Olympiad problems

2015 India PRMO, 15

$15.$ Let $n$ be the largest integer that is the product of exactly $3$ distinct prime numbers, $x,y,$ and $10x+y,$ where $x$ and $y$ are digits. What is the sum of digits of $n ?$

2013 Junior Balkan Team Selection Tests - Romania, 2

Tags: number theory , sum of digits , digit , mutliple

Call the number $\overline{a_1a_2... a_m}$ ($a_1 \ne 0,a_m \ne 0$) the reverse of the number $\overline{a_m...a_2a_1}$. Prove that the sum between a number $n$ and its reverse is a multiple of $81$ if and only if the sum of the digits of $n$ is a multiple of $81$.

1990 IMO Longlists, 65

Tags: pigeonhole principle , decimal representation , divisibility , multiple , digit , number theory

Prove that every integer $ k$ greater than 1 has a multiple that is less than $ k^4$ and can be written in the decimal system with at most four different digits.

2009 Puerto Rico Team Selection Test, 2

Tags: digit , perfect square , last digit , number theory

The last three digits of $ N$ are $ x25$. For how many values of $ x$ can $ N$ be the square of an integer?

2018 Brazil EGMO TST, 1

Tags: number theory , perfect square , digit

(a) Let $m$ and $n$ be positive integers and $p$ a positive rational number, with $m > n$, such that $\sqrt{m} -\sqrt{n}= p$. Prove that $m$ and $n$ are perfect squares. (b) Find all four-digit numbers $\overline{abcd}$, where each letter $a, b, c$ and $d$ represents a digit, such that $\sqrt{\overline{abcd}} -\sqrt{\overline{acd}}= \overline{bb}$.

1998 Estonia National Olympiad, 2

Tags: prime , number theory , digit

Find all prime numbers of the form $10101...01$.

1989 Austrian-Polish Competition, 3

Tags: prime , digit , number theory

Find all natural numbers $N$ (in decimal system) with the following properties: (i) $N =\overline{aabb}$, where $\overline{aab}$ and $\overline{abb}$ are primes, (ii) $N = P_1P_2P_3$, where $P_k (k = 1,2,3)$ is a prime consisting of $k$ (decimal) digits.

2001 Portugal MO, 6

Tags: number theory , digit

Let $n$ be a natural number. Prove that there is a multiple of $n$ that can be written only with the digits $0$ and $1$.

2015 Caucasus Mathematical Olympiad, 4

Tags: digit , remainder , number theory

Is there a nine-digit number without zero digits, the remainder of dividing which on each of its digits is different?

2013 Dutch Mathematical Olympiad, 5

Tags: number theory , sum , 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?

2019 Bundeswettbewerb Mathematik, 2

Tags: digit , divisible , divisibility , number theory

The lettes $A,C,F,H,L$ and $S$ represent six not necessarily distinct decimal digits so that $S \ne 0$ and $F \ne 0$. We form the two six-digit numbers $SCHLAF$ and $FLACHS$. Show that the difference of these two numbers is divisible by $271$ if and only if $C=L$ and $H=A$. [i]Remark:[/i] The words "Schlaf" and "Flachs" are German for "sleep" and "flax".

2006 Cuba MO, 8

Tags: power of 2 , digit , last digit , number theory

Prove that for any integer $k$ ($k \ge 2$) there exists a power of $2$ that among its last $k$ digits, the nines constitute no less than half. For example, for $k = 2$ and $k = 3$ we have the powers $2^{12} = ... 96$ and $2^{53} = ... 992$. [hide=original wording] Probar que para cualquier k entero existe una potencia de 2 que entre sus ultimos k dıgitos, los nueves constituyen no menos de la mitad. [/hide]

2013 Argentina National Olympiad, 3

Tags: digit , remainder , number theory

Find how many are the numbers of $2013$ digits $d_1d_2…d_{2013}$ with odd digits $d_1,d_2,…,d_{2013}$ such that the sum of $1809$ terms $$d_1 \cdot d_2+d_2\cdot d_3+…+d_{1809}\cdot d_{1810}$$ has remainder $1$ when divided by $4$ and the sum of $203$ terms $$d_{1810}\cdot d_{1811}+d_{1811}\cdot d_{1812}+…+d_{2012}\cdot d_{2013}$$ has remainder $1$ when dividing by $4$.

2016 Kyiv Mathematical Festival, P5

Tags: combinatorics , digit , divisibility

On the board a 20-digit number which have 10 ones and 10 twos in its decimal form is written. It is allowed to choose two different digits and to reverse the order of digits in the interval between them. Is it always possible to get a number divisible by 11 using such operations?

VMEO III 2006 Shortlist, N14

Tags: digit , number theory , sum of digits

For any natural number $n = \overline{a_i...a_2a_1}$, consider the number $$T(n) =10 \sum_{i \,\, even} a_i+\sum_{i \,\, odd} a_i.$$ Let's find the smallest positive integer $A$ such that is sum of the natural numbers $n_1,n_2,...,n_{148}$ as well as of $m_1,m_2,...,m_{149}$ and matches the pattern: $A=n_1+n_2+...+n_{148}=m_1+m_2+...+m_{149}$ $T(n_1)=T(n_2)=...=T(n_{148})$ $T(m_1)=T(m_2)=...=T(m_{148})$

1985 Swedish Mathematical Competition, 2

Tags: digit , number theory

Find the least natural number such that if the first digit (in the decimal system) is placed last, the new number is $7/2 $ times as large as the original number.

1998 May Olympiad, 1

Tags: digit , number theory

Inés chose four different digits from the set $\{1,2,3,4,5,6,7,8,9\}$. He formed with them all possible four-digit numbers and added all those four-digit numbers. The result is $193314$. Find the four digits Inés chose.

1992 Czech And Slovak Olympiad IIIA, 3

Tags: number theory , sum , digit , sum of digits

Let $S(n)$ denote the sum of digits of $n \in N$. Find all $n$ such that $S(n) = S(2n) = S(3n) =... = S(n^2)$

2004 Thailand Mathematical Olympiad, 16

Tags: number theory , digit

What are last three digits of $2^{2^{2004}}$ ?

1978 IMO, 1

Tags: modular arithmetic , digit , decimal representation , number theory

Let $ m$ and $ n$ be positive integers such that $ 1 \le m < n$. In their decimal representations, the last three digits of $ 1978^m$ are equal, respectively, to the last three digits of $ 1978^n$. Find $ m$ and $ n$ such that $ m \plus{} n$ has its least value.

1987 Austrian-Polish Competition, 7

Tags: number theory , sum of digits , product of digits , digit

For any natural number $n= \overline{a_k...a_1a_0}$ $(a_k \ne 0)$ in decimal system write $p(n)=a_0 \cdot a_1 \cdot ... \cdot a_k$, $s(n)=a_0+ a_1+ ... + a_k$, $n^*= \overline{a_0a_1...a_k}$. Consider $P=\{n | n=n^*, \frac{1}{3} p(n)= s(n)-1\}$ and let $Q$ be the set of numbers in $P$ with all digits greater than $1$. (a) Show that $P$ is infinite. (b) Show that $Q$ is finite. (c) Write down all the elements of $Q$.

1970 All Soviet Union Mathematical Olympiad, 139

Tags: digit , number theory

Prove that for every natural number $k$ there exists an infinite set of such natural numbers $t$, that the decimal notation of $t$ does not contain zeroes and the sums of the digits of the numbers $t$ and $kt$ are equal.

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?

1978 IMO Longlists, 7

Tags: modular arithmetic , number theory , digit , decimal representation

Let $ m$ and $ n$ be positive integers such that $ 1 \le m < n$. In their decimal representations, the last three digits of $ 1978^m$ are equal, respectively, to the last three digits of $ 1978^n$. Find $ m$ and $ n$ such that $ m \plus{} n$ has its least value.

2022 Peru MO (ONEM), 4

Tags: digit , perfect power , number theory

For each positive integer n, the number $R(n) = 11 ... 1$ is defined, which is made up of exactly $n$ digits equal to $1$. For example, $R(5) = 11111$. Let $n > 4$ be an integer for which, by writing all the positive divisors of $R(n)$, it is true that each written digit belongs to the set $\{0, 1\}$. Show that $n$ is a power of an odd prime number. Clarification: A power of an odd prime number is a number of the form $p^a$, where $p$ is an odd prime number and $a$ is a positive integer.