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

2012 Romania National Olympiad, 4
Tags: number theory , digit
[i]Reduced name[/i] of a natural number $A$ with $n$ digits ($n \ge 2$) a number of $n-1$ digits obtained by deleting one of the digits of $A$: For example, the [i]reduced names[/i] of $1024$ is $124$, $104$ and $120$. Determine how many seven-digit numbers cannot be written as the sum of one natural numbers $A$ and a [i]reduced name[/i] of $A$.

2007 Puerto Rico Team Selection Test, 5
Tags: number theory , digit
Juan wrote a natural number and Maria added a digit $ 1$ to the left and a digit $ 1$ to the right. Maria's number exceeds to the number of Juan in $14789$. Find the number of Juan.

1989 Swedish Mathematical Competition, 1
Tags: number theory , digit , perfect square
Let $n$ be a positive integer. Prove that the numbers $n^2(n^2 + 2)^2$ and $n^4(n^2 + 2)^2$ are written in base $n^2 +1$ with the same digits but in opposite order.

2018 Malaysia National Olympiad, A2
Tags: digit , number theory
An integer has $2018$ digits and is divisible by $7$. The first digit is $d$, while all the other digits are $2$. What is the value of $d$?

2001 Mexico National Olympiad, 1
Tags: number theory , digit
Find all $7$-digit numbers which are multiples of $21$ and which have each digit $3$ or $7$.

1994 Tournament Of Towns, (440) 6
Tags: digit , number theory
Let $c_n$ be the first digit of $2^n$ (in decimal representation). Prove that the number of different $13$-tuples $< c_k$,$...$, $c_{k+12}>$ is equal to $57$. (AY Belov,)

1994 Chile National Olympiad, 3
Tags: prime , number theory , digit
Let $x$ be an integer of $n$ digits, all equal to $ 1$. Show that if $x$ is prime, then $n$ is also prime.

2018 Flanders Math Olympiad, 4
Tags: number theory , digit
Determine all three-digit numbers N such that $N^2$ has six digits and so that the sum of the number formed by the first three digits of $N^2$ and the number formed by the latter three digits of $N^2$ equals $N$.

2001 Estonia Team Selection Test, 5
Tags: product , power , digit , prime , number theory
Find the exponent of $37$ in the representation of the number $111...... 11$ with $3\cdot 37^{2000}$ digits equals to $1$, as product of prime powers

2009 Regional Olympiad of Mexico Center Zone, 4
Tags: consecutive , digit , number theory , perfect square
Let $N = 2 \: \: \underbrace {99… 9} _{n \,\,\text {times}} \: \: 82 \: \: \underbrace {00… 0} _{n \,\, \text {times} } \: \: 29$. Prove that $N$ can be written as the sum of the squares of $3$ consecutive natural numbers.

2000 Belarus Team Selection Test, 6.2
Tags: digit , perfect square , number theory
A positive integer $A_k...A_1A_0$ is called monotonic if $A_k \le ..\le A_1 \le A_0$. Show that for any $n \in N$ there is a monotonic perfect square with $n$ digits.

2010 Saudi Arabia Pre-TST, 1.3
Tags: divide , digit , divisible , number theory
1) Let $a$ and $b$ be relatively prime positive integers. Prove that there is a positive integer $n$ such that $1 \le n \le b$ and $b$ divides $a^n - 1$. 2) Prove that there is a multiple of $7^{2010}$ of the form $99... 9$ ($n$ nines), for some positive integer $n$ not exceeding $7^{2010}$.

1972 IMO Shortlist, 6
Tags: modular arithmetic , number theory , decimal representation , digit
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.

2009 Switzerland - Final Round, 2
Tags: number theory , digit
A [i]palindrome [/i] is a natural number that works in the decimal system forwards and backwards read is the same size (e.g. $1129211$ or $7337$). Determine all pairs $(m, n)$ of natural numbers, such that $$(\underbrace{11... 11}_{m}) \cdot (\underbrace{11... 11}_{n})$$ is a palindrome.

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 \[ (\sqrt{2} + \sqrt{3})^{1980}. \]

2024 Bundeswettbewerb Mathematik, 2
Tags: digit , perfect square , number theory
Can a number of the form $44\dots 41$, with an odd number of decimal digits $4$ followed by a digit $1$, be a perfect square?

1986 Tournament Of Towns, (119) 1
Tags: number theory , digit , 2-digit , sum , difference
We are given two two-digit numbers , $x$ and $y$. It is known that $x$ is twice as big as $y$. One of the digits of $y$ is the sum, while the other digit of $y$ is the difference, of the digits of $x$ . Find the values of $x$ and $y$, proving that there are no others.

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

1964 Bulgaria National Olympiad, Problem 1
Tags: number theory , digit
A $6n$-digit number is divisible by $7$. Prove that if its last digit is moved to the beginning of the number then the new number is also divisible by $7$.

2016 India PRMO, 3
Tags: number theory , algebra , sum of digits , digit
Suppose $N$ is any positive integer. Add the digits of $N$ to obtain a smaller integer. Repeat this process of digit-addition till you get a single digit numbem. Find the number of positive integers $N \le 1000$, such that the final single-digit number $n$ is equal to $5$. Example: $N = 563\to (5 + 6 + 3) = 14 \to(1 + 4) = 5$ will be counted as one such integer.

2004 German National Olympiad, 3
Tags: number theory , divisibility , digit , sum of digits
Prove that for every positive integer $n$ there is an $n$-digit number $z$ with none of its digits $0$ and such that $z$ is divisible by its sum of digits.

II Soros Olympiad 1995 - 96 (Russia), 10.2
Tags: number theory , digit
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.

2014 Contests, 1
Tags: number theory , modular arithmetic , digit
Consider the number $\left(101^2-100^2\right)\cdot\left(102^2-101^2\right)\cdot\left(103^2-102^2\right)\cdot...\cdot\left(200^2-199^2\right)$. [list=a] [*] Determine its units digit. [*] Determine its tens digit. [/list]

2019 Durer Math Competition Finals, 12
Tags: number theory , polynomial , last digit , digit
$P$ and $Q$ are two different non-constant polynomials such that $P(Q(x)) = P(x)Q(x)$ and $P(1) = P(-1) = 2019$. What are the last four digits of $Q(P(-1))$?

2017 Finnish National High School Mathematics Comp, 3
Tags: number theory , digit , last digit
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 $n> 7$.