Olympiad problems

1987 IMO Longlists, 19

Tags: symmetry , combinatorics , Combinatorics of words , counting , Digits , recurrence relation , IMO Shortlist

How many words with $n$ digits can be formed from the alphabet $\{0, 1, 2, 3, 4\}$, if neighboring digits must differ by exactly one? [i]Proposed by Germany, FR.[/i]

2017 Lusophon Mathematical Olympiad, 4

Tags: number theory , Digits

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

1993 Tournament Of Towns, (394) 2

Tags: number theory , Digits

The decimal representation of all integers from $1$ to an arbitrary integer $n$ are written one after another as such: $$123... 91011... 99100... (n).$$ Does there exist $n$ such that each of the digits $0,1,2,...,9$ appears the same number of times in the given sequence? (A Andzans)

2024 Polish Junior MO Finals, 5

Tags: number theory , number theory proposed , Divisibility , multiple , Digits

Let $S=\underbrace{111\dots 1}_{19}\underbrace{999\dots 9}_{19}$. Show that the $2S$-digit number \[\underbrace{111\dots 1}_{S}\underbrace{999\dots 9}_{S}\] is a multiple of $19$.

1974 Swedish Mathematical Competition, 3

Tags: number theory , recurrence relation , last digits , Digits

Let $a_1=1$, $a_2=2^{a_1}$, $a_3=3^{a_2}$, $a_4=4^{a_3}$, $\dots$, $a_9 = 9^{a_8}$. Find the last two digits of $a_9$.

2016 Costa Rica - Final Round, F3

Tags: Digits , functional equation , functional , algebra

Let $f: Z^+ \to Z^+ \cup \{0\}$ a function that meets the following conditions: a) $f (a b) = f (a) + f (b)$, b) $f (a) = 0$ provided that the digits of the unit of $a$ are $7$, c) $f (10) = 0$. Find $f (2016).$

1970 All Soviet Union Mathematical Olympiad, 142

Tags: number theory , Digits

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.

1977 Germany Team Selection Test, 4

Tags: modular arithmetic , Divisibility , Digits , decimal representation , IMO , IMO 1975 , 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.)

2014 Singapore Junior Math Olympiad, 2

Tags: Digits , Perfect Square , number theory

Let $a$ be a positive integer such that the last two digits of $a^2$ are both non-zero. When the last two digits of $a^2$ are deleted, the resulting number is still a perfect square. Find, with justification, all possible values of $a$.

2024 Kyiv City MO Round 1, Problem 1

Tags: number theory , Digits

Find the number of positive integers for which the product of digits and the sum of digits are the same and equal to $8$.

2020 Polish Junior MO First Round, 6.

Tags: number theory , Digits

Let $a$, $b$ $c$ be the natural numbers, such that every digit occurs exactly the same number of times in each of the numbers $a$, $b$, $c$. Is it possible that $a + b + c = 10^{1001}$? Justify your answer.

1992 IMO Longlists, 69

Tags: inequalities , algebra , Digits , binary representation , combinatorics , Sequence , IMO Shortlist

Let $ \alpha(n)$ be the number of digits equal to one in the binary representation of a positive integer $ n.$ Prove that: (a) the inequality $ \alpha(n) (n^2 ) \leq \frac{1}{2} \alpha(n)(\alpha(n) + 1)$ holds; (b) the above inequality is an equality for infinitely many positive integers, and (c) there exists a sequence $ (n_i )^{\infty}_1$ such that $ \frac{\alpha ( n^2_i )}{\alpha (n_i }$ goes to zero as $ i$ goes to $ \infty.$ [i]Alternative problem:[/i] Prove that there exists a sequence a sequence $ (n_i )^{\infty}_1$ such that $ \frac{\alpha ( n^2_i )}{\alpha (n_i )}$ (d) $ \infty;$ (e) an arbitrary real number $ \gamma \in (0,1)$; (f) an arbitrary real number $ \gamma \geq 0$; as $ i$ goes to $ \infty.$

1975 Chisinau City MO, 102

Tags: combinatorics , number theory , divisible , divides , Digits

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 $k = 15$.

1962 Dutch Mathematical Olympiad, 3

Tags: number theory , recurrence relation , Digits

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 $u_n$.

1997 VJIMC, Problem 4-M

Tags: limits , real analysis , number theory , Digits

Find all real numbers $a>0$ for which the series $$\sum_{n=1}^\infty\frac{a^{f(n)}}{n^2}$$is convergent; $f(n)$ denotes the number of $0$'s in the decimal expansion of $f$.

2007 Regional Olympiad of Mexico Center Zone, 6

Tags: number theory , Digits

Certain tickets are numbered as follows: $1, 2, 3, \dots, N $. Exactly half of the tickets have the digit $ 1$ on them. If $N$ is a three-digit number, determine all possible values of $N $.

1983 All Soviet Union Mathematical Olympiad, 370

Tags: decimal representation , number theory , algebra , Digits

The infinite decimal notation of the real number $x$ contains all the digits. Let $u_n$ be the number of different $n$-digit segments encountered in $x$ notation. Prove that if for some $n$, $u_n \le (n+8)$, than $x$ is a rational number.

1991 Tournament Of Towns, (284) 4

Tags: number theory , Digits

The number $123$ is shown on the screen of a computer. Each minute the computer adds $102$ to the number on the screen. The computer expert Misha may change the order of digits in the number on the screen whenever he wishes. Can he ensure that no four-digit number ever appears on the screen? (F.L. Nazarov, Leningrad)

2004 Greece JBMO TST, 3

Tags: number theory , Digits

If in a $3$-digit number we replace with each other it's last two digits, and add the resulting number to the starting one, we find sum a $4$-digit number that starts with $173$. Which is the starting number?

1998 Tournament Of Towns, 2

Tags: number theory , Sum , Product , Digits , 4-digit

For every four-digit number, we take the product of its four digits. Then we add all of these products together . What is the result? ( G Galperin)

1970 All Soviet Union Mathematical Olympiad, 139

Tags: number theory , Digits

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.

2015 Singapore Junior Math Olympiad, 1

Tags: multiple , number theory , divides , Digits

Consider the integer $30x070y03$ where $x, y$ are unknown digits. Find all possible values of $x, y$ so that the given integer is a multiple of $37$.

2015 CHMMC (Fall), 1

Tags: CHMMC , number theory , last digits , Digits

Call a positive integer $x$ $n$-[i]cube-invariant[/i] if the last $n$ digits of $x$ are equal to the last $n$ digits of $x^3$. For example, $1$ is $n$-cube invariant for any integer $n$. How many $2015$-cube-invariant numbers $x$ are there such that $x < 10^{2015}$?

1968 IMO, 2

Tags: number theory , decimal representation , Digits , equation , IMO , IMO 1968

Find all natural numbers $n$ the product of whose decimal digits is $n^2-10n-22$.

2020 HK IMO Preliminary Selection Contest, 1

Tags: algebra , Digits

Let $n=(10^{2020}+2020)^2$. Find the sum of all the digits of $n$.