Olympiad problems

2024 Germany Team Selection Test, 2

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]

1998 Moldova Team Selection Test, 10

Tags: bounded , sequence , unbounded , digit , product of digits , recurrence relation , number theory

Let $P(x)$ denote the product of all (decimal) digits of a natural number $x$. For any positive integer $x_1$, define the sequence $(x_n)$ recursively by $x_{n+1} = x_n + P(x_n)$. Prove or disprove that the sequence $(x_n)$ is necessarily bounded.

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 $256$?

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?

2020 Regional Olympiad of Mexico West, 4

Tags: product of digits , number theory

Given a positive integer $ n $, we denote by $ P(n) $ the result of multiplying all the digits of $ n $. Find a number $ m $ with ten digits, none of them zero, with the following property: $$P\left(m+P(m)\right)= P (m)$$

1980 All Soviet Union Mathematical Olympiad, 297

Tags: number theory , product of digits , sequence

Let us denote with $P(n)$ the product of all the digits of $n$. Consider the sequence $$n_{k+1} = n_k + P(n_k)$$ Can it be unbounded for some $n_1$?

1987 Austrian-Polish Competition, 7

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

For any natural number $n= \overline{a_k...a_1a_0}$ $(a_k \ne 0)$ in decimal system write $p(n)=a_0 \cdot a_1 \cdot ... \cdot a_k$, $s(n)=a_0+ a_1+ ... + a_k$, $n^*= \overline{a_0a_1...a_k}$. Consider $P=\{n | n=n^*, \frac{1}{3} p(n)= s(n)-1\}$ and let $Q$ be the set of numbers in $P$ with all digits greater than $1$. (a) Show that $P$ is infinite. (b) Show that $Q$ is finite. (c) Write down all the elements of $Q$.

1984 Tournament Of Towns, (056) O4

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

The product of the digits of the natural number $N$ is denoted by $P(N)$ whereas the sum of these digits is denoted by $S(N)$. How many solutions does the equation $P(P(N)) + P(S(N)) + S(P(N)) + S(S(N)) = 1984$ have?

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 $10! = 1 \cdot 2 \cdot 3\cdot ... \cdot9\cdot10$.

1982 Austrian-Polish Competition, 4

Tags: sequence , bounded , unbounded , digit , product of digits , recurrence relation , number theory

Let $P(x)$ denote the product of all (decimal) digits of a natural number $x$. For any positive integer $x_1$, define the sequence $(x_n)$ recursively by $x_{n+1} = x_n + P(x_n)$. Prove or disprove that the sequence $(x_n)$ is necessarily bounded.

1964 Dutch Mathematical Olympiad, 5

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

Consider a sequence of non-negative integers g$_1,g_2,g_3,...$ each consisting of three digits (numbers smaller than $100$ are also written with three digits; the number $27$, for example, is written as $027$). Each number consists of the preceding by taking the product of the three digits that make up the preceding. The resulting sequence is of course dependent on the choice of $g_1$ (e.g. $g_1 = 359$ leads to $g_2= 135$, $g_3= 015$, $g_4 = 000$).Prove that independent of the choice of $g_1$: (a) $g_{n+1}\le g_n$ (b) $g_{10}= 000$.