Olympiad problems

2013 NZMOC Camp Selection Problems, 7

In a sequence of positive integers an inversion is a pair of positions such that the element in the position to the left is greater than the element in the position to the right. For instance the sequence $2,5,3,1,3$ has five inversions - between the first and fourth positions, the second and all later positions, and between the third and fourth positions. What is the largest possible number of inversions in a sequence of positive integers whose sum is $2014$?

2017 Czech-Polish-Slovak Junior Match, 3

Tags: number theory , digit , divisible

How many $8$-digit numbers are $*2*0*1*7$, where four unknown numbers are replaced by stars, which are divisible by $7$?

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.

1991 Tournament Of Towns, (307) 4

Tags: number theory , recurrence relation , digit

A sequence $a_n$ is determined by the rules $a_0 = 9$ and for any nonnegative $k$, $$a_{k+1}=3a_k^4+4a_k^3.$$ Prove that $a_{10}$ contains more than $1000$ nines in decimal notation. (Yao)

2013 Israel National Olympiad, 4

Tags: number theory , digit

Determine the number of positive integers $n$ satisfying: [list] [*] $n<10^6$ [*] $n$ is divisible by 7 [*] $n$ does not contain any of the digits 2,3,4,5,6,7,8. [/list]

2011 May Olympiad, 5

Tags: number theory , digit , divisible

We consider all $14$-digit positive integers, divisible by $18$, whose digits are exclusively $ 1$ and $2$, but there are no consecutive digits $2$. How many of these numbers are there?

2019 Paraguay Mathematical Olympiad, 3

Tags: number theory , digit

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

2018 Junior Regional Olympiad - FBH, 4

Tags: digit , last digit , number theory

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

1956 Putnam, A2

Tags: number theory , digit

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

2011 May Olympiad, 2

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

Using only once each of the digits $1, 2, 3, 4, 5, 6, 7$ and $ 8$, write the square and the cube of a positive integer. Determine what that number can be.

1999 Tournament Of Towns, 5

Tags: combinatorics , sequence , digit

Tireless Thomas and Jeremy construct a sequence. At the beginning there is one positive integer in the sequence. Then they successively write new numbers in the sequence in the following way: Thomas obtains the next number by adding to the previous number one of its (decimal) digits, while Jeremy obtains the next number by subtracting from the previous number one of its digits. Prove that there is a number in this sequence which will be repeated at least $100$ times. (A Shapovalov)

2018 Federal Competition For Advanced Students, P2, 6

Tags: number theory , decimal representation , digit

Determine all digits $z$ such that for each integer $k \ge 1$ there exists an integer $n\ge 1$ with the property that the decimal representation of $n^9$ ends with at least $k$ digits $z$. [i](Proposed by Walther Janous)[/i]

2020 Tournament Of Towns, 1

Tags: number theory , digit , divisible

Does there exist a positive integer that is divisible by $2020$ and has equal numbers of digits $0, 1, 2, . . . , 9$ ? Mikhail Evdokimov

1983 IMO Longlists, 70

Tags: number theory , digit , decimal representation , periodic sequence , non-periodical functions

Let $d_n$ be the last nonzero digit of the decimal representation of $n!$. Prove that $d_n$ is aperiodic; that is, there do not exist $T$ and $n_0$ such that for all $n \geq n_0, d_{n+T} = d_n.$

2016 KOSOVO TST, 2

Tags: algebra , induction , digit

Show that for any $n\geq 2$, $2^{2^n}+1$ ends with 7

1999 Bundeswettbewerb Mathematik, 2

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

For every natural number $n$, let $Q(n)$ denote the sum of the decimal digits of $n$. Prove that there are infinitely many positive integers $k$ with $Q(3^k) \ge Q(3^{k+1})$.

2002 Germany Team Selection Test, 3

Tags: decimal representation , number theory , factorial , digit

Prove that there is no positive integer $n$ such that, for $k = 1,2,\ldots,9$, the leftmost digit (in decimal notation) of $(n+k)!$ equals $k$.

2000 Belarus Team Selection Test, 6.2

Tags: digit , perfect square , number theory

A positive integer $A_k...A_1A_0$ is called monotonic if $A_k \le ..\le A_1 \le A_0$. Show that for any $n \in N$ there is a monotonic perfect square with $n$ digits.

2015 India PRMO, 6

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

$6.$ How many two digit positive integers $N$ have the property that the sum of $N$ and the number obtained by reversing the order of the digits of $N$ is a perfect square $?$

2023 Bulgaria EGMO TST, 4

Tags: combinatorics , pigeonhole principle , coloring , digit

Each two-digit is number is coloured in one of $k$ colours. What is the minimum value of $k$ such that, regardless of the colouring, there are three numbers $a$, $b$ and $c$ with different colours with $a$ and $b$ having the same units digit (second digit) and $b$ and $c$ having the same tens digit (first digit)?

2018 Dutch Mathematical Olympiad, 1

Tags: number theory , digit , divisible

We call a positive integer a [i]shuffle[/i] number if the following hold: (1) All digits are nonzero. (2) The number is divisible by $11$. (3) The number is divisible by $12$. If you put the digits in any other order, you again have a number that is divisible by $12$. How many $10$-digit [i]shuffle[/i] numbers are there?

1996 Austrian-Polish Competition, 1

Tags: number theory , divide , divisible , digit

Let $k \ge 1$ be a positive integer. Prove that there exist exactly $3^{k-1}$ natural numbers $n$ with the following properties: (i) $n$ has exactly $k$ digits (in decimal representation), (ii) all the digits of $n$ are odd, (iii) $n$ is divisible by $5$, (iv) the number $m = n/5$ has $k$ odd digits

2017 May Olympiad, 1

Tags: number theory , digit

We shall call a positive integer [i]ascending [/i] if its digits read from left to right they are in strictly increasing order. For example, $458$ is ascending and $2339$ is not. Find the largest ascending number that is a multiple of $56$.

1967 All Soviet Union Mathematical Olympiad, 085

Tags: digit , number theory

a) The digits of a natural number were rearranged. Prove that the sum of given and obtained numbers can't equal $999...9$ ($1967$ of nines). b) The digits of a natural number were rearranged. Prove that if the sum of the given and obtained numbers equals $1010$, than the given number was divisible by $10$.

1998 Mexico National Olympiad, 1

Tags: number theory , digit , consecutive

A number is called lucky if computing the sum of the squares of its digits and repeating this operation sufficiently many times leads to number $1$. For example, $1900$ is lucky, as $1900 \to 82 \to 68 \to 100 \to 1$. Find infinitely many pairs of consecutive numbers each of which is lucky.