Olympiad problems

2013 Tournament of Towns, 2

There is a positive integer $A$. Two operations are allowed: increasing this number by $9$ and deleting a digit equal to $1$ from any position. Is it always possible to obtain $A+1$ by applying these operations several times?

2009 Switzerland - Final Round, 2

Tags: number theory , digit

A [i]palindrome [/i] is a natural number that works in the decimal system forwards and backwards read is the same size (e.g. $1129211$ or $7337$). Determine all pairs $(m, n)$ of natural numbers, such that $$(\underbrace{11... 11}_{m}) \cdot (\underbrace{11... 11}_{n})$$ is a palindrome.

1930 Eotvos Mathematical Competition, 1

Tags: digit , number theory

How many five-digit multiples of 3 end with the digit 6 ?

2016 Ecuador Juniors, 6

Tags: number theory , digit

Determine the number of positive integers $N = \overline{abcd}$, with $a, b, c, d$ nonzero digits, which satisfy $(2a -1) (2b -1) (2c- 1) (2d - 1) = 2abcd -1$.

2019 Ecuador Juniors, 1

Tags: digit , number theory

A three-digit $\overline{abc}$ number is called [i]Ecuadorian [/i] if it meets the following conditions: $\bullet$ $\overline{abc}$ does not end in $0$. $\bullet$ $\overline{abc}$ is a multiple of $36$. $\bullet$ $\overline{abc} - \overline{cba}$ is positive and a multiple of $36$. Determine all the Ecuadorian numbers.

1962 Dutch Mathematical Olympiad, 4

Tags: digit , last digit , number theory , floor function

Write using with the floor function: the last, the second last, and the first digit of the number $n$ written in the decimal system.

2020 HK IMO Preliminary Selection Contest, 1

Tags: digit , algebra

Let $n=(10^{2020}+2020)^2$. Find the sum of all the digits of $n$.

1986 Brazil National Olympiad, 4

Tags: digit , number theory

Find all $10$ digit numbers $a_0a_1...a_9$ such that for each $k, a_k$ is the number of times that the digit $k$ appears in the number.

2003 Paraguay Mathematical Olympiad, 1

Tags: product of digits , digit , number theory

How many numbers greater than $1.000$ but less than $10.000$ have as a product of their digits $256$?

2019 Lusophon Mathematical Olympiad, 1

Tags: number theory , digit , divisible

Find a way to write all the digits of $1$ to $9$ in a sequence and without repetition, so that the numbers determined by any two consecutive digits of the sequence are divisible by $7$ or $13$.

1952 Moscow Mathematical Olympiad, 216

Tags: number theory , integer sequence , sequence , sum of squares , digit

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 $a_1, a_2, a_3, ....$

2018 Flanders Math Olympiad, 4

Tags: number theory , digit

Determine all three-digit numbers N such that $N^2$ has six digits and so that the sum of the number formed by the first three digits of $N^2$ and the number formed by the latter three digits of $N^2$ equals $N$.

1991 Swedish Mathematical Competition, 5

Tags: digit , number theory

Show that there are infinitely many odd positive integers $n$ such that in binary $n$ has more $1$s than $n^2$.

2000 Austria Beginners' Competition, 3

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

A two-digit number is [i]nice [/i] if it is both a multiple of the product of its digits and a multiple of the sum of its digits. How many numbers satisfy this property? What is the ratio of the number to the sum of digits for each of the nice numbers?

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 \[ (\sqrt{2} + \sqrt{3})^{1980}. \]

1962 Poland - Second Round, 6

Tags: digit , number theory

Find a three-digit number with the property that the number represented by these digits and in the same order, but with a numbering base different than $ 10 $, is twice as large as the given number.

1972 All Soviet Union Mathematical Olympiad, 168

Tags: digit , game strategy

A game for two. One gives a digit and the second substitutes it instead of a star in the following difference: $$**** - **** = $$ Then the first gives the next digit, and so on $8$ times. The first wants to obtain the greatest possible difference, the second -- the least. Prove that: 1. The first can operate in such a way that the difference would be not less than $4000$, not depending on the second's behaviour. 2. The second can operate in such a way that the difference would be not greater than $4000$, not depending on the first's behaviour.

1993 Nordic, 3

Tags: digit , positive integer , system of equations , number theory

Find all solutions of the system of equations $\begin{cases} s(x) + s(y) = x \\ x + y + s(z) = z \\ s(x) + s(y) + s(z) = y - 4 \end{cases}$ where $x, y$, and $z$ are positive integers, and $s(x), s(y)$, and $s(z)$ are the numbers of digits in the decimal representations of $x, y$, and $z$, respectively.

2009 Tournament Of Towns, 2

Tags: digit , number theory

Let $a^b$ denote the number $ab$. The order of operations in the expression 7^7^7^7^7^7^7 must be determined by parentheses ($5$ pairs of parentheses are needed). Is it possible to put parentheses in two distinct ways so that the value of the expression be the same?

2016 Ecuador Juniors, 1

Tags: digit , number theory

A natural number of five digits is called [i]Ecuadorian [/i]if it satisfies the following conditions: $\bullet$ All its digits are different. $\bullet$ The digit on the far left is equal to the sum of the other four digits. Example: $91350$ is an Ecuadorian number since $9 = 1 + 3 + 5 + 0$, but $54210$ is not since $5 \ne 4 + 2 + 1 + 0$. Find how many Ecuadorian numbers exist.

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.

2023 Chile Junior Math Olympiad, 1

Tags: digit , number theory

Determine the number of three-digit numbers with the following property: The number formed by the first two digits is prime and the number formed by the last two digits is prime.

2019 Brazil National Olympiad, 1

Tags: number theory , digit

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.

2013 Thailand Mathematical Olympiad, 5

Tags: number theory , sum of digits , digit , divide , divisible

Find a five-digit positive integer $n$ (in base $10$) such that $n^3 - 1$ is divisible by $2556$ and which minimizes the sum of digits of $n$.

1989 Tournament Of Towns, (236) 4

Tags: number theory , digit

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)