Olympiad problems

2001 IMO Shortlist, 1

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

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 $k$.

1998 All-Russian Olympiad Regional Round, 8.1

Tags: number theory , Digits

Are there $n$-digit numbers M and N such that all digits $M$ are even, all $N$ digits are odd, every digit from $0$ to $9$ occurs in decimal notation M or N at least once, and $M$ is divisible by $N$?

2018 Hanoi Open Mathematics Competitions, 13

Tags: sum of digits , Product , Digits , number theory , 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 $n = S(n) + P(n)$.

2011 Denmark MO - Mohr Contest, 5

Tags: number theory , Digits , last digits , Perfect Square

Determine all sets $(a, b, c)$ of positive integers where one obtains $b^2$ by removing the last digit in $c^2$ and one obtains $a^2$ by removing the last digit in $b^2$. .

1997 Denmark MO - Mohr Contest, 1

Tags: number theory , algebra , Digits

Let $n =123456789101112 ... 998999$ be the natural number where is obtained by writing the natural numbers from $1$ to $999$ one after the other. What is the $1997$-th digit number in $n$?

1955 Kurschak Competition, 2

Tags: number theory , divides , Digits

How many five digit numbers are divisible by $3$ and contain the digit $6$?

2002 Belarusian National Olympiad, 8

Tags: number theory , Digits

The set of three-digit natural numbers formed from digits $1,2, 3, 4, 5, 6$ is called [i]nice [/i] if it satisfies the following condition: for any two different digits from $1, 2, 3, 4, 5, 6$ there exists a number from the set which contains both of them. For any nice set we calculate the sum of all its elements. Determine the smallest possible value of these sums. (E. Barabanov)

1991 Tournament Of Towns, (287) 3

Tags: number theory , Digits

We are looking for numbers ending with the digit $5$ such that in their decimal expansion each digit beginning with the second digit is no less than the previous one. Moreover the squares of these numbers must also possess the same property. (a) Find four such numbers. (b) Prove that there are infinitely many. (A. Andjans, Riga)

2002 Greece JBMO TST, 2

Tags: number theory , Digits , sum of digits

Let $A$ be a $3$-digit positive integer and $B$ be the positive integer that comes from $A$ be replacing with each other the digits of hundreds with the digit of the units. It is also given that $B$ is a $3$-digit number. Find numbers $A$ and $B$ if it is known that $A$ divided by $B$ gives quotient $3$ and remainder equal to seven times the sum of it's digits.

1998 Austrian-Polish Competition, 7

Tags: number theory , Digits

Consider all pairs $(a, b)$ of natural numbers such that the product $a^ab^b$ written in decimal system ends with exactly $98$ zeros. Find the pair $(a, b)$ for which the product $ab$ is the smallest.

2018 Hanoi Open Mathematics Competitions, 5

Tags: number theory , Digits

Find all $3$-digit numbers $\overline{abc}$ ($a,b \ne 0$) such that $\overline{bcd} \times a = \overline{1a4d}$ for some integer $d$ from $1$ to $9$

2000 Slovenia National Olympiad, Problem 1

Tags: Digits

In the expression $4\cdot\text{RAKEC}=\text{CEKAR}$, each letter represents a (decimal) digit. Replace the letters so that the equality is true.

2021 Middle European Mathematical Olympiad, 8

Tags: number theory , Digits , Perfect Squares , memo , MEMO 2021

Prove that there are infinitely many positive integers $n$ such that $n^2$ written in base $4$ contains only digits $1$ and $2$.

2019 Istmo Centroamericano MO, 1

Tags: number theory , remainder , Digits

Determine all the numbers formed by three different and non-zero digits, such that the six numbers obtained by permuting these digits leaves the same remainder after the division by $4$.

2006 VTRMC, Problem 1

Tags: number theory , Digits

Find, with proof, all positive integers $n$ such that neither $n$ nor $n^2$ contain a $1$ when written in base $3$.

2013 Argentina National Olympiad, 3

Tags: Digits , number theory , remainder

Find how many are the numbers of $2013$ digits $d_1d_2…d_{2013}$ with odd digits $d_1,d_2,…,d_{2013}$ such that the sum of $1809$ terms $$d_1 \cdot d_2+d_2\cdot d_3+…+d_{1809}\cdot d_{1810}$$ has remainder $1$ when divided by $4$ and the sum of $203$ terms $$d_{1810}\cdot d_{1811}+d_{1811}\cdot d_{1812}+…+d_{2012}\cdot d_{2013}$$ has remainder $1$ when dividing by $4$.

1992 All Soviet Union Mathematical Olympiad, 577

Tags: number theory , Digits

Find all integers $k > 1$ such that for some distinct positive integers $a, b$, the number $k^a + 1$ can be obtained from $k^b + 1$ by reversing the order of its (decimal) digits.

2017 Dutch Mathematical Olympiad, 1

Tags: number theory , Digits

We consider positive integers written down in the (usual) decimal system. Within such an integer, we number the positions of the digits from left to right, so the leftmost digit (which is never a $0$) is at position $1$. An integer is called [i]even-steven[/i] if each digit at an even position (if there is one) is greater than or equal to its neighbouring digits (if these exist). An integer is called [i]oddball[/i] if each digit at an odd position is greater than or equal to its neighbouring digits (if these exist). For example, $3122$ is [i]oddball[/i] but not [i]even-steven[/i], $7$ is both [i]even-steven[/i] and [i]oddball[/i], and $123$ is neither [i]even-steven[/i] nor [i]oddball[/i]. (a) Prove: every oddball integer greater than $9$ can be obtained by adding two [i]oddball [/i] integers. (b) Prove: there exists an oddball integer greater than $9$ that cannot be obtained by adding two [i]even-steven[/i] integers.

2021 Irish Math Olympiad, 3

Tags: number theory , Digits , 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 $n + T(n) = 10n$.

2005 Singapore Senior Math Olympiad, 1

Tags: Perfect Square , Digits , Difference , number theory

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.

2016 CentroAmerican, 1

Tags: number theory , Digits

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.

2023 Ukraine National Mathematical Olympiad, 10.1

Tags: number theory , Digits

Find all positive integers $k$, for which the product of some consecutive $k$ positive integers ends with $k$. [i]Proposed by Oleksiy Masalitin[/i]

1952 Moscow Mathematical Olympiad, 216

Tags: Integer sequence , Sequence , Sum of Squares , Digits , number theory

A sequence of integers is constructed as follows: $a_1$ is an arbitrary three-digit number, $a_2$ is the sum of squares of the digits of $a_1, a_3$ is the sum of squares of the digits of $a_2$, etc. Prove that either $1$ or $4$ must occur in the sequence $a_1, a_2, a_3, ....$

1981 Brazil National Olympiad, 2

Tags: number theory , Digits , power of 2

Show that there are at least $3$ and at most $4$ powers of $2$ with $m$ digits. For which $m$ are there $4$?

2021 Kyiv City MO Round 1, 8.2

Tags: number theory , Digits

Oleksiy writes all the digits from $0$ to $9$ on the board, after which Vlada erases one of them. Then he writes $10$ nine-digit numbers on the board, each consisting of all the nine digits written on the board (they don't have to be distinct). It turned out that the sum of these $10$ numbers is a ten-digit number, all of whose digits are distinct. Which digit could have been erased by Vlada? [i]Proposed by Oleksii Masalitin[/i]