2003)$.) Find all numbers $a$ for which $d\left( a\right) =a^2$. [i]Proposed by Zoran Sunic, USA[/i]

1999 Greece JBMO TST, 3

Tags: decimal representation , digit , number theory

Find digits $a,b,c,x$ ($a>0$) such that $\overline{abc}+\overline{acb}=\overline{199x}$

1970 IMO, 2

Tags: inequalities , decimal representation , digit , algebra

We have $0\le x_i<b$ for $i=0,1,\ldots,n$ and $x_n>0,x_{n-1}>0$. If $a>b$, and $x_nx_{n-1}\ldots x_0$ represents the number $A$ base $a$ and $B$ base $b$, whilst $x_{n-1}x_{n-2}\ldots x_0$ represents the number $A'$ base $a$ and $B'$ base $b$, prove that $A'B<AB'$.

1967 IMO Longlists, 14

Tags: number theory , approximation , decimal representation , rational number , irrational number

Which fractions $ \dfrac{p}{q},$ where $p,q$ are positive integers $< 100$, is closest to $\sqrt{2} ?$ Find all digits after the point in decimal representation of that fraction which coincide with digits in decimal representation of $\sqrt{2}$ (without using any table).

2019 India PRMO, 7

Tags: algebra , decimal representation , number theory

Let $s(n)$ denote the sum of digits of a positive integer $n$ in base $10$. If $s(m)=20$ and $s(33m)=120$, what is the value of $s(3m)$?

1989 Mexico National Olympiad, 4

Tags: number theory , decimal representation

Find the smallest possible natural number $n = \overline{a_m ...a_2a_1a_0} $ (in decimal system) such that the number $r = \overline{a_1a_0a_m ..._20} $ equals $2n$.

1996 Estonia National Olympiad, 4

Tags: decimal representation , prime , power

Prove that for each prime number $p > 5$ there exists a positive integer n such that $p^n$ ends in $001$ in decimal representation.

1990 IMO Longlists, 98

Tags: number theory , decimal representation , prime number , digit

Find all natural numbers $ n$ for which every natural number whose decimal representation has $ n \minus{} 1$ digits $ 1$ and one digit $ 7$ is prime.

1977 Germany Team Selection Test, 4

Tags: modular arithmetic , divisibility , digit , decimal representation , digit sum

When $4444^{4444}$ is written in decimal notation, the sum of its digits is $ A.$ Let $B$ be the sum of the digits of $A.$ Find the sum of the digits of $ B.$ ($A$ and $B$ are written in decimal notation.)

2016 Bundeswettbewerb Mathematik, 1

Tags: number theory , prime number , decimal representation

A number with $2016$ zeros that is written as $101010 \dots 0101$ is given, in which the zeros and ones alternate. Prove that this number is not prime.

1967 IMO Shortlist, 2

Tags: number theory , approximation , decimal representation , rational number , irrational number

Which fractions $ \dfrac{p}{q},$ where $p,q$ are positive integers $< 100$, is closest to $\sqrt{2} ?$ Find all digits after the point in decimal representation of that fraction which coincide with digits in decimal representation of $\sqrt{2}$ (without using any table).

2007 Germany Team Selection Test, 3

Tags: rational , decimal representation , number theory

For $ x \in (0, 1)$ let $ y \in (0, 1)$ be the number whose $ n$-th digit after the decimal point is the $ 2^{n}$-th digit after the decimal point of $ x$. Show that if $ x$ is rational then so is $ y$. [i]Proposed by J.P. Grossman, Canada[/i]

1985 All Soviet Union Mathematical Olympiad, 396

Tags: sum of digits , sum of squares , sum , decimal representation , digit , number theory

Is there any numbber $n$, such that the sum of its digits in the decimal notation is $1000$, and the sum of its square digits in the decimal notation is $1000000$?

2023 CIIM, 2

Tags: combinatorics , dice , decimal representation

A toymaker has $k$ dice at his disposal, each with $6$ blank sides. On each side of each of these dice, the toymaker must draw one of the digits $0, 1, 2, \ldots , 9$. Determine (in terms of $k$) the largest integer $n$ such that the toymaker can draw digits on the $k$ dice such that, for any positive integer $r \leq n$, it is possible to choose some of the $k$ dice and form with them the decimal representation of $r$. [b]Note:[/b] The digits 6 and 9 are distinguishable: they appear as [u]6[/u] and [u]9[/u].

2022/2023 Tournament of Towns, P3

Tags: number theory , decimal representation

Let us call a positive integer [i]pedestrian[/i] if all its decimal digits are equal to 0 or 1. Suppose that the product of some two pedestrian integers also is pedestrian. Is it necessary in this case that the sum of digits of the product equals the product of the sums of digits of the factors? [i]Viktor Kleptsyn, Konstantin Knop[/i]

1990 IMO Shortlist, 27

Tags: number theory , decimal representation , prime number , digit

Find all natural numbers $ n$ for which every natural number whose decimal representation has $ n \minus{} 1$ digits $ 1$ and one digit $ 7$ is prime.

2015 Bundeswettbewerb Mathematik Germany, 2

Tags: number theory , fraction , decimal representation

In the decimal expansion of a fraction $\frac{m}{n}$ with positive integers $m$ and $n$ you can find a string of numbers $7143$ after the comma. Show $n>1250$. [i]Example:[/i] I mean something like $0.7143$.

2014 Taiwan TST Round 3, 2

Tags: number theory , perfect square , decimal representation

Determine whether there exists an infinite sequence of nonzero digits $a_1 , a_2 , a_3 , \cdots $ and a positive integer $N$ such that for every integer $k > N$, the number $\overline{a_k a_{k-1}\cdots a_1 }$ is a perfect square.

1997 Moldova Team Selection Test, 3

Tags: number theory , pigeonhole principle , decimal representation , divisibility , multiple , digit

Prove that every integer $ k$ greater than 1 has a multiple that is less than $ k^4$ and can be written in the decimal system with at most four different digits.

2001 IMO Shortlist, 1

Tags: number theory , decimal representation , digit , factorial

Prove that there is no positive integer $n$ such that, for $k = 1,2,\ldots,9$, the leftmost digit (in decimal notation) of $(n+k)!$ equals $k$.

2004 Germany Team Selection Test, 1

Tags: number theory , decimal representation , algorithm , combinatorics

Each positive integer $a$ undergoes the following procedure in order to obtain the number $d = d\left(a\right)$: (i) move the last digit of $a$ to the first position to obtain the numb er $b$; (ii) square $b$ to obtain the number $c$; (iii) move the first digit of $c$ to the end to obtain the number $d$. (All the numbers in the problem are considered to be represented in base $10$.) For example, for $a=2003$, we get $b=3200$, $c=10240000$, and $d = 02400001 = 2400001 = d(2003)$.) Find all numbers $a$ for which $d\left( a\right) =a^2$. [i]Proposed by Zoran Sunic, USA[/i]

2018 Federal Competition For Advanced Students, P2, 6

Tags: number theory , decimal representation , digit

Determine all digits $z$ such that for each integer $k \ge 1$ there exists an integer $n\ge 1$ with the property that the decimal representation of $n^9$ ends with at least $k$ digits $z$. [i](Proposed by Walther Janous)[/i]

1983 IMO Longlists, 70

Tags: number theory , digit , decimal representation , periodic sequence , non-periodical functions

Let $d_n$ be the last nonzero digit of the decimal representation of $n!$. Prove that $d_n$ is aperiodic; that is, there do not exist $T$ and $n_0$ such that for all $n \geq n_0, d_{n+T} = d_n.$