This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 159

2007 Junior Balkan Team Selection Tests - Moldova, 1
Tags: number theory , digit , sum , product , sum of digits
The numbers $d_1, d_2,..., d_6$ are distinct digits of the decimal number system other than $6$. Prove that $d_1+d_2+...+d_6= 36$ if and only if $(d_1-6) (d_2-6) ... (d_6 -6) = -36$.

2015 Romania National Olympiad, 4
Tags: number theory , sum of digits
A positive integer will be called [i]typical[/i] if the sum of its decimal digits is a multiple of $2011$. a) Show that there are infinitely many [i]typical[/i] numbers, each having at least $2011$ multiples which are also typical numbers. b) Does there exist a positive integer such that each of its multiples is typical?

2021 Final Mathematical Cup, 3
Tags: inequalities , number theory , algebra , sum of digits
For every positive integer $n$, $s(n)$ denotes the sum of the digits in the decimal representation of $n$. Prove that for every integer $n \ge 5$, we have $$S(1)S(3)...S(2n-1) \ge S(2)S(4)...S(2n)$$

2021 Girls in Mathematics Tournament, 3
Tags: number theory , divisible , sum of digits
A natural number is called [i]chaotigal [/i] if it and its successor both have the sum of their digits divisible by $2021$. How many digits are in the smallest chaotigal number?

2005 Paraguay Mathematical Olympiad, 1
Tags: number theory , sum of digits
With the digits $1, 2, 3,. . . . . . , 9$ three-digit numbers are written such that the sum of the three digits is $17$. How many numbers can be written?

1995 Tournament Of Towns, (482) 6
Tags: number theory , arithmetic sequence , sum of digits
Does there exist an increasing arithmetic progression of (a) $11$ (b) $10000$ (c) infinitely many positive integers such that the sums of their digits in base $10$ also form an increasing arithmetic progression? (A Shapovalov)

2000 Belarus Team Selection Test, 4.3
Tags: algebra , arithmetic sequence , divisibility , sum of digits , modular arithmetic
Prove that for every real number $M$ there exists an infinite arithmetic progression such that: - each term is a positive integer and the common difference is not divisible by 10 - the sum of the digits of each term (in decimal representation) exceeds $M$.

Oliforum Contest III 2012, 2
Tags: number theory , polynomial , sum of digits
Show that for every polynomial $f(x)$ with integer coefficients, there exists a integer $C$ such that the set $\{n \in Z :$ the sum of digits of $f(n)$ is $C\}$ is not finite.

2016 Rioplatense Mathematical Olympiad, Level 3, 6
Tags: number theory , sum of digits , sequence , divisibility
When the natural numbers are written one after another in an increasing way, you get an infinite succession of digits $123456789101112 ....$ Denote $A_k$ the number formed by the first $k$ digits of this sequence . Prove that for all positive integer $n$ there is a positive integer $m$ which simultaneously verifies the following three conditions: (i) $n$ divides $A_m$, (ii) $n$ divides $m$, (iii) $n$ divides the sum of the digits of $A_m$.