Olympiad problems

1998 All-Russian Olympiad Regional Round, 11.5

A whole number is written on the board. Its last digit is remembered is then erased and multiplied by $5$ added to the number that remained on the board after erasing. The number was originally written $7^{1998}$. After applying several such operations, can one get the number $1998^7$?

2002 Estonia National Olympiad, 2

Tags: digit , divisible , number theory

Do there exist distinct non-zero digits $a, b$ and $c$ such that the two-digit number $\overline{ab}$ is divisible by $c$, the number $\overline{bc}$ is divisible by $a$ and $\overline{ca}$, is divisible by $b$?

2013 Junior Balkan Team Selection Tests - Romania, 2

Tags: sum of digits , digit , 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 $81$.

1996 Denmark MO - Mohr Contest, 4

Tags: number theory , perfect square , last digit , digit

Regarding a natural number $n$, it is stated that the number $n^2$ has $7$ as the second to last digit. What is the last digit of $n^2$?

II Soros Olympiad 1995 - 96 (Russia), 11.5

Tags: number theory , digit

Let's consider all possible natural seven-digit numbers, in the decimal notation of which the numbers $1$, $2$, $3$, $4$, $5$, $6$, $7$ are used once each. Let's number these numbers in ascending order. What number will be the $1995th$ ?

2015 CHMMC (Fall), 1

Tags: number theory , last digit , digit

Call a positive integer $x$ $n$-[i]cube-invariant[/i] if the last $n$ digits of $x$ are equal to the last $n$ digits of $x^3$. For example, $1$ is $n$-cube invariant for any integer $n$. How many $2015$-cube-invariant numbers $x$ are there such that $x < 10^{2015}$?

2016 Brazil Team Selection Test, 1

Tags: digit , number theory

For each positive integer $n$, determine the digits of units and hundreds of the decimal representation of the number $$\frac{1 + 5^{2n+1}}{6}$$

1998 IMO Shortlist, 7

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

Prove that for each positive integer $n$, there exists a positive integer with the following properties: It has exactly $n$ digits. None of the digits is 0. It is divisible by the sum of its digits.

2018 Federal Competition For Advanced Students, P1, 3

Tags: combinatorics , game , number theory , decimal representation , digit

Alice and Bob determine a number with $2018$ digits in the decimal system by choosing digits from left to right. Alice starts and then they each choose a digit in turn. They have to observe the rule that each digit must differ from the previously chosen digit modulo $3$. Since Bob will make the last move, he bets that he can make sure that the final number is divisible by $3$. Can Alice avoid that? [i](Proposed by Richard Henner)[/i]

2020 Austrian Junior Regional Competition, 2

Tags: combinatorics , digit

How many positive five-digit integers are there that have the product of their five digits equal to $900$? (Karl Czakler)

2016 Hanoi Open Mathematics Competitions, 1

Tags: digit , number theory

How many are there $10$-digit numbers composed from the digits $1, 2, 3$ only and in which, two neighbouring digits differ by $1$ : (A): $48$ (B): $64$ (C): $72$ (D): $128$ (E): None of the above.

1970 All Soviet Union Mathematical Olympiad, 142

Tags: number theory , digit

All natural numbers containing not more than $n$ digits are divided onto two groups. The first contains the numbers with the even sum of the digits, the second -- with the odd sum. Prove that if $0<k<n$ than the sum of the $k$-th powers of the numbers in the first group equals to the sum of the $k$-th powers of the numbers in the second group.

2018 India PRMO, 20

Tags: product , digit , number theory

Determine the sum of all possible positive integers $n, $ the product of whose digits equals $n^2 -15n -27$.

2001 Mexico National Olympiad, 1

Tags: number theory , digit

Find all $7$-digit numbers which are multiples of $21$ and which have each digit $3$ or $7$.

1991 Tournament Of Towns, (319) 6

Tags: number theory , arithmetic progression , digit

An arithmetical progression (whose difference is not equal to zero) consists of natural numbers without any nines in its decimal notation. (a) Prove that the number of its terms is less than $100$. (b) Give an example of such a progression with $72$ terms. (c) Prove that the number of terms in any such progression does not exceed $72$. (V. Bugaenko and Tarasov, Moscow)

1960 Polish MO Finals, 5

Tags: algebra , number theory , digit

From the digits $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$ all possible four-digit numbers with different digits are formed. Find the sum of these numbers.

2017 Romania National Olympiad, 1

Tags: number theory , digit

Consider the set $$M = \left\{\frac{a}{\overline{ba}}+\frac{b}{\overline{ab}} \, | a,b\in\{1,2,3,4,5,6,7,8,9\} \right\}.$$ a) Show that the set $M$ contains no integer. b) Find the smallest and the largest element of $M$

2021 Puerto Rico Team Selection Test, 4

Tags: digit , multiple , divisible , number theory

How many numbers $\overline{abcd}$ with different digits satisfy the following property: if we replace the largest digit with the digit $1$ results in a multiple of $30$?

2021 German National Olympiad, 3

Tags: digit , sum of cubes , combinatorics , algebra

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 Tournament Of Towns, (307) 4

Tags: number theory , digit , recurrence relation

A sequence $a_n$ is determined by the rules $a_0 = 9$ and for any nonnegative $k$, $$a_{k+1}=3a_k^4+4a_k^3.$$ Prove that $a_{10}$ contains more than $1000$ nines in decimal notation. (Yao)

1998 Tournament Of Towns, 4

Tags: 3-digit , sum , product , digit , number theory

For every three-digit number, we take the product of its three digits. Then we add all of these products together. What is the result? (G Galperin)

1994 Tournament Of Towns, (440) 6

Tags: digit , number theory

Let $c_n$ be the first digit of $2^n$ (in decimal representation). Prove that the number of different $13$-tuples $< c_k$,$...$, $c_{k+12}>$ is equal to $57$. (AY Belov,)

1983 Tournament Of Towns, (035) O4

Tags: permutation , number theory , digit , sum of digits , combinatorics

The natural numbers $M$ and $K$ are represented by different permutations of the same digits. Prove that (a) The sum of the digits of $2M$ equals the sum of the digits of $2K$. (b) The sum of the digits of $M/2$ equals the sum of the digits of $K/2$ ($M, K$ both even). (c) The sum of the digits of $5M$ equals the sum of the digits of $5 K$. (AD Lisitskiy)

2023 Indonesia MO, 4

Tags: number theory , digit , nt construction

Determine whether or not there exists a natural number $N$ which satisfies the following three criteria: 1. $N$ is divisible by $2^{2023}$, but not by $2^{2024}$, 2. $N$ only has three different digits, and none of them are zero, 3. Exactly 99.9% of the digits of $N$ are odd.

2020 Flanders Math Olympiad, 2

Tags: combinatorics , number theory , digit

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.