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: 546

1999 Greece JBMO TST, 3
Tags: number theory , decimal representation , Digits
Find digits $a,b,c,x$ ($a>0$) such that $\overline{abc}+\overline{acb}=\overline{199x}$

2005 iTest, 27
Tags: number theory , Perfect Square , Digits
Find the sum of all non-zero digits that can repeat at the end of a perfect square. (For example, if $811$ were a perfect square, $1$ would be one of these non-zero digits.)

2019 Lusophon Mathematical Olympiad, 1
Tags: number theory , Digits , divisible
Find a way to write all the digits of $1$ to $9$ in a sequence and without repetition, so that the numbers determined by any two consecutive digits of the sequence are divisible by $7$ or $13$.

1969 Poland - Second Round, 2
Tags: number theory , Digits
Find all four-digit numbers in which the thousands digit is equal to the hundreds digit and the tens digit is equal to the units digit and which are squares of integers.

2021 Pan-American Girls' Math Olympiad, Problem 4
Tags: Digits , number theory , PAGMO
Lucía multiplies some positive one-digit numbers (not necessarily distinct) and obtains a number $n$ greater than 10. Then, she multiplies all the digits of $n$ and obtains an odd number. Find all possible values of the units digit of $n$. $\textit{Proposed by Pablo Serrano, Ecuador}$

1986 Brazil National Olympiad, 4
Tags: number theory , Digits
Find all $10$ digit numbers $a_0a_1...a_9$ such that for each $k, a_k$ is the number of times that the digit $k$ appears in the number.

2008 May Olympiad, 1
Tags: number theory , multiple , Digits
How many different numbers with $6$ digits and multiples of $45$ can be written by adding one digit to the left and one to the right of $2008$?

1925 Eotvos Mathematical Competition, 2
Tags: number theory , Digits , last digits
How maay zeros are there at the end of the number $$1000! = 1 \cdot 2 \cdot 3 \cdot ... \cdot 999 \cdot 1000?$$

1996 Austrian-Polish Competition, 1
Tags: number theory , divides , divisible , Digits
Let $k \ge 1$ be a positive integer. Prove that there exist exactly $3^{k-1}$ natural numbers $n$ with the following properties: (i) $n$ has exactly $k$ digits (in decimal representation), (ii) all the digits of $n$ are odd, (iii) $n$ is divisible by $5$, (iv) the number $m = n/5$ has $k$ odd digits

2018 Junior Regional Olympiad - FBH, 2
Tags: blackboard , Digits , original number
On blackboard is written $3$ digit number so all three digits are distinct than zero. Out of it, we made three $2$ digit numbers by crossing out first digit of original number, crossing out second digit of original number and crossing out third digit of original number. Sum of those three numbers is $293$. Which number is written on blackboard?

1954 Moscow Mathematical Olympiad, 268
Tags: maximum , Digits , number theory
Delete $100$ digits from the number $1234567891011... 9899100$ so that the remaining number were as big as possible.

2016 KOSOVO TST, 2
Tags: Digits , induction , algebra
Show that for any $n\geq 2$, $2^{2^n}+1$ ends with 7

2013 Tournament of Towns, 2
Tags: Digits , number theory , Composite
Does there exist a ten-digit number such that all its digits are different and after removing any six digits we get a composite four-digit number?

1994 All-Russian Olympiad Regional Round, 9.3
Tags: number theory , Digits , trinomial
Does there exist a quadratic trinomial $p(x)$ with integer coefficients such that, for every natural number $n$ whose decimal representation consists of digits $1$, $p(n)$ also consists only of digits $1$?

2020 Tuymaada Olympiad, 7
Tags: number theory , Digits
How many positive integers $N$ in the segment $\left[10, 10^{20} \right]$ are such that if all their digits are increased by $1$ and then multiplied, the result is $N+1$? [i](F. Bakharev)[/i]

1969 IMO Longlists, 40
Tags: combinatorics , decimal representation , counting , Digits , IMO Shortlist , IMO Longlist
$(MON 1)$ Find the number of five-digit numbers with the following properties: there are two pairs of digits such that digits from each pair are equal and are next to each other, digits from different pairs are different, and the remaining digit (which does not belong to any of the pairs) is different from the other digits.

1998 Tournament Of Towns, 4
Tags: 3-digit , Sum , Product , Digits , 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)

2011 May Olympiad, 5
Tags: number theory , Digits , divisible
We consider all $14$-digit positive integers, divisible by $18$, whose digits are exclusively $ 1$ and $2$, but there are no consecutive digits $2$. How many of these numbers are there?

2014 Contests, 2
Tags: algebra , sum of cubes , Sequence , Digits , number theory
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?

2010 Contests, 1
Tags: number theory , Sum of Squares , algebra , Digits , Digit
Nine positive integers $a_1,a_2,...,a_9$ have their last $2$-digit part equal to $11,12,13,14,15,16,17,18$ and $19$ respectively. Find the last $2$-digit part of the sum of their squares.

2023 Czech-Polish-Slovak Junior Match, 5
Tags: number theory , Digits
Mazo performs the following operation on triplets of non-negative integers: If at least one of them is positive, it chooses one positive number, decreases it by one, and replaces the digits in the units place with the other two numbers. It starts with the triple $x$, $y$, $z$. Find a triple of positive integers $x$, $y$, $z$ such that $xy + yz + zx = 1000$ (*) and the number of operations that Mazo can subsequently perform with the triple $x, y, z$ is (a) maximal (i.e. there is no triple of positive integers satisfying (*) that would allow him to do more operations); (b) minimal (i.e. every triple of positive integers satisfying (*) allows him to perform at least so many operations).

1980 All Soviet Union Mathematical Olympiad, 303
Tags: limit , irrational , Sequence , algebra , Digits
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.

II Soros Olympiad 1995 - 96 (Russia), 10.2
Tags: number theory , Digits
Find a number that increases by a factor of $1996$ if the digits in the first and fifth places after the decimal place are swapped in its decimal notation.

2015 India PRMO, 7
Tags: number theory , Digits
$7.$ Let $E(n)$ denote the sum of even digits of $n.$ For example, $E(1243)=2+4=6.$ What is the value of $E(1)+E(2)+E(3)+...+E(100) ?$

2001 Denmark MO - Mohr Contest, 2
Tags: number theory , factorial , Digits , last digits
If there is a natural number $n$ such that the number $n!$ has exactly $11$ zeros at the end? (With $n!$ is denoted the number $1\cdot 2\cdot 3 \cdot ... (n - )1 \cdot n$).