Olympiad problems

1968 German National Olympiad, 4

Sixteen natural numbers written in the decimal system may form a geometric sequence, of which the first five members have nine digits, five further members have ten digits, four members have eleven digits and two terms have twelve digits. Prove that there is exactly one sequence with these properties.

1999 Cono Sur Olympiad, 4

Tags: number theory , coloring , multiple , digit

Let $A$ be a six-digit number, three of which are colored and equal to $1, 2$, and $4$. Prove that it is always possible to obtain a number that is a multiple of $7$, by performing only one of the following operations: either delete the three colored figures, or write all the numbers of $A$ in some order.

2022 IFYM, Sozopol, 4

Tags: number theory , digit , sum of digits

A natural number $x$ is written on the board. In one move, we can take the number on the board and between any two of its digits in its decimal notation we can we put a sign $+$, or we may not put it, then we calculate the obtained result and we write it on the board in place of $x$. For example, from the number $819$. we can get $18$ by $8 + 1 + 9$, $90$ by $81 + 9$, and $27$ by $8 + 19$. Prove that no matter what $x$ is, we can reach a single digit number with at most $4$ moves.

2005 May Olympiad, 1

Tags: number theory , digit

Find the smallest $3$-digit number that is the product of two $2$-digit numbers , so that the seven digits of these three numbers are all different.

2014 Contests, 2

Tags: number theory , sum of cubes , sequence , digit , algebra

The first term of a sequence is $2014$. Each succeeding term is the sum of the cubes of the digits of the previous term. What is the $2014$ th term of the sequence?

1997 VJIMC, Problem 4-M

Tags: real analysis , limit , digit , number theory

Find all real numbers $a>0$ for which the series $$\sum_{n=1}^\infty\frac{a^{f(n)}}{n^2}$$is convergent; $f(n)$ denotes the number of $0$'s in the decimal expansion of $f$.

2007 Regional Olympiad of Mexico Center Zone, 6

Tags: number theory , digit

Certain tickets are numbered as follows: $1, 2, 3, \dots, N $. Exactly half of the tickets have the digit $ 1$ on them. If $N$ is a three-digit number, determine all possible values of $N $.

1988 Tournament Of Towns, (175) 1

Tags: number theory , digit

Is it possible to select two natural numbers $m$ and $n$ so that the number $n$ results from a permutation of the digits of $m$, and $m+n =999 . . . 9$ ?

1989 Tournament Of Towns, (217) 1

Tags: divide , divisible , digit , number theory

Find a pair of $2$ six-digit numbers such that, if they are written down side by side to form a twelve-digit number , this number is divisible by the product of the two original numbers. Find all such pairs of six-digit numbers. ( M . N . Gusarov, Leningrad)

2003 May Olympiad, 1

Tags: number theory , digit , algebra

Pedro writes all the numbers with four different digits that can be made with digits $a, b, c, d$, that meet the following conditions: $$ a\ne 0 \, , \, b=a+2 \, , \, c=b+2 \, , \, d=c+2$$ Find the sum of all the numbers Pedro wrote.

2017 Romania National Olympiad, 1

Tags: digit , number theory

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 $M$

1990 IMO Shortlist, 8

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}).$

2018 India PRMO, 20

Tags: number theory , product , digit

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

1968 IMO, 2

Tags: decimal representation , digit , equation , number theory

Find all natural numbers $n$ the product of whose decimal digits is $n^2-10n-22$.

2024 Kyiv City MO Round 2, Problem 1

Tags: number theory , digit

For some positive integer $n$, Katya wrote on the board next to each other numbers $2^n$ and $14^n$ (in this order), thus forming a new number $A$. Can the number $A - 1$ be prime? [i]Proposed by Oleksii Masalitin[/i]

2007 Junior Balkan Team Selection Tests - Moldova, 5

Tags: number theory , digit , product of digits

Determine the smallest natural number written in the decimal system with the product of the digits equal to $10! = 1 \cdot 2 \cdot 3\cdot ... \cdot9\cdot10$.

1994 Chile National Olympiad, 3

Tags: prime , digit , number theory

Let $x$ be an integer of $n$ digits, all equal to $ 1$. Show that if $x$ is prime, then $n$ is also prime.

2022 Durer Math Competition Finals, 7

Tags: number theory , digit

The [i]fragments [/i] of a positive integer are the numbers seen when reading one or more of its digits in order. The [i]fragment sum[/i] equals the sum of all the fragments, including the number itself. For example, the fragment sum of $2022$ is $2022+202+022+20+02+22+2+0+2+2 = 2296$. There is another four-digit number with the same fragment sum. What is it? As the example shows, if a fragment occurs multiple times, then all its occurrences are added, and the fragments beginning with $0$ also count (for instance, $022$ is worth $22$).

1998 Tuymaada Olympiad, 6

Tags: digit , periodic , periodic sequence , number theory

Prove that the sequence of the first digits of the numbers in the form $2^n+3^n$ is nonperiodic.

2010 Singapore Junior Math Olympiad, 2

Tags: number theory , digit , multiple

Find the sum of all the $5$-digit integers which are not multiples of $11$ and whose digits are $1, 3, 4, 7, 9$.

2016 Puerto Rico Team Selection Test, 2

Tags: number theory , digit

Determine all $6$-digit numbers $(abcdef)$ such that $(abcdef) = (def)^2$ where $(x_1x_2...x_n)$ is not a multiplication but a number of $n$ digits.

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.

2019 Durer Math Competition Finals, 12

Tags: last digit , digit , number theory , polynomial

$P$ and $Q$ are two different non-constant polynomials such that $P(Q(x)) = P(x)Q(x)$ and $P(1) = P(-1) = 2019$. What are the last four digits of $Q(P(-1))$?

1998 Israel National Olympiad, 2

Tags: decimal representation , digit , number theory

Show that there is a multiple of $2^{1998}$ whose decimal representation consists only of the digits $1$ and $2$.

1991 Tournament Of Towns, (287) 3

Tags: number theory , digit

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)