Olympiad problems

2016 Croatia Team Selection Test, Problem 4

Let $p > 10^9$ be a prime number such that $4p + 1$ is also prime. Prove that the decimal expansion of $\frac{1}{4p+1}$ contains all the digits $0,1, \ldots, 9$.

1968 IMO Shortlist, 22

Tags: decimal representation , digit , equation , number theory

Find all natural numbers $n$ the product of whose decimal digits is $n^2-10n-22$.

1951 Moscow Mathematical Olympiad, 193

Tags: decimal representation , algebra

Prove that the first 3 digits after the decimal point in the decimal expression of the number $\frac{0.123456789101112 . . . 495051}{0.515049 . . . 121110987654321}$ are $239$.

1994 IMO Shortlist, 7

Tags: decimal representation , divisibility , number theory

A wobbly number is a positive integer whose digits are alternately zero and non-zero with the last digit non-zero (for example, 201). Find all positive integers which do not divide any wobbly number.

2024 CAPS Match, 1

Tags: decimal representation , number theory , intersection , fraction

Determine whether there exist 2024 distinct positive integers satisfying the following: if we consider every possible ratio between two distinct numbers (we include both $a/b$ and $b/a$), we will obtain numbers with finite decimal expansions (after the decimal point) of mutually distinct non-zero lengths.

1994 Bundeswettbewerb Mathematik, 1

Tags: algebra , pigeonhole , decimal representation , real number , digit

Given eleven real numbers, prove that there exist two of them such that their decimal representations agree infinitely often.

1980 IMO Shortlist, 6

Tags: decimal representation , combinatorics , digit , modular arithmetic

Find the digits left and right of the decimal point in the decimal form of the number \[ (\sqrt{2} + \sqrt{3})^{1980}. \]

1986 IMO Longlists, 67

Tags: number theory , decimal representation , rational number , algebra

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 $a = 0.1d_1(2)d_2(2) \cdots d_{r(2)}(2)d_1(3) . . . d_{r(3)}(3)d_1(4) \cdots .$

PEN A Problems, 103

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

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 Croatia Team Selection Test, Problem 4

Tags: decimal representation , prime , prime number , primitive root , number theory

Let $p > 10^9$ be a prime number such that $4p + 1$ is also prime. Prove that the decimal expansion of $\frac{1}{4p+1}$ contains all the digits $0,1, \ldots, 9$.

1962 IMO Shortlist, 1

Tags: decimal representation , divisibility , number theory

Find the smallest natural number $n$ which has the following properties: a) Its decimal representation has a 6 as the last digit. b) If the last digit 6 is erased and placed in front of the remaining digits, the resulting number is four times as large as the original number $n$.

2006 IMO Shortlist, 2

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]

1983 All Soviet Union Mathematical Olympiad, 370

Tags: decimal representation , digit , algebra , number theory

The infinite decimal notation of the real number $x$ contains all the digits. Let $u_n$ be the number of different $n$-digit segments encountered in $x$ notation. Prove that if for some $n$, $u_n \le (n+8)$, than $x$ is a rational number.

1994 Bundeswettbewerb Mathematik, 1

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

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.

2019 Mediterranean Mathematics Olympiad, 3

Tags: number theory , decimal representation , sum of digits

Prove that there exist infinitely many positive integers $x,y,z$ for which the sum of the digits in the decimal representation of $~4x^4+y^4-z^2+4xyz$ $~$ is at most $2$. (Proposed by Gerhard Woeginger, Austria)

2021 Irish Math Olympiad, 3

Tags: number theory , digit , decimal representation

For each integer $n \ge 100$ we define $T(n)$ to be the number obtained from $n$ by moving the two leading digits to the end. For example, $T(12345) = 34512$ and $T(100) = 10$. Find all integers $n \ge 100$ for which $n + T(n) = 10n$.

2021 China Team Selection Test, 3

Tags: number theory , sum of digits , decimal representation

Find all positive integer $n(\ge 2)$ and rational $\beta \in (0,1)$ satisfying the following: There exist positive integers $a_1,a_2,...,a_n$, such that for any set $I \subseteq \{1,2,...,n\}$ which contains at least two elements, $$ S(\sum_{i\in I}a_i)=\beta \sum_{i\in I}S(a_i). $$ where $S(n)$ denotes sum of digits of decimal representation of $n$.

1982 Bundeswettbewerb Mathematik, 1

Tags: algebra , divisor , decimal representation

Max divided a natural number $p$ by a natural number $q \leq 100$. In the decimal representation of the quotient he calculated, the sequence of digits $1982$ occurs somewhere after the decimal point. Show that Max made a computational mistake.

2003 IMO Shortlist, 6

Tags: function , combinatorics , modular arithmetic , counting , decimal representation

Let $f(k)$ be the number of integers $n$ satisfying the following conditions: (i) $0\leq n < 10^k$ so $n$ has exactly $k$ digits (in decimal notation), with leading zeroes allowed; (ii) the digits of $n$ can be permuted in such a way that they yield an integer divisible by $11$. Prove that $f(2m) = 10f(2m-1)$ for every positive integer $m$. [i]Proposed by Dirk Laurie, South Africa[/i]

1999 IMO Shortlist, 5

Tags: number theory , decimal representation , sum of digits

Let $n,k$ be positive integers such that n is not divisible by 3 and $k \geq n$. Prove that there exists a positive integer $m$ which is divisible by $n$ and the sum of its digits in decimal representation is $k$.

1986 China Team Selection Test, 3

Tags: number theoretic functions , decimal representation , digit , algebra

Given a positive integer $A$ written in decimal expansion: $(a_{n},a_{n-1}, \ldots, a_{0})$ and let $f(A)$ denote $\sum^{n}_{k=0} 2^{n-k}\cdot a_k$. Define $A_1=f(A), A_2=f(A_1)$. Prove that: [b]I.[/b] There exists positive integer $k$ for which $A_{k+1}=A_k$. [b]II.[/b] Find such $A_k$ for $19^{86}.$

1975 IMO Shortlist, 6

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.)

1990 IMO Longlists, 65

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

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.

1978 IMO, 1

Tags: modular arithmetic , digit , decimal representation , number theory

Let $ m$ and $ n$ be positive integers such that $ 1 \le m < n$. In their decimal representations, the last three digits of $ 1978^m$ are equal, respectively, to the last three digits of $ 1978^n$. Find $ m$ and $ n$ such that $ m \plus{} n$ has its least value.

2014 India IMO Training Camp, 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.