Olympiad problems

2008 May Olympiad, 1

How many different numbers with $6$ digits and multiples of $45$ can be written by adding one digit to the left and one to the right of $2008$?

2014 Bosnia and Herzegovina Junior BMO TST, 1

Tags: number theory , digit , divisible

Let $x$, $y$ and $z$ be nonnegative integers. Find all numbers in form $\overline{13xy45z}$ divisible with $792$, where $x$, $y$ and $z$ are digits.

2017 Czech-Polish-Slovak Junior Match, 3

Tags: digit , divisible , number theory

How many $8$-digit numbers are $*2*0*1*7$, where four unknown numbers are replaced by stars, which are divisible by $7$?

2019 Portugal MO, 2

Tags: number theory , digit

A five-digit integer is said to be [i]balanced [/i]i f the sum of any three of its digits is divisible by any of the other two. How many [i]balanced [/i] numbers are there?

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.)

1998 Mexico National Olympiad, 1

Tags: number theory , digit , consecutive

A number is called lucky if computing the sum of the squares of its digits and repeating this operation sufficiently many times leads to number $1$. For example, $1900$ is lucky, as $1900 \to 82 \to 68 \to 100 \to 1$. Find infinitely many pairs of consecutive numbers each of which is lucky.

2020 Regional Competition For Advanced Students, 2

Tags: digit , number theory

The set $M$ consists of all $7$-digit positive integer numbers that contain (in decimal notation) each of the digits $1, 3, 4, 6, 7, 8$ and $9$ exactly once. (a) Find the smallest positive difference $d$ of two numbers from $M$. (b) How many pairs $(x, y)$ with $x$ and $y$ from M are there for which $x - y = d$? (Gerhard Kirchner)

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 $n + T(n) = 10n$.

2000 Greece JBMO TST, 1

Tags: irreducible , number theory , digit

a) Prove that the fraction $\frac{3n+5}{2n+3}$ is irreducible for every $n \in N$ b) Let $x,y$ be digits of decimal representation system with $x>0$, and $\frac{\overline{xy}+12}{\overline{xy}-3}\in N$, prove that $x+y=9$. Is the converse true?

2020 Polish Junior MO First Round, 1.

Tags: number theory , digit , easy

Determine all natural numbers $n$, such that it's possible to insert one digit at the right side of $n$ to obtain $13n$.

2019 Saudi Arabia JBMO TST, 3

Tags: number theory , digit

Find all positive integers of form abcd such that $$\overline{abcd} = a^{a+b+c+d} - a^{-a+b-c+d} + a$$

1974 All Soviet Union Mathematical Olympiad, 201

Tags: digit , 3-digit , number theory

Find all the three-digit numbers such that it equals to the arithmetic mean of the six numbers obtained by rearranging its digits.

2012 VJIMC, Problem 4

Tags: digit , number theory

Find all positive integers $n$ for which there exists a positive integer $k$ such that the decimal representation of $n^k$ starts and ends with the same digit.

1997 VJIMC, Problem 4-M

Tags: limit , real analysis , number theory , digit

Find all real numbers $a>0$ for which the series $$\sum_{n=1}^\infty\frac{a^{f(n)}}{n^2}$$is convergent; $f(n)$ denotes the number of $0$'s in the decimal expansion of $f$.

2019 Saint Petersburg Mathematical Olympiad, 1

Tags: number theory , digit , palindrome

A natural number is called a palindrome if it is read in the same way. from left to right and from right to left (in particular, the last digit of the palindrome coincides with the first and therefore not equal to zero). Squares of two different natural numbers have $1001$ digits. Prove that strictly between these squares, there is one palindrome.

1986 China Team Selection Test, 3

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

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 $19^{86}.$

2005 Estonia National Olympiad, 5

Tags: number theory , digit , combinatorics

How many positive integers less than $10,000$ have an even number of even digits and an odd number of odd digits ? (Assume no number starts with zero.)

2003 May Olympiad, 1

Tags: number theory , digit , algebra

Pedro writes all the numbers with four different digits that can be made with digits $a, b, c, d$, that meet the following conditions: $$ a\ne 0 \, , \, b=a+2 \, , \, c=b+2 \, , \, d=c+2$$ Find the sum of all the numbers Pedro wrote.

1912 Eotvos Mathematical Competition, 1

Tags: number theory , digit

How many positive integers of $n$ digits exist such that each digit is $1, 2$, or $3$? How many of these contain all three of the digits $1, 2$, and $3$ at least once?

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, N14

Tags: digit , sum of digits , number theory

For any natural number $n = \overline{a_i...a_2a_1}$, consider the number $$T(n) =10 \sum_{i \,\, even} a_i+\sum_{i \,\, odd} a_i.$$ Let's find the smallest positive integer $A$ such that is sum of the natural numbers $n_1,n_2,...,n_{148}$ as well as of $m_1,m_2,...,m_{149}$ and matches the pattern: $A=n_1+n_2+...+n_{148}=m_1+m_2+...+m_{149}$ $T(n_1)=T(n_2)=...=T(n_{148})$ $T(m_1)=T(m_2)=...=T(m_{148})$

2019 IberoAmerican, 1

Tags: number theory , digit

For each positive integer $n$, let $s(n)$ be the sum of the squares of the digits of $n$. For example, $s(15)=1^2+5^2=26$. Determine all integers $n\geq 1$ such that $s(n)=n$.

2018 Czech-Polish-Slovak Junior Match, 4

Tags: digit , number theory , divisible

Determine the smallest positive integer $A$ with an odd number of digits and this property, that both $A$ and the number $B$ created by removing the middle digit of the number $A$ are divisible by $2018$.

2009 Peru MO (ONEM), 1

Tags: number theory , digit

For each positive integer $n$, let $c(n)$ be the number of digits of $n$. Let $A$ be a set of positive integers with the following property: If $a$ and $b$ are two distinct elements in $A$, then $c(a +b)+2 > c(a)+c(b)$. Find the largest number of elements that $A$ can have. PS. In the original wording: c(n) = ''cantidad de dıgitos''

1974 Vietnam National Olympiad, 1

Tags: number theory , digit , positive integer

Find all positive integers $n$ and $b$ with $0 < b < 10$ such that if $a_n$ is the positive integer with $n$ digits, all of them $1$, then $a_{2n} - b a_n$ is a square.