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

2006 Switzerland - Final Round, 3
Tags: digit , sum of digits , number theory
Calculate the sum of digit of the number $$9 \times 99 \times 9999 \times ... \times \underbrace{ 99...99}_{2^n}$$ where the number of nines doubles in each factor.

2010 Princeton University Math Competition, 3
Tags: number theory , digit , decimal representation
Find (with proof) all natural numbers $n$ such that, for some natural numbers $a$ and $b$, $a\ne b$, the digits in the decimal representations of the two numbers $n^a+1$ and $n^b+1$ are in reverse order.

2020 China Northern MO, P5
Tags: digit , multiple , number theory
Find all positive integers $a$ so that for any $\left \lfloor \frac{a+1}{2} \right \rfloor$-digit number that is composed of only digits $0$ and $2$ (where $0$ cannot be the first digit) is not a multiple of $a$.

1980 Bulgaria National Olympiad, Problem 1
Tags: number theory , algebra , digit
Show that there exists a unique sequence of decimal digits $p_0=5,p_1,p_2,\ldots$ such that, for any $k$, the square of any positive integer ending with $\overline{p_kp_{k-1}\cdots p_0}$ ends with the same digits.

2025 Bundeswettbewerb Mathematik, 2
Tags: factorial , digit , number theory
For each integer $n \ge 2$ we consider the last digit different from zero in the decimal expansion of $n!$. The infinite sequence of these digits starts with $2,6,4,2,2$. Determine all digits which occur at least once in this sequence, and show that each of those digits occurs in fact infinitely often.

2010 CHMMC Fall, 1
Tags: digit , number theory
The numbers $25$ and $76$ have the property that when squared in base $10$, their squares also end in the same two digits. A positive integer is called [i]amazing [/i] if it has at most $3$ digits when expressed in base $21$ and also has the property that its square expressed in base $21$ ends in the same $3$ digits. (For this problem, the last three digits of a one-digit number b are 00b, and the last three digits of a two-digit number $\underline{ab}$ are $0\underline{ab}$.) Compute the sum of all amazing numbers. Express your answer in base $21$.

1968 Spain Mathematical Olympiad, 7
Tags: number theory , digit , power of 2
In the sequence of powers of $2$ (written in the decimal system, beginning with $2^1 = 2$) there are three terms of one digit, another three of two digits, another three of $3$, four out of $4$, three out of $5$, etc. Clearly reason the answers to the following questions: a) Can there be only two terms with a certain number of digits? b) Can there be five consecutive terms with the same number of digits? c) Can there be four terms of n digits, followed by four with $n + 1$ digits? d) What is the maximum number of consecutive powers of $2$ that can be found without there being four among them with the same number of digits?

1975 All Soviet Union Mathematical Olympiad, 210
Tags: number theory , combinatorics , digit
Prove that it is possible to find $2^{n+1}$ of $2^n$ digit numbers containing only "$1$" and "$2$" as digits, such that every two of them distinguish at least in $2^{n-1}$ digits.

2017 Israel National Olympiad, 2
Tags: number theory , product , digit , maximize
Denote by $P(n)$ the product of the digits of a positive integer $n$. For example, $P(1948)=1\cdot9\cdot4\cdot8=288$. [list=a] [*] Evaluate the sum $P(1)+P(2)+\dots+P(2017)$. [*] Determine the maximum value of $\frac{P(n)}{n}$ where $2017\leq n\leq5777$. [/list]

1984 Tournament Of Towns, (056) O4
Tags: sum of digits , product of digits , digit , number theory
The product of the digits of the natural number $N$ is denoted by $P(N)$ whereas the sum of these digits is denoted by $S(N)$. How many solutions does the equation $P(P(N)) + P(S(N)) + S(P(N)) + S(S(N)) = 1984$ have?

2019 Istmo Centroamericano MO, 1
Tags: remainder , digit , number theory
Determine all the numbers formed by three different and non-zero digits, such that the six numbers obtained by permuting these digits leaves the same remainder after the division by $4$.

2009 Singapore Junior Math Olympiad, 3
Tags: prime , digit , number theory
Suppose $\overline{a_1a_2...a_{2009}}$ is a $2009$-digit integer such that for each $i = 1,2,...,2007$, the $2$-digit integer $\overline{a_ia_{i+1}}$ contains $3$ distinct prime factors. Find $a_{2008}$ (Note: $\overline{xyz...}$ denotes an integer whose digits are $x, y,z,...$.)

2001 All-Russian Olympiad Regional Round, 9.8
Tags: number theory , perfect square , digit
Sasha wrote a non-zero number on the board and added it to it on the right, one non-zero digit at a time, until he writes out a million digits. Prove that an exact square has been written on the board no more than $100$ times.

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?

2014 Singapore Junior Math Olympiad, 1
Tags: number theory , multiple , digit
Consider the integers formed using the digits $0,1,2,3,4,5,6$, without repetition. Find the largest multiple of $55$. Justify your answer.

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

2008 Hanoi Open Mathematics Competitions, 7
Tags: digit , number theory
Find all triples $(a, b,c)$ of consecutive odd positive integers such that $a < b < c$ and $a^2 + b^2 + c^2$ is a four digit number with all digits equal.

1970 IMO Longlists, 42
Tags: inequalities , decimal representation , digit , algebra
We have $0\le x_i<b$ for $i=0,1,\ldots,n$ and $x_n>0,x_{n-1}>0$. If $a>b$, and $x_nx_{n-1}\ldots x_0$ represents the number $A$ base $a$ and $B$ base $b$, whilst $x_{n-1}x_{n-2}\ldots x_0$ represents the number $A'$ base $a$ and $B'$ base $b$, prove that $A'B<AB'$.

2020-21 IOQM India, 8
Tags: perfect square , digit , number theory
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 $m$.

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

2024 Malaysian IMO Training Camp, 2
Tags: number theory , digit
A finite sequence of decimal digits from $\{0,1,\cdots, 9\}$ is said to be [i]common[/i] if for each sufficiently large positive integer $n$, there exists a positive integer $m$ such that the expansion of $n$ in base $m$ ends with this sequence of digits. For example, $0$ is common because for any large $n$, the expansion of $n$ in base $n$ is $10$, whereas $00$ is not common because for any squarefree $n$, the expansion of $n$ in any base cannot end with $00$. Determine all common sequences. [i]Proposed by Wong Jer Ren[/i]

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

2011 May Olympiad, 2
Tags: digit , number theory
We say that a four-digit number $\overline{abcd}$ ($a \ne 0$) is [i]pora [/i] if the following terms are true : $\bullet$ $a\ge b$ $\bullet$ $ab - cd = cd -ba$. For example, $2011$ is pora because $20-11 = 11-02$ Find all the numbers around.

2019 Costa Rica - Final Round, 3
Tags: number theory , digit
Let $x, y$ be two positive integers, with $x> y$, such that $2n = x + y$, where n is a number two-digit integer. If $\sqrt{xy}$ is an integer with the digits of $n$ but in reverse order, determine the value of $x - y$

1966 IMO Shortlist, 54
Tags: decimal representation , digit , last digit , number theory , modular arithmetic
We take $100$ consecutive natural numbers $a_{1},$ $a_{2},$ $...,$ $a_{100}.$ Determine the last two digits of the number $a_{1}^{8}+a_{2}^{8}+...+a_{100}^{8}.$