Olympiad problems

1996 Argentina National Olympiad, 2

Decide if there exists any number of $10$ digits such that rearranging $10,000$ times its digits results in $10,000$ different numbers that are multiples of $7$.

2000 Tuymaada Olympiad, 1

Tags: number theory , digit , multiple , maximum

Given the number $188188...188$ (number $188$ is written $101$ times). Some digits of this number are crossed out. What is the largest multiple of $7$, that could happen?

1970 IMO Shortlist, 7

Tags: number theory , digit , equation , modular arithmetic

For which digits $a$ do exist integers $n \geq 4$ such that each digit of $\frac{n(n+1)}{2}$ equals $a \ ?$

1955 Poland - Second Round, 2

Tags: number theory , digit

Find the natural number $ n $ knowing that the sum $$ 1 + 2 + 3 + \ldots + n$$ is a three-digit number with identical digits.

1973 Chisinau City MO, 64

Tags: number theory , algebra , digit

Prove that in the decimal notation of the number $(5+\sqrt{26})^{-1973}$ immediately after the decimal point there are at least $1973$ zeros.

2017 Costa Rica - Final Round, N2

Tags: digit , number theory

A positive integer is said to be "nefelibata" if, upon taking its last digit and placing it as the first digit, keeping the order of all the remaining digits intact (for example, 312 -> 231), the resulting number is exactly double the original number. Find the smallest possible nefelibata number.

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 $A'B<AB'$.

1993 Tournament Of Towns, (369) 1

Tags: power of 2 , digit , number theory

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 $2$.

2004 Estonia National Olympiad, 3

Tags: number theory , digit , perfect square

The teacher had written on the board a positive integer consisting of a number of $4$s followed by the same number of $8$s followed . During the break, Juku stepped up to the board and added to the number one more $4$ at the start and a $9$ at the end. Prove that the resulting number is an a square. of an integer.

1987 IMO Longlists, 19

Tags: combinatorics , recurrence relation , symmetry , counting , combinatorics of words , digit

How many words with $n$ digits can be formed from the alphabet $\{0, 1, 2, 3, 4\}$, if neighboring digits must differ by exactly one? [i]Proposed by Germany, FR.[/i]

1947 Moscow Mathematical Olympiad, 134

Tags: digit , number theory

How many digits are there in the decimal expression of $2^{100}$ ?

2013 Flanders Math Olympiad, 1

Tags: divisible , digit , sum , number theory

A six-digit number is [i]balanced [/i] when all digits are different from zero and the sum of the first three digits is equal to the sum of the last three digits. Prove that the sum of all six-digit balanced numbers is divisible by $13$.

1996 All-Russian Olympiad Regional Round, 8.2

Tags: number theory , digit

Let's call a ticket with a number from $000000$ to $999999$ [i]excellent [/i] if the difference between some two adjacent digits is $5$. Find the number of excellent tickets.

2022 Kazakhstan National Olympiad, 2

Tags: number theory , function , digit

We define the function $Z(A)$ where we write the digits of $A$ in base $10$ form in reverse. (For example: $Z(521)=125$). Call a number $A$ $good$ if the first and last digits of $A$ are different, none of it's digits are $0$ and the equality: $$Z(A^2)=(Z(A))^2$$ happens. Find all such good numbers greater than $10^6$.\\

2005 Singapore Senior Math Olympiad, 1

Tags: number theory , perfect square , digit , difference

The digits of a $3$-digit number are interchanged so that none of the digits retain their original position. The difference of the two numbers is a $2$-digit number and is a perfect square. Find the difference.

1981 Bundeswettbewerb Mathematik, 1

Tags: number theory , digit , power

Let $a$ and $n$ be positive integers and $s = a + a^2 + \cdots + a^n$. Prove that the last digit of $s$ is $1$ if and only if the last digits of $a$ and $n$ are both equal to $1$.

1991 Tournament Of Towns, (319) 6

Tags: number theory , arithmetic progression , digit

An arithmetical progression (whose difference is not equal to zero) consists of natural numbers without any nines in its decimal notation. (a) Prove that the number of its terms is less than $100$. (b) Give an example of such a progression with $72$ terms. (c) Prove that the number of terms in any such progression does not exceed $72$. (V. Bugaenko and Tarasov, Moscow)

2017 Lusophon Mathematical Olympiad, 4

Tags: number theory , digit

Find how many multiples of 360 are of the form $\overline{ab2017cd}$, where a, b, c, d are digits, with a > 0.

2018 Hanoi Open Mathematics Competitions, 13

Tags: number theory , sum of digits , product , digit , sum

For a positive integer $n$, let $S(n), P(n)$ denote the sum and the product of all the digits of $n$ respectively. 1) Find all values of n such that $n = P(n)$: 2) Determine all values of n such that $n = S(n) + P(n)$.

2018 Estonia Team Selection Test, 6

Tags: number theory , digit

We call a positive integer $n$ whose all digits are distinct [i]bright[/i], if either $n$ is a one-digit number or there exists a divisor of $n$ which can be obtained by omitting one digit of $n$ and which is bright itself. Find the largest bright positive integer. (We assume that numbers do not start with zero.)

2003 Cuba MO, 1

Tags: number theory , digit , prime

Given the following list of numbers: $$1990, 1991, 1992, ..., 2002, 2003, 2003, 2003, ..., 2003$$ where the number $2003$ appears $12$ times. Is it possible to write these numbers in some order so that the $100$-digit number that we get is prime?

2016 CentroAmerican, 1

Tags: number theory , digit

Find all positive integers $n$ that have 4 digits, all of them perfect squares, and such that $n$ is divisible by 2, 3, 5 and 7.

2018 Israel National Olympiad, 4

Tags: digit , digit sum , divisibility , number theory

The three-digit number 999 has a special property: It is divisible by 27, and its digit sum is also divisible by 27. The four-digit number 5778 also has this property, as it is divisible by 27 and its digit sum is also divisible by 27. How many four-digit numbers have this property?

1965 German National Olympiad, 5

Tags: number theory , digit

Determine all triples of nonzero decimal digits $(x,y,z)$ for which the equality $\sqrt{ \underbrace{xxx...x}_{2n}- \underbrace{yy...y}_{n}}= \underbrace{zzz...z}_{n}$ holds for at least two different natural numbers $n$.