Olympiad problems

1991 IMO Shortlist, 15

Tags: periodic sequence , digit , decimal representation , factorial , modular arithmetic

Let $ a_n$ be the last nonzero digit in the decimal representation of the number $ n!.$ Does the sequence $ a_1, a_2, \ldots, a_n, \ldots$ become periodic after a finite number of terms?

2020 Flanders Math Olympiad, 2

Tags: digit , combinatorics , number theory

Every officially published book used to have an ISBN code (International Standard Book Number) which consisted of $10$ symbols. Such code looked like this: $$a_1a_2 . . . a_9a_{10}$$ with $a_1, . . . , a_9 \in \{0, 1, . . . , 9\}$ and $a_{10} \in \{0, 1, . . . , 9, X\}$. The symbol $X$ stood for the number $10$. With a valid ISBN code was $$a_1 + 2a2 + . . . + 9a_9 + 10a_{10}$$ a multiple of $11$. Prove the following statements. (a) If one symbol is changed in a valid ISBN code, the result is no valid ISBN code. (b) When two different symbols swap places in a valid ISBN code then the result is not a valid ISBN.

1999 Denmark MO - Mohr Contest, 5

Tags: digit , divide , divisible , number theory

Is there a number whose digits are only $1$'s and which is divided by $1999$?

2015 Cuba MO, 3

Tags: number theory , digit

Determine the smallest integer of the form $\frac{ \overline{AB}}{B}$ .where $A$ and $B$ are three-digit positive integers and $\overline{AB}$ denotes the six-digit number that is form by writing the numbers $A$ and $B$ consecutively.

2013 Chile National Olympiad, 1

Tags: number theory , digit , combinatorics

Find the sum of all $5$-digit positive integers that they have only the digits $1, 2$, and $5$, none repeated more than three consecutive times.

1990 IMO Longlists, 23

Tags: digit , calculate , decimal representation , sum of digits , number theory

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}).$

2010 Hanoi Open Mathematics Competitions, 3

Tags: digit , last , number theory

Find $5$ last digits of the number $M = 5^{2010}$ . (A): $65625$, (B): $45625$, (C): $25625$, (D): $15625$, (E) None of the above.

2015 Caucasus Mathematical Olympiad, 1

Tags: digit , divisible , remainder , number theory

Is there an eight-digit number without zero digits, which when divided by the first digit gives the remainder $1$, when divided by the second digit will give the remainder $2$, ..., when divided by the eighth digit will give the remainder $8$?

1989 Chile National Olympiad, 1

Tags: digit , number theory , sum of digits

Writing $1989$ in base $b$, we obtain a three-digit number: $xyz$. It is known that the sum of the digits is the same in base $10$ and in base $b$, that is, $1 + 9 + 8 + 9 = x + y + z$. Determine $x,y,z,b.$

2018 Auckland Mathematical Olympiad, 2

Tags: digit , number theory

Consider a positive integer, $N = 9 + 99 + 999 + ... +\underbrace{999...9}_{2018}$. How many times does the digit $1$ occur in its decimal representation?

2024 Malaysian Squad Selection Test, 2

Tags: digit , number theory

A finite sequence of decimal digits from $\{0,1,\cdots, 9\}$ is said to be [i]common[/i] if for each sufficiently large positive integer $n$, there exists a positive integer $m$ such that the expansion of $n$ in base $m$ ends with this sequence of digits. For example, $0$ is common because for any large $n$, the expansion of $n$ in base $n$ is $10$, whereas $00$ is not common because for any squarefree $n$, the expansion of $n$ in any base cannot end with $00$. Determine all common sequences. [i]Proposed by Wong Jer Ren[/i]

1992 IMO Longlists, 54

Tags: decimal representation , digit , number theory

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 $n > 125.$

2015 Caucasus Mathematical Olympiad, 1

Tags: digit , sum , divisible , number theory

Does there exist a four-digit positive integer with different non-zero digits, which has the following property: if we add the same number written in the reverse order, then we get a number divisible by $101$?

2025 Israel National Olympiad (Gillis), P3

Tags: digit , perfect square , number theory

Bart wrote the digit "$1$" $2024$ times in a row. Then, Lisa wrote an additional $2024$ digits to the right of the digits Bart wrote, such that the resulting number is a square of an integer. Find all possibilities for the digits Lisa wrote.

2001 Regional Competition For Advanced Students, 1

Tags: digit , power , sum of powers , sum , number theory

Let $n$ be an integer. We consider $s (n)$, the sum of the $2001$ powers of $n$ with the exponents $0$ to $2000$. So $s (n) = \sum_{k=0}^{2000}n ^k$ . What is the unit digit of $s (n)$ in the decimal system?

2007 Dutch Mathematical Olympiad, 3

Tags: digit , number theory , divisible

Does there exist an integer having the form $444...4443$ (all fours, and ending with a three) that is divisible by $13$? If so, give an integer having that form that is divisible by $13$, if not, prove that such an integer cannot exist.

2000 Korea Junior Math Olympiad, 5

Tags: digit , number theory

$a$ is a $2000$ digit natural number of the form $$a=2(A)99…99(B)(C)$$ expressed in base $10$. $a$ is not a multiple of $10$, and $2(A)+(B)(C)=99$. $a=2899..9971$ is a possible example of $a$. $b$ is a number you earn when you write the digits of $a$ in a reverse order(Writing the digits of some number in a reverse order means like reordering $1234$ into $4321$). Find every positive integer $a$ that makes $ab$ a square number.

1968 Bulgaria National Olympiad, Problem 3

Tags: digit , bases , number theory

Prove that a binomial coefficient $\binom nk$ is odd if and only if all digits $1$ of $k$, when $k$ is written in binary, are on the same positions when $n$ is written in binary. [i]I. Dimovski[/i]

1996 Tournament Of Towns, (512) 5

Tags: last digit , digit , number theory

Does there exist a $6$-digit number $A$ such that none of its $500 000$ multiples $A$, $2A$, $3A$, ..., $500 000A$ ends in $6$ identical digits? (S Tokarev)

2019 Saudi Arabia JBMO TST, 3

Tags: digit , combinatorics

Let $6$ pairwise different digits are given and all of them are different from $0$. Prove that there exist $2$ six-digit integers, such that their difference is equal to $9$ and each of them contains all given $6$ digits.

1995 Bundeswettbewerb Mathematik, 4

Tags: digit , multiple , number theory

Prove that every integer $k > 1$ has a multiple less than $k^4$ whose decimal expension has at most four distinct digits.

2000 Czech And Slovak Olympiad IIIA, 6

Tags: digit , number theory

Find all four-digit numbers $\overline{abcd}$ (in decimal system) such that $\overline{abcd}= (\overline{ac}+1).(\overline{bd} +1)$

2018 Bundeswettbewerb Mathematik, 1

Tags: arithmetic , digit , number theory

Find the largest positive integer with the property that each digit apart from the first and the last one is smaller than the arithmetic mean of her neighbours.

2019 Durer Math Competition Finals, 8

Tags: digit , last digit , number theory

Let $N$ be a positive integer such that $N$ and $N^2$ both end in the same four digits $\overline{abcd}$, where $a \ne 0$. What is the four-digit number $\overline{abcd}$?

2015 May Olympiad, 1

Tags: digit , number theory

The teacher secretly thought of a three-digit $S$ number. Students $A, B, C$ and $D$ tried to guess, saying, respectively, $541$, $837$, $291$ and $846$. The teacher told them, “Each of you got it right exactly one digit of $S$ and in the correct position ”. What is the number $S$?