Olympiad problems

1998 Estonia National Olympiad, 2

Find all prime numbers of the form $10101...01$.

1990 IMO Longlists, 23

Tags: number theory , Digits , sum of digits , Calculate , decimal representation , IMO Shortlist

For a given positive integer $ k$ denote the square of the sum of its digits by $ f_1(k)$ and let $ f_{n\plus{}1}(k) \equal{} f_1(f_n(k)).$ Determine the value of $ f_{1991}(2^{1990}).$

1989 Tournament Of Towns, (236) 4

Tags: number theory , Digits

The numbers $2^{1989}$ and $5^{1989}$ are written out one after the other (in decimal notation). How many digits are written altogether? (G. Galperin)

1998 IMO Shortlist, 7

Tags: number theory , sum of digits , Digits , Divisibility , IMO Shortlist

Prove that for each positive integer $n$, there exists a positive integer with the following properties: It has exactly $n$ digits. None of the digits is 0. It is divisible by the sum of its digits.

2007 Junior Balkan Team Selection Tests - Moldova, 5

Tags: number theory , Digits , product of digits

Determine the smallest natural number written in the decimal system with the product of the digits equal to $10! = 1 \cdot 2 \cdot 3\cdot ... \cdot9\cdot10$.

2014 India PRMO, 2

Tags: algebra , sum of cubes , Sequence , Digits , number theory

The first term of a sequence is $2014$. Each succeeding term is the sum of the cubes of the digits of the previous term. What is the $2014$ th term of the sequence?

2009 BAMO, 3

Tags: summer program , Digits , Last digit , number theory

There are many sets of two different positive integers $a$ and $b$, both less than $50$, such that $a^2$ and $b^2$ end in the same last two digits. For example, $35^2 = 1225$ and $45^2 = 2025$ both end in $25$. What are all possible values for the average of $a$ and $b$? For the purposes of this problem, single-digit squares are considered to have a leading zero, so for example we consider $2^2$ to end with the digits 04, not $4$.

2024 Ukraine National Mathematical Olympiad, Problem 1

Tags: number theory , Digits

Oleksiy wrote several distinct positive integers on the board and calculated all their pairwise sums. It turned out that all digits from $0$ to $9$ appear among the last digits of these sums. What could be the smallest number of integers that Oleksiy wrote? [i]Proposed by Oleksiy Masalitin[/i]

2019 Brazil National Olympiad, 1

Tags: Brazilian Math Olympiad , Digits , number theory

An eight-digit number is said to be 'robust' if it meets both of the following conditions: (i) None of its digits is $0$. (ii) The difference between two consecutive digits is $4$ or $5$. Answer the following questions: (a) How many are robust numbers? (b) A robust number is said to be 'super robust' if all of its digits are distinct. Calculate the sum of all the super robust numbers.

2006 Junior Balkan Team Selection Tests - Romania, 4

Tags: Digits , number theory , Perfect Square

For a positive integer $n$ denote $r(n)$ the number having the digits of $n$ in reverse order- for example, $r(2006) = 6002$. Prove that for any positive integers a and b the numbers $4a^2 + r(b)$ and $4b^2 + r(a)$ can not be simultaneously squares.

1995 Czech And Slovak Olympiad IIIA, 4

Tags: Digits , number theory , divisible

Do there exist $10000$ ten-digit numbers divisible by $7$, all of which can be obtained from one another by a reordering of their digits?

2001 German National Olympiad, 5

Tags: fibonacci number , number theory , Digits

The Fibonacci sequence is given by $x_1 = x_2 = 1$ and $x_{k+2} = x_{k+1} + x_k$ for each $k \in N$. (a) Prove that there are Fibonacci numbes that end in a $9$ in the decimal system. (b) Determine for which $n$ can a Fibonacci number end in $n$ $9$-s in the decimal system.

2004 Paraguay Mathematical Olympiad, 5

Tags: number theory , Perfect Square , Digits

We have an integer $A$ such that $A^2$ is a four digit number, with $5$ in the ten's place . Find all possible values of $A$.

1968 Leningrad Math Olympiad, 8.6*

Tags: number theory , combinatorics , Digits

All $10$-digit numbers consisting of digits $1, 2$ and $3$ are written one under the other. Each number has one more digit added to the right. $1$, $2$ or $3$, and it turned out that to the number $111. . . 11$ added $1$ to the number $ 222. . . 22$ was assigned $2$, and the number $333. . . 33$ was assigned $3$. It is known that any two numbers that differ in all ten digits have different digits assigned to them. Prove that the assigned column of numbers matches with one of the ten columns written earlier.

2005 Estonia National Olympiad, 3

Tags: Digits , number theory , divisible

How many such four-digit natural numbers divisible by $7$ exist such when changing the first and last number we also get a four-digit divisible by $7$?

1990 IMO Shortlist, 27

Tags: number theory , decimal representation , prime numbers , Digits , IMO Shortlist

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.

2009 Postal Coaching, 4

Tags: Digits , remainder , number theory

A four - digit natural number which is divisible by $7$ is given. The number obtained by writing the digits in reverse order is also divisible by $7$. Furthermore, both the numbers leave the same remainder when divided by $37$. Find the 4-digit number.

1965 Spain Mathematical Olympiad, 2

Tags: Digits

How many numbers of $3$ digits have their central digit greater than any of the other two? How many of them have also three different digits?

2017 Abels Math Contest (Norwegian MO) Final, 3a

Tags: combinatorics , number theory , Digits

Nils has a telephone number with eight different digits. He has made $28$ cards with statements of the type “The digit $a$ occurs earlier than the digit $b$ in my telephone number” – one for each pair of digits appearing in his number. How many cards can Nils show you without revealing his number?

1992 IMO Longlists, 54

Tags: number theory , decimal representation , Digits , IMO Shortlist , IMO Longlist

Suppose that $n > m \geq 1$ are integers such that the string of digits $143$ occurs somewhere in the decimal representation of the fraction $\frac{m}{n}$. Prove that $n > 125.$

1969 All Soviet Union Mathematical Olympiad, 122

Tags: number theory , Digits , 3-digit

Find four different three-digit decimal numbers starting with the same digit, such that their sum is divisible by three of them.

1970 IMO Longlists, 42

Tags: inequalities , algebra , decimal representation , Digits , IMO , IMO 1970

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 $A'B<AB'$.

2020 Colombia National Olympiad, 1

Tags: number theory , Digits , Perfect Squares , Perfect Square

A positive integer is called [i]sabroso [/i]if when it is added to the number obtained when its digits are interchanged from one side of its written form to the other, the result is a perfect square. For example, $143$ is sabroso, since $143 + 341 =484 = 22^2$. Find all two-digit sabroso numbers.

1975 All Soviet Union Mathematical Olympiad, 210

Tags: number theory , combinatorics , Digits

Prove that it is possible to find $2^{n+1}$ of $2^n$ digit numbers containing only "$1$" and "$2$" as digits, such that every two of them distinguish at least in $2^{n-1}$ digits.

1983 All Soviet Union Mathematical Olympiad, 354

Tags: inequalities , algebra , Digits

Natural number $k$ has $n$ digits in its decimal notation. It was rounded up to tens, then the obtained number was rounded up to hundreds, and so on $(n-1)$ times. Prove that the obtained number $m$ satisfies inequality $m < \frac{18k}{13}$. (Examples of rounding: $191\to190\to 200, 135\to140\to 100$.)