Olympiad problems

2019 Gulf Math Olympiad, 2

1. Find $N$, the smallest positive multiple of $45$ such that all of its digits are either $7$ or $0$. 2. Find $M$, the smallest positive multiple of $32$ such that all of its digits are either $6$ or $1$. 3. How many elements of the set $\{1,2,3,...,1441\}$ have a positive multiple such that all of its digits are either $5$ or $2$?

1969 Poland - Second Round, 2

Tags: number theory , digit

Find all four-digit numbers in which the thousands digit is equal to the hundreds digit and the tens digit is equal to the units digit and which are squares of integers.

PEN A Problems, 103

Tags: modular arithmetic , divisibility , digit , decimal representation , digit sum

When $4444^{4444}$ is written in decimal notation, the sum of its digits is $ A.$ Let $B$ be the sum of the digits of $A.$ Find the sum of the digits of $ B.$ ($A$ and $B$ are written in decimal notation.)

2018 Israel National Olympiad, 4

Tags: number theory , digit , digit sum , divisibility

The three-digit number 999 has a special property: It is divisible by 27, and its digit sum is also divisible by 27. The four-digit number 5778 also has this property, as it is divisible by 27 and its digit sum is also divisible by 27. How many four-digit numbers have this property?

2018 Saudi Arabia GMO TST, 2

Tags: number theory , digit , divisible

Two positive integers $m$ and $n$ are called [i]similar [/i] if one of them can be obtained from the other one by swapping two digits (note that a $0$-digit cannot be swapped with the leading digit). Find the greatest integer $N$ such that N is divisible by $13$ and any number similar to $N$ is not divisible by $13$.

1987 Austrian-Polish Competition, 7

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

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 $Q$.

2018 Federal Competition For Advanced Students, P1, 3

Tags: combinatorics , game , number theory , decimal representation , digit

Alice and Bob determine a number with $2018$ digits in the decimal system by choosing digits from left to right. Alice starts and then they each choose a digit in turn. They have to observe the rule that each digit must differ from the previously chosen digit modulo $3$. Since Bob will make the last move, he bets that he can make sure that the final number is divisible by $3$. Can Alice avoid that? [i](Proposed by Richard Henner)[/i]

2019 IberoAmerican, 1

Tags: number theory , digit

For each positive integer $n$, let $s(n)$ be the sum of the squares of the digits of $n$. For example, $s(15)=1^2+5^2=26$. Determine all integers $n\geq 1$ such that $s(n)=n$.

1990 Greece National Olympiad, 4

Tags: number theory , last digit , digit

Since this is the $6$th Greek Math Olympiad and the year is $1989$, can you find the last two digits of $6^{1989}$?

1970 All Soviet Union Mathematical Olympiad, 142

Tags: number theory , digit

All natural numbers containing not more than $n$ digits are divided onto two groups. The first contains the numbers with the even sum of the digits, the second -- with the odd sum. Prove that if $0<k<n$ than the sum of the $k$-th powers of the numbers in the first group equals to the sum of the $k$-th powers of the numbers in the second group.

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.

2008 Flanders Math Olympiad, 1

Tags: number theory , digit

Determine all natural numbers $n$ of $4$ digits whose quadruple minus the number formed by the digits of $n$ in reverse order equals $30$.

Oliforum Contest V 2017, 1

Tags: number theory , digit , multiple

We know that there exists a positive integer with $7$ distinct digits which is multiple of each of them. What are its digits? (Paolo Leonetti)

2010 Singapore Junior Math Olympiad, 2

Tags: multiple , number theory , digit

Find the sum of all the $5$-digit integers which are not multiples of $11$ and whose digits are $1, 3, 4, 7, 9$.

2009 Tournament Of Towns, 3

Tags: minimum , digit , number theory , combinatorics

Alex is going to make a set of cubical blocks of the same size and to write a digit on each of their faces so that it would be possible to form every $30$-digit integer with these blocks. What is the minimal number of blocks in a set with this property? (The digits $6$ and $9$ do not turn one into another.)

1956 Putnam, A2

Tags: number theory , digit

Prove that every positive integer has a multiple whose decimal representation involves all ten digits.

2011 Junior Balkan Team Selection Tests - Romania, 1

Tags: number theory , decimal representation , digit

Call a positive integer [i]balanced [/i] if the number of its distinct prime factors is equal to the number of its digits in the decimal representation; for example, the number $385 = 5 \cdot 7 \cdot 11$ is balanced, while $275 = 5^2 \cdot 11$ is not. Prove that there exist only a finite number of balanced numbers.

1988 Tournament Of Towns, (175) 1

Tags: number theory , digit

Is it possible to select two natural numbers $m$ and $n$ so that the number $n$ results from a permutation of the digits of $m$, and $m+n =999 . . . 9$ ?

2018 May Olympiad, 1

Tags: number theory , digit , perfect square

You have a $4$-digit whole number that is a perfect square. Another number is built adding $ 1$ to the unit's digit, subtracting $ 1$ from the ten's digit, adding $ 1$ to the hundred's digit and subtracting $ 1$ from the ones digit of one thousand. If the number you get is also a perfect square, find the original number. It's unique?

1977 Germany Team Selection Test, 4

Tags: modular arithmetic , divisibility , digit , decimal representation , digit sum

When $4444^{4444}$ is written in decimal notation, the sum of its digits is $ A.$ Let $B$ be the sum of the digits of $A.$ Find the sum of the digits of $ B.$ ($A$ and $B$ are written in decimal notation.)

1993 Tournament Of Towns, (369) 1

Tags: number theory , power of 2 , digit

Find all integers of the form $2^n$ (where $n$ is a natural number) such that after deleting the first digit of its decimal representation we again get a power of $2$.

2019 Cono Sur Olympiad, 2

Tags: number theory , digit

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.

1985 All Soviet Union Mathematical Olympiad, 396

Tags: sum of digits , sum of squares , sum , number theory , decimal representation , digit

Is there any numbber $n$, such that the sum of its digits in the decimal notation is $1000$, and the sum of its square digits in the decimal notation is $1000000$?

2011 May Olympiad, 2

Tags: number theory , digit

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.

1970 IMO Longlists, 42

Tags: inequalities , algebra , decimal representation , digit

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 $A'B<AB'$.