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: 38

2020 AMC 10, 19
Tags: AMC , AMC 10 , AMC 10 B , 2020 AMC 10B , 2020 AMC , Digit
In a certain card game, a player is dealt a hand of $10$ cards from a deck of $52$ distinct cards. The number of distinct (unordered) hands that can be dealt to the player can be written as $158A00A4AA0$. What is the digit $A$? $\textbf{(A) } 2 \qquad\textbf{(B) } 3 \qquad\textbf{(C) } 4 \qquad\textbf{(D) } 6 \qquad\textbf{(E) } 7$

1998 Tuymaada Olympiad, 6
Tags: number theory , Digits , Digit , periodic , Periodic sequence
Prove that the sequence of the first digits of the numbers in the form $2^n+3^n$ is nonperiodic.

2010 Saudi Arabia IMO TST, 3
Tags: number theory , Digit
Consider the arithmetic sequence $8, 21,34,47,....$ a) Prove that this sequence contains infinitely many integers written only with digit $9$. b) How many such integers less than $2010^{2010}$ are in the se­quence?

2017 Czech-Polish-Slovak Junior Match, 3
Tags: Digit , divisible , number theory
How many $8$-digit numbers are $*2*0*1*7$, where four unknown numbers are replaced by stars, which are divisible by $7$?

2015 Caucasus Mathematical Olympiad, 1
Tags: number theory , Digit , divisible , Sum
Does there exist a four-digit positive integer with different non-zero digits, which has the following property: if we add the same number written in the reverse order, then we get a number divisible by $101$?

2019 Hanoi Open Mathematics Competitions, 2
Tags: number theory , last digits , Digit
What is the last digit of $4^{3^{2019}}$? [b]A.[/b] $0$ [b]B.[/b] $2$ [b]C.[/b] $4$ [b]D.[/b] $6$ [b]E.[/b] $8$

2013 Dutch Mathematical Olympiad, 5
Tags: Sum , number theory , Digit
The number $S$ is the result of the following sum: $1 + 10 + 19 + 28 + 37 +...+ 10^{2013}$ If one writes down the number $S$, how often does the digit `$5$' occur in the result?

2010 Singapore Junior Math Olympiad, 2
Tags: Digit , multiple , number theory
Find the sum of all the $5$-digit integers which are not multiples of $11$ and whose digits are $1, 3, 4, 7, 9$.

1964 German National Olympiad, 4
Tags: number theory , last digits , Digit , periodic
Denote by $a_n$ the last digit of the number $n^{(n^n)}$ (let $n\ne 0$ be a natural number ). Prove that the numbers $a_n$ form a periodic sequence and state this period!

1987 Bundeswettbewerb Mathematik, 1
Tags: number theory , Digit , prime , Power
Let $p>3$ be a prime and $n$ a positive integer such that $p^n$ has $20$ digits. Prove that at least one digit appears more than twice in this number.

1975 Vietnam National Olympiad, 4
Tags: Arithmetic Progression , Digit , number theory
Find all terms of the arithmetic progression $-1, 18, 37, 56, ...$ whose only digit is $5$.

2015 Hanoi Open Mathematics Competitions, 2
Tags: number theory , Digit
The last digit of number $2017^{2017} - 2013^{2015}$ is (A): $2$, (B): $4$, (C): $6$, (D): $8$, (E): None of the above.

2001 Regional Competition For Advanced Students, 1
Tags: number theory , Digit , Sum of powers , Sum , powers
Let $n$ be an integer. We consider $s (n)$, the sum of the $2001$ powers of $n$ with the exponents $0$ to $2000$. So $s (n) = \sum_{k=0}^{2000}n ^k$ . What is the unit digit of $s (n)$ in the decimal system?

1983 IMO Shortlist, 24
Tags: number theory , Digit , decimal representation , Periodic sequence , non-periodical functions , IMO Shortlist
Let $d_n$ be the last nonzero digit of the decimal representation of $n!$. Prove that $d_n$ is aperiodic; that is, there do not exist $T$ and $n_0$ such that for all $n \geq n_0, d_{n+T} = d_n.$

2013 Saudi Arabia IMO TST, 3
Tags: number theory , Digit , Divisors
For a positive integer $n$, we consider all its divisors (including $1$ and itself). Suppose that $p\%$ of these divisors have their unit digit equal to $3$. (For example $n = 117$, has six divisors, namely $1,3,9,13,39,117$. Two of these divisors namely $3$ and $13$, have unit digits equal to $3$. Hence for $n = 117$, $p =33.33...$). Find, when $n$ is any positive integer, the maximum possible value of $p$.

2020 AMC 10, 12
Tags: AMC , AMC 10 , AMC 10 B , 2020 AMC 10B , 2020 AMC , Digit
The decimal representation of $$\dfrac{1}{20^{20}}$$ consists of a string of zeros after the decimal point, followed by a 9 and then several more digits. How many zeros are in that initial string of zeros after the decimal point? $\textbf{(A) }23\qquad\textbf{(B) }24\qquad\textbf{(C) }25\qquad\textbf{(D) }26\qquad\textbf{(E) }27$

1994 Swedish Mathematical Competition, 1
Tags: equation , algebra , Digit
$x\sqrt8 + \frac{1}{x\sqrt8} = \sqrt8$ has two real solutions $x_1, x_2$. The decimal expansion of $x_1$ has the digit $6$ in place $1994$. What digit does $x_2$ have in place $1994$?

2006 Tournament of Towns, 3
Tags: Integer Part , algebra , Digit , irrational
The $n$-th digit of number $a = 0.12457...$ equals the first digit of the integer part of the number $n\sqrt2$. Prove that $a$ is irrational number. (6)

1983 IMO Longlists, 70
Tags: number theory , Digit , decimal representation , Periodic sequence , non-periodical functions , IMO Shortlist
Let $d_n$ be the last nonzero digit of the decimal representation of $n!$. Prove that $d_n$ is aperiodic; that is, there do not exist $T$ and $n_0$ such that for all $n \geq n_0, d_{n+T} = d_n.$

2020-21 IOQM India, 8
Tags: number theory , Perfect Square , Digit
A $5$-digit number (in base $10$) has digits $k, k + 1, k + 2, 3k, k + 3$ in that order, from left to right. If this number is $m^2$ for some natural number $m$, find the sum of the digits of $m$.

2013 Czech And Slovak Olympiad IIIA, 4
Tags: combinatorics , game , Digits , Digit
On the board is written in decimal the integer positive number $N$. If it is not a single digit number, wipe its last digit $c$ and replace the number $m$ that remains on the board with a number $m -3c$. (For example, if $N = 1,204$ on the board, $120 - 3 \cdot 4 = 108$.) Find all the natural numbers $N$, by repeating the adjustment described eventually we get the number $0$.

2017 Hanoi Open Mathematics Competitions, 4
Tags: number theory , last , Digit
Put $S = 2^1 + 3^5 + 4^9 + 5^{13} + ... + 505^{2013} + 506^{2017}$. The last digit of $S$ is (A): $1$ (B): $3$ (C): $5$ (D): $7$ (E): None of the above.

2013 Tournament of Towns, 2
Tags: combinatorics , Digit , number theory
There is a positive integer $A$. Two operations are allowed: increasing this number by $9$ and deleting a digit equal to $1$ from any position. Is it always possible to obtain $A+1$ by applying these operations several times?

1984 Spain Mathematical Olympiad, 7
Tags: number theory , decimal representation , Digit
Consider the natural numbers written in the decimal system. (a) Find the smallest number which decreases five times when its first digit is erased. Which form do all numbers with this property have? (b) Prove that there is no number that decreases $12$ times when its first digit is erased. (c) Find the necessary and sufficient condition on $k$ for the existence of a natural number which is divided by $k$ when its first digit is erased.

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.