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: 573

2003 Cuba MO, 1
Tags: number theory , digit , prime
Given the following list of numbers: $$1990, 1991, 1992, ..., 2002, 2003, 2003, 2003, ..., 2003$$ where the number $2003$ appears $12$ times. Is it possible to write these numbers in some order so that the $100$-digit number that we get is prime?

2009 Tournament Of Towns, 4
Tags: digit , number theory
We increased some positive integer by $10\%$ and obtained a positive integer. Is it possible that in doing so we decreased the sum of digits exactly by $10\%$ ?

1952 Moscow Mathematical Olympiad, 216
Tags: integer sequence , sequence , sum of squares , digit , number theory
A sequence of integers is constructed as follows: $a_1$ is an arbitrary three-digit number, $a_2$ is the sum of squares of the digits of $a_1, a_3$ is the sum of squares of the digits of $a_2$, etc. Prove that either $1$ or $4$ must occur in the sequence $a_1, a_2, a_3, ....$

1980 All Soviet Union Mathematical Olympiad, 303
Tags: limit , irrational , sequence , algebra , digit
The number $x$ from $[0,1]$ is written as an infinite decimal fraction. Having rearranged its first five digits after the point we can obtain another fraction that corresponds to the number $x_1$. Having rearranged five digits of $x_k$ from $(k+1)$-th till $(k+5)$-th after the point we obtain the number $x_{k+1}$. a) Prove that the sequence $x_i$ has limit. b) Can this limit be irrational if we have started with the rational number? c) Invent such a number, that always produces irrational numbers, no matter what digits were transposed.

1999 Portugal MO, 1
Tags: number theory , digit
A number is said to be [i]balanced [/i] if one of its digits is average of the others. How many [i]balanced [/i]$3$-digit numbers are there?

1980 IMO, 3
Tags: modular arithmetic , combinatorics , decimal representation , digit
Find the digits left and right of the decimal point in the decimal form of the number \[ (\sqrt{2} + \sqrt{3})^{1980}. \]

2001 German National Olympiad, 5
Tags: fibonacci number , number theory , digit
The Fibonacci sequence is given by $x_1 = x_2 = 1$ and $x_{k+2} = x_{k+1} + x_k$ for each $k \in N$. (a) Prove that there are Fibonacci numbes that end in a $9$ in the decimal system. (b) Determine for which $n$ can a Fibonacci number end in $n$ $9$-s in the decimal system.

2024 Ukraine National Mathematical Olympiad, Problem 1
Tags: number theory , digit
Oleksiy wrote several distinct positive integers on the board and calculated all their pairwise sums. It turned out that all digits from $0$ to $9$ appear among the last digits of these sums. What could be the smallest number of integers that Oleksiy wrote? [i]Proposed by Oleksiy Masalitin[/i]

1999 Bundeswettbewerb Mathematik, 2
Tags: number theory , sum of digits , digit , inequalities
For every natural number $n$, let $Q(n)$ denote the sum of the decimal digits of $n$. Prove that there are infinitely many positive integers $k$ with $Q(3^k) \ge Q(3^{k+1})$.

1970 All Soviet Union Mathematical Olympiad, 139
Tags: number theory , digit
Prove that for every natural number $k$ there exists an infinite set of such natural numbers $t$, that the decimal notation of $t$ does not contain zeroes and the sums of the digits of the numbers $t$ and $kt$ are equal.

1975 IMO, 4
Tags: modular arithmetic , divisibility , digit , decimal representation , digit sum
When $4444^{4444}$ is written in decimal notation, the sum of its digits is $ A.$ Let $B$ be the sum of the digits of $A.$ Find the sum of the digits of $ B.$ ($A$ and $B$ are written in decimal notation.)

2004 Estonia National Olympiad, 3
Tags: number theory , digit , 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.

2016 Ecuador Juniors, 6
Tags: number theory , digit
Determine the number of positive integers $N = \overline{abcd}$, with $a, b, c, d$ nonzero digits, which satisfy $(2a -1) (2b -1) (2c- 1) (2d - 1) = 2abcd -1$.

2009 Junior Balkan Team Selection Tests - Romania, 1
Tags: number theory , digit , multiple
For all positive integers $n$ define $a_n=2 \underbrace{33...3}_{n \, times}$, where digit $3$ occurs $n$ times. Show that the number $a_{2009}$ has infinitely many multiples in the set $\{a_n | n \in N*\}$.

1975 IMO Shortlist, 5
Tags: decimal representation , digit , series summation , inequality , algebra
Let $M$ be the set of all positive integers that do not contain the digit $9$ (base $10$). If $x_1, \ldots , x_n$ are arbitrary but distinct elements in $M$, prove that \[\sum_{j=1}^n \frac{1}{x_j} < 80 .\]

2017 Ecuador Juniors, 1
Tags: number theory , digit
An ancient Inca legend tells that a monster lives among the mountains that when wakes up, eats everyone who read this issue. After such a task, the monster returns to the mountains and sleeps for a number of years equal to the sum of its digits of the year in which you last woke up. The monster woke up for the first time in the year $234$. a) Would the monster have woken up between the years $2005$ and $2015$? b) Will we be safe in the next $10$ years?

2018 Hanoi Open Mathematics Competitions, 5
Tags: number theory , digit
Find all $3$-digit numbers $\overline{abc}$ ($a,b \ne 0$) such that $\overline{bcd} \times  a = \overline{1a4d}$ for some integer $d$ from $1$ to $9$

2024 Czech-Polish-Slovak Junior Match, 5
Tags: palindrome , digit , powers of 2 , number theory
Is there a positive integer $n$ such that when we write the decimal digits of $2^n$ in opposite order, we get another integer power of $2$?

2018 Bundeswettbewerb Mathematik, 1
Tags: digit , number theory , arithmetic
Find the largest positive integer with the property that each digit apart from the first and the last one is smaller than the arithmetic mean of her neighbours.

1989 Tournament Of Towns, (205) 3
Tags: number theory , divisible , divide , digit
What digit must be put in place of the "$?$" in the number $888...88?999...99$ (where the $8$ and $9$ are each written $50$ times) in order that the resulting number is divisible by $7$? (M . I. Gusarov)

VII Soros Olympiad 2000 - 01, 11.4
Tags: number theory , digit
Let $a$ be the largest root of the equation $x^3 - 3x^2 + 1 = 0$. Find the first $200$ decimal digits for the number $a^{2000}$.

1993 Tournament Of Towns, (394) 2
Tags: number theory , digit
The decimal representation of all integers from $1$ to an arbitrary integer $n$ are written one after another as such: $$123... 91011... 99100... (n).$$ Does there exist $n$ such that each of the digits $0,1,2,...,9$ appears the same number of times in the given sequence? (A Andzans)

2002 Estonia National Olympiad, 2
Tags: power , digit , number theory
Does there exist an integer containing only digits $2$ and $0$ which is a $k$-th power of a positive integer ($k \ge2$)?

2020 HK IMO Preliminary Selection Contest, 1
Tags: algebra , digit
Let $n=(10^{2020}+2020)^2$. Find the sum of all the digits of $n$.

2018 Auckland Mathematical Olympiad, 1
Tags: number theory , digit , multiple
Find a multiple of $2018$ whose decimal expansion's first four digits are $2017$.