Olympiad problems

2020 Puerto Rico Team Selection Test, 2

The cost of $1000$ grams of chocolate is $x$ dollars and the cost of $1000$ grams of potatoes is $y$ dollars, the numbers $x$ and $y$ are positive integers and have not more than $2$ digits. Mother said to Maria to buy $200$ grams of chocolate and $1000$ grams of potatoes that cost exactly $N$ dollars. Maria got confused and bought $1000$ grams of chocolate and $200$ grams of potatoes that cost exactly $M$ dollars ($M >N$). It turned out that the numbers $M$ and $N$ have no more than two digits and are formed of the same digits but in a different order. Find $x$ and $y$.

2020 Malaysia IMONST 1, 9

Tags: number theory , Digits

What is the smallest positive multiple of $225$ that can be written using digits $0$ and $1$ only?

2013 Junior Balkan Team Selection Tests - Romania, 2

Tags: sum of digits , Digits , mutliple , number theory

Call the number $\overline{a_1a_2... a_m}$ ($a_1 \ne 0,a_m \ne 0$) the reverse of the number $\overline{a_m...a_2a_1}$. Prove that the sum between a number $n$ and its reverse is a multiple of $81$ if and only if the sum of the digits of $n$ is a multiple of $81$.

1978 Swedish Mathematical Competition, 2

Tags: Sum , number theory , Digits , algebra

Let $s_m$ be the number $66\cdots 6$ with $m$ digits $6$. Find \[ s_1 + s_2 + \cdots + s_n \]

2004 Estonia National Olympiad, 3

Tags: number theory , Digits , 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.

2021 German National Olympiad, 3

Tags: algebra , sum of cubes , algebra proposed , combinatorics proposed , Digits , Digit

For a fixed $k$ with $4 \le k \le 9$ consider the set of all positive integers with $k$ decimal digits such that each of the digits from $1$ to $k$ occurs exactly once. Show that it is possible to partition this set into two disjoint subsets such that the sum of the cubes of the numbers in the first set is equal to the sum of the cubes in the second set.

1991 IMO Shortlist, 15

Tags: factorial , modular arithmetic , Periodic sequence , IMO Shortlist , decimal representation , Digits

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?

2022 Peru MO (ONEM), 4

Tags: number theory , Perfect Powers , Perfect power , Digits

For each positive integer n, the number $R(n) = 11 ... 1$ is defined, which is made up of exactly $n$ digits equal to $1$. For example, $R(5) = 11111$. Let $n > 4$ be an integer for which, by writing all the positive divisors of $R(n)$, it is true that each written digit belongs to the set $\{0, 1\}$. Show that $n$ is a power of an odd prime number. Clarification: A power of an odd prime number is a number of the form $p^a$, where $p$ is an odd prime number and $a$ is a positive integer.

2018 Pan-African Shortlist, N4

Tags: number theory , Divisibility , Digits

Let $S$ be a set of $49$-digit numbers $n$, with the property that each of the digits $1, 2, 3, \dots, 7$ appears in the decimal expansion of $n$ seven times (and $8, 9$ and $0$ do not appear). Show that no two distinct elements of $S$ divide each other.

2000 Singapore MO Open, 3

Tags: number theory , Digits

Is there a positive integer with at most four digits whose value is increased by exactly $60\%$ when the first digit is moved to the end of the number? For example, when the first digit of $1234$ is moved to the end of the number, the result is the integer $2341$.

2020 China Northern MO, P5

Tags: number theory , Digits , multiple

Find all positive integers $a$ so that for any $\left \lfloor \frac{a+1}{2} \right \rfloor$-digit number that is composed of only digits $0$ and $2$ (where $0$ cannot be the first digit) is not a multiple of $a$.

2006 Hanoi Open Mathematics Competitions, 1

Tags: number theory , Digits

What is the last two digits of the number $(11 + 12 + 13 + ... + 2006)^2$?

2024 Germany Team Selection Test, 2

Tags: number theory , number theory proposed , Digits , product of digits , Divisibility

Show that there exists a real constant $C>1$ with the following property: For any positive integer $n$, there are at least $C^n$ positive integers with exactly $n$ decimal digits, which are divisible by the product of their digits. (In particular, these $n$ digits are all non-zero.) [i]Proposed by Jean-Marie De Koninck and Florian Luca[/i]

1965 All Russian Mathematical Olympiad, 059

Tags: number theory , Digits

A bus ticket is considered to be lucky if the sum of the first three digits equals to the sum of the last three ($6$ digits in Russian buses). Prove that the sum of all the lucky numbers is divisible by $13$.

2019 Saudi Arabia JBMO TST, 3

Tags: number theory , Digits

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

2013 Costa Rica - Final Round, 6

Tags: number theory , Digits , divisible

Let $a$ and $ b$ be positive integers (of one or more digits) such that $ b$ is divisible by $a$, and if we write $a$ and $ b$, one after the other in this order, we get the number $(a + b)^2$. Prove that $\frac{b}{a}= 6$.

2000 Austria Beginners' Competition, 3

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

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?

1997 Israel Grosman Mathematical Olympiad, 1

Tags: number theory , Digits , primes

Prove that there are at most three primes between $10$ and $10^{10}$ all of whose decimal digits are $1$.

2019 Ecuador Juniors, 1

Tags: number theory , Digits

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.

1947 Moscow Mathematical Olympiad, 134

Tags: Digits , number theory

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

2021 Israel National Olympiad, P1

Tags: combinatorics , number theory , Digits

Sophie wrote on a piece of paper every integer number from 1 to 1000 in decimal notation (including both endpoints). [b]a)[/b] Which digit did Sophie write the most? [b]b)[/b] Which digit did Sophie write the least?

2017 Israel National Olympiad, 2

Tags: number theory , Product , Digits , maximize

Denote by $P(n)$ the product of the digits of a positive integer $n$. For example, $P(1948)=1\cdot9\cdot4\cdot8=288$. [list=a] [*] Evaluate the sum $P(1)+P(2)+\dots+P(2017)$. [*] Determine the maximum value of $\frac{P(n)}{n}$ where $2017\leq n\leq5777$. [/list]

2002 Estonia National Olympiad, 2

Tags: Power , Digits , number theory

Does there exist an integer containing only digits $2$ and $0$ which is a $k$-th power of a positive integer ($k \ge2$)?

2022 Mediterranean Mathematics Olympiad, 2

Tags: number theory , divisible , Digits

(a) Decide whether there exist two decimal digits $a$ and $b$, such that every integer with decimal representation $ab222 ... 231$ is divisible by $73$. (b) Decide whether there exist two decimal digits $c$ and $d$, such that every integer with decimal representation $cd222... 231$ is divisible by $79$.

1970 IMO, 2

Tags: inequalities , algebra , decimal representation , Digits , IMO , IMO 1970

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 $A'B<AB'$.