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

2000 Slovenia National Olympiad, Problem 1
Tags: digit
In the expression $4\cdot\text{RAKEC}=\text{CEKAR}$, each letter represents a (decimal) digit. Replace the letters so that the equality is true.

2019 Cono Sur Olympiad, 2
Tags: digit , number theory
We say that a positive integer $M$ with $2n$ digits is [i]hypersquared[/i] if the following three conditions are met: [list] [*]$M$ is a perfect square. [*]The number formed by the first $n$ digits of $M$ is a perfect square. [*]The number formed by the last $n$ digits of $M$ is a perfect square and has exactly $n$ digits (its first digit is not zero). [/list] Find a hypersquared number with $2000$ digits.

2019 Bundeswettbewerb Mathematik, 4
Tags: digit , sqrt , number theory
In the decimal expansion of $\sqrt{2}=1.4142\dots$, Isabelle finds a sequence of $k$ successive zeroes where $k$ is a positive integer. Show that the first zero of this sequence can occur no earlier than at the $k$-th position after the decimal point.

2017 Saudi Arabia BMO TST, 3
Tags: digit , multiple , number theory
How many ways are there to insert plus signs $+$ between the digits of number $111111 ...111$ which includes thirty of digits $1$ so that the result will be a multiple of $30$?

2001 Estonia National Olympiad, 2
Tags: digit , remainder , number theory , sum of digits
Dividing a three-digit number by the number obtained from it by swapping its first and last digit we get $3$ as the quotient and the sum of digits of the original number as the remainder. Find all three-digit numbers with this property.

1984 All Soviet Union Mathematical Olympiad, 386
Tags: 3-digit , digit , prime number , prime , number theory
Let us call "absolutely prime" the prime number, if having transposed its digits in an arbitrary order, we obtain prime number again. Prove that its notation cannot contain more than three different digits.

2012 Singapore Junior Math Olympiad, 2
Tags: digit , number theory
Does there exist an integer $A$ such that each of the ten digits $0, 1, . . . , 9$ appears exactly once as a digit in exactly one of the numbers $A, A^2, A^ 3$ ?

2007 Estonia National Olympiad, 1
Tags: divide , digit , number theory
The seven-digit integer numbers are different in pairs and this number is divided by each of its own numbers. a) Find all possibilities for the three numbers that are not included in this number. b) Give an example of such a number.

1972 IMO Longlists, 22
Tags: digit , decimal representation , number theory , modular arithmetic
Show that for any $n \not \equiv 0 \pmod{10}$ there exists a multiple of $n$ not containing the digit $0$ in its decimal expansion.

2020 Puerto Rico Team Selection Test, 2
Tags: system of equations , digit , diophantine equation , diophantine , number theory
The cost of $1000$ grams of chocolate is $x$ dollars and the cost of $1000$ grams of potatoes is $y$ dollars, the numbers $x$ and $y$ are positive integers and have not more than $2$ digits. Mother said to Maria to buy $200$ grams of chocolate and $1000$ grams of potatoes that cost exactly $N$ dollars. Maria got confused and bought $1000$ grams of chocolate and $200$ grams of potatoes that cost exactly $M$ dollars ($M >N$). It turned out that the numbers $M$ and $N$ have no more than two digits and are formed of the same digits but in a different order. Find $x$ and $y$.

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 ($k \ge2$)?

1985 Tournament Of Towns, (089) 5
Tags: digit , square table , combinatorics
The digits $0, 1 , 2, ..., 9$ are written in a $10 x 10$ table , each number appearing $10$ times . (a) Is it possible to write them in such a way that in any row or column there would be not more than $4$ different digits? (b) Prove that there must be a row or column containing more than $3$ different digits . { L . D . Kurlyandchik , Leningrad)

2024 Irish Math Olympiad, P4
Tags: digit , number theory
How many 4-digit numbers $ABCD$ are there with the property that $|A-B|= |B-C|= |C-D|$? Note that the first digit $A$ of a four-digit number cannot be zero.

2017 May Olympiad, 1
Tags: digit , odd , number theory
To each three-digit number, Matías added the number obtained by inverting its digits. For example, he added $729$ to the number $927$. Calculate in how many cases the result of the sum of Matías is a number with all its digits odd.

1972 IMO Shortlist, 6
Tags: digit , decimal representation , modular arithmetic , number theory
Show that for any $n \not \equiv 0 \pmod{10}$ there exists a multiple of $n$ not containing the digit $0$ in its decimal expansion.

2019 Durer Math Competition Finals, 14
Tags: last digit , digit , number theory
Let $S$ be the set of all positive integers less than $10,000$ whose last four digits in base $2$ are the same as its last four digits in base $5$. What remainder do we get if we divide the sum of all elements of $S$ by $10000$?

2017 Finnish National High School Mathematics Comp, 3
Tags: digit , last digit , number theory
Consider positive integers $m$ and $n$ for which $m> n$ and the number $22 220 038^m-22 220 038^n$ has are eight zeros at the end. Show that $n> 7$.

2017 Romania National Olympiad, 4
Tags: digit , prime number , prime , number theory
Find all prime numbers with $n \ge 3$ digits, having the property: for every $k \in \{1, 2, . . . , n -2\}$, deleting any $k$ of its digits leaves a prime number.

2022 Denmark MO - Mohr Contest, 2
Tags: palindrome , digit , number theory
A positive integer is a [i]palindrome [/i] if it is written identically forwards and backwards. For example, $285582$ is a palindrome. A six digit number $ABCDEF$, where $A, B, C, D, E, F$ are digits, is called [i]cozy [/i] if $AB$ divides $CD$ and $CD$ divides $EF$. For example, $164896$ is cozy. Determine all cozy palindromes.

2020 Polish Junior MO First Round, 6.
Tags: digit , number theory
Let $a$, $b$ $c$ be the natural numbers, such that every digit occurs exactly the same number of times in each of the numbers $a$, $b$, $c$. Is it possible that $a + b + c = 10^{1001}$? Justify your answer.

2014 Hanoi Open Mathematics Competitions, 3
Tags: sequence , digit , algebra
How many $0$'s are there in the sequence $x_1, x_2,..., x_{2014}$ where $x_n =\big[ \frac{n + 1}{\sqrt{2015}}\big] -\big[ \frac{n }{\sqrt{2015}}\big]$ , $n = 1, 2,...,2014$ ? (A): $1128$, (B): $1129$, (C): $1130$, (D): $1131$, (E) None of the above.

2010 Puerto Rico Team Selection Test, 2
Tags: digit , number theory
Find two three-digit numbers $x$ and $y$ such that the sum of all other three digit numbers is equal to $600x$.

2004 Austria Beginners' Competition, 1
Tags: digit , number theory
Find the smallest four-digit number that when divided by $3$ gives a four-digit number with the same digits. (Note: Four digits means that the thousand Unit digit must not be $0$.)

2018 Singapore Junior Math Olympiad, 1
Tags: divide , digit , multiple , number theory
Consider the integer $30x070y03$ where $x, y$ are unknown digits. Find all possible values of $x, y$ so that the given integer is a multiple of $37$.

1969 All Soviet Union Mathematical Olympiad, 117
Tags: algebra , digit , sequence
Given a finite sequence of zeros and ones, which has two properties: a) if in some arbitrary place in the sequence we select five digits in a row and also select five digits in any other place in a row, then these fives will be different (they may overlap); b) if you add any digit to the right of the sequence, then property (a) will no longer hold true. Prove that the first four digits of our sequence coincide with the last four