Olympiad problems

2016 India Regional Mathematical Olympiad, 4

Tags: combinatorics , digit

Find all $6$ digit natural numbers, which consist of only the digits $1,2,$ and $3$, in which $3$ occurs exactly twice and the number is divisible by $9$.

2020 Puerto Rico Team Selection Test, 2

Tags: diophantine equation , system of equations , number theory , digit , diophantine

The cost of $1000$ grams of chocolate is $x$ dollars and the cost of $1000$ grams of potatoes is $y$ dollars, the numbers $x$ and $y$ are positive integers and have not more than $2$ digits. Mother said to Maria to buy $200$ grams of chocolate and $1000$ grams of potatoes that cost exactly $N$ dollars. Maria got confused and bought $1000$ grams of chocolate and $200$ grams of potatoes that cost exactly $M$ dollars ($M >N$). It turned out that the numbers $M$ and $N$ have no more than two digits and are formed of the same digits but in a different order. Find $x$ and $y$.

1997 Moldova Team Selection Test, 3

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.

2024 Malaysian IMO Training Camp, 2

Tags: digit , number theory

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]

2023 Durer Math Competition Finals, 3

Tags: digit , number theory

Which is the largest four-digit number that has all four of its digits among its divisors and its digits are all different?

2019 Paraguay Mathematical Olympiad, 3

Tags: digit , number theory

Let $\overline{ABCD}$ be a $4$-digit number. What is the smallest possible positive value of $\overline{ABCD}- \overline{DCBA}$?

2010 CHMMC Fall, 1

Tags: number theory , digit

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 $21$.

2018 Pan-African Shortlist, N4

Tags: number theory , divisibility , digit

Let $S$ be a set of $49$-digit numbers $n$, with the property that each of the digits $1, 2, 3, \dots, 7$ appears in the decimal expansion of $n$ seven times (and $8, 9$ and $0$ do not appear). Show that no two distinct elements of $S$ divide each other.

1964 Dutch Mathematical Olympiad, 5

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

Consider a sequence of non-negative integers g$_1,g_2,g_3,...$ each consisting of three digits (numbers smaller than $100$ are also written with three digits; the number $27$, for example, is written as $027$). Each number consists of the preceding by taking the product of the three digits that make up the preceding. The resulting sequence is of course dependent on the choice of $g_1$ (e.g. $g_1 = 359$ leads to $g_2= 135$, $g_3= 015$, $g_4 = 000$).Prove that independent of the choice of $g_1$: (a) $g_{n+1}\le g_n$ (b) $g_{10}= 000$.

1927 Eotvos Mathematical Competition, 2

Tags: number theory , digit

Find the sum of all distinct four-digit numbers that contain only the digits $1, 2, 3, 4,5$, each at most once.

2019 Brazil National Olympiad, 1

Tags: digit , number theory

An eight-digit number is said to be 'robust' if it meets both of the following conditions: (i) None of its digits is $0$. (ii) The difference between two consecutive digits is $4$ or $5$. Answer the following questions: (a) How many are robust numbers? (b) A robust number is said to be 'super robust' if all of its digits are distinct. Calculate the sum of all the super robust numbers.

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.)

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.

1983 All Soviet Union Mathematical Olympiad, 354

Tags: inequalities , algebra , digit

Natural number $k$ has $n$ digits in its decimal notation. It was rounded up to tens, then the obtained number was rounded up to hundreds, and so on $(n-1)$ times. Prove that the obtained number $m$ satisfies inequality $m < \frac{18k}{13}$. (Examples of rounding: $191\to190\to 200, 135\to140\to 100$.)

2009 Puerto Rico Team Selection Test, 2

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

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

2017 Lusophon Mathematical Olympiad, 4

Tags: number theory , digit

Find how many multiples of 360 are of the form $\overline{ab2017cd}$, where a, b, c, d are digits, with a > 0.

1995 Czech And Slovak Olympiad IIIA, 4

Tags: digit , number theory , divisible

Do there exist $10000$ ten-digit numbers divisible by $7$, all of which can be obtained from one another by a reordering of their digits?

2022 Kazakhstan National Olympiad, 2

Tags: number theory , digit , function

We define the function $Z(A)$ where we write the digits of $A$ in base $10$ form in reverse. (For example: $Z(521)=125$). Call a number $A$ $good$ if the first and last digits of $A$ are different, none of it's digits are $0$ and the equality: $$Z(A^2)=(Z(A))^2$$ happens. Find all such good numbers greater than $10^6$.\\

2009 Tournament Of Towns, 2

Tags: number theory , digit

Let $a^b$ denote the number $ab$. The order of operations in the expression 7^7^7^7^7^7^7 must be determined by parentheses ($5$ pairs of parentheses are needed). Is it possible to put parentheses in two distinct ways so that the value of the expression be the same?

1962 Dutch Mathematical Olympiad, 3

Tags: number theory , recurrence relation , digit

Consider the positive integers written in the decimal system with $n$ digits, the start of which is not zero and where there are no two sevens next to each other. The number of these numbers is called $u_n$. Derive a relation that expresses $u_{n+2}$ in terms of $u_{n+1}$ and $u_n$.

2018 Junior Regional Olympiad - FBH, 4

Tags: last digit , digit , number theory

Determine the last digit of number $18^1+18^2+...+18^{19}+18^{20}$

2014 Israel National Olympiad, 1

Tags: modular arithmetic , digit , number theory

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]

2000 Tuymaada Olympiad, 1

Tags: number theory , digit , multiple , maximum

Given the number $188188...188$ (number $188$ is written $101$ times). Some digits of this number are crossed out. What is the largest multiple of $7$, that could happen?

2002 Junior Balkan Team Selection Tests - Romania, 2

Tags: number theory , perfect square , digit

The last four digits of a perfect square are equal. Prove that all of them are zeros.

1975 Chisinau City MO, 102

Tags: combinatorics , number theory , divisible , divide , digit

Two people write a $2k$-digit number, using only the numbers $1, 2, 3, 4$ and $5$. The first number on the left is written by the first of them, the second - the second, the third - the first, etc. Can the second one achieve this so that the resulting number is divisible by $9$, if the first seeks to interfere with it? Consider the cases $k = 10$ and $k = 15$.