Olympiad problems

2011 Regional Olympiad of Mexico Center Zone, 4

Show that if a $6n$-digit number is divisible by $7$, then the number that results from moving the ones digit to the beginning of the number is also a multiple of $7$.

1962 Dutch Mathematical Olympiad, 4

Tags: floor function , number theory , digit , last digit

Write using with the floor function: the last, the second last, and the first digit of the number $n$ written in the decimal system.

2000 Czech And Slovak Olympiad IIIA, 6

Tags: number theory , digit

Find all four-digit numbers $\overline{abcd}$ (in decimal system) such that $\overline{abcd}= (\overline{ac}+1).(\overline{bd} +1)$

2010 Greece JBMO TST, 1

Tags: digit , number theory , sum of squares , algebra

Nine positive integers $a_1,a_2,...,a_9$ have their last $2$-digit part equal to $11,12,13,14,15,16,17,18$ and $19$ respectively. Find the last $2$-digit part of the sum of their squares.

1983 IMO Longlists, 70

Tags: digit , number theory , 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 $n \geq n_0, d_{n+T} = d_n.$

2017 Macedonia JBMO TST, 5

Tags: exponent , prime factor , number theory , digit

Find all the positive integers $n$ so that $n$ has the same number of digits as its number of different prime factors and the sum of these different prime factors is equal to the sum of exponents of all these primes in factorization of $n$.

1970 All Soviet Union Mathematical Olympiad, 132

Tags: number theory , digit , 17-digit

The digits of the $17$-digit number are rearranged in the reverse order. Prove that at list one digit of the sum of the new and the initial number is even.

2000 Argentina National Olympiad, 1

Tags: digit , number theory

The natural numbers are written in succession, forming a sequence of digits$$12345678910111213141516171819202122232425262728293031\ldots$$Determine how many digits the natural number has that contributes to this sequence with the digit in position $10^{2000}$. Clarification: The natural number that contributes to the sequence with the digit in position $10$ has $2$ digits, because it is $10$; The natural number that contributes to the sequence with the digit at position $10^2$ has $2$ digits, because it is $55$.

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.

2013 Czech And Slovak Olympiad IIIA, 4

Tags: digit , combinatorics , game

On the board is written in decimal the integer positive number $N$. If it is not a single digit number, wipe its last digit $c$ and replace the number $m$ that remains on the board with a number $m -3c$. (For example, if $N = 1,204$ on the board, $120 - 3 \cdot 4 = 108$.) Find all the natural numbers $N$, by repeating the adjustment described eventually we get the number $0$.

2016 Regional Olympiad of Mexico Northeast, 6

Tags: number theory , digit

A positive integer $N$ is called [i]northern[/i] if for each digit $d > 0$, there exists a divisor of $N$ whose last digit is $d$. How many [i]northern [/i] numbers less than $2016$ are there with the fewest number of divisors as possible?

2024 Kyiv City MO Round 1, Problem 1

Tags: number theory , digit

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

2001 Estonia Team Selection Test, 5

Tags: product , power , number theory , digit , prime

Find the exponent of $37$ in the representation of the number $111...... 11$ with $3\cdot 37^{2000}$ digits equals to $1$, as product of prime powers

1999 Denmark MO - Mohr Contest, 5

Tags: number theory , digit , divide , divisible

Is there a number whose digits are only $1$'s and which is divided by $1999$?

2015 May Olympiad, 4

Tags: sum of digits , digit , number theory

We say that a number is [i]superstitious [/i] when it is equal to $13$ times the sum of its digits . Find all superstitious numbers.

2016 Ecuador Juniors, 1

Tags: number theory , digit

A natural number of five digits is called [i]Ecuadorian [/i]if it satisfies the following conditions: $\bullet$ All its digits are different. $\bullet$ The digit on the far left is equal to the sum of the other four digits. Example: $91350$ is an Ecuadorian number since $9 = 1 + 3 + 5 + 0$, but $54210$ is not since $5 \ne 4 + 2 + 1 + 0$. Find how many Ecuadorian numbers exist.

2004 German National Olympiad, 3

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

Prove that for every positive integer $n$ there is an $n$-digit number $z$ with none of its digits $0$ and such that $z$ is divisible by its sum of digits.

1990 IMO Longlists, 65

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

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.

2012 Denmark MO - Mohr Contest, 4

Tags: number theory , digit

Two two-digit numbers $a$ and b satisfy that the product $a \cdot b$ divides the four-digit number one gets by writing the two digits in $a$ followed by the two digits in $b$. Determine all possible values of $a$ and $b$.

2019 Durer Math Competition Finals, 12

Tags: number theory , polynomial , last digit , digit

$P$ and $Q$ are two different non-constant polynomials such that $P(Q(x)) = P(x)Q(x)$ and $P(1) = P(-1) = 2019$. What are the last four digits of $Q(P(-1))$?

2009 Mathcenter Contest, 5

Tags: number theory , digit , power of 2

For $n\in\mathbb{N}$, prove that $2^n$ can begin with any sequence of digits. Hint: $\log 2$ is irrational number.

2012 Bosnia and Herzegovina Junior BMO TST, 2

Tags: transformation , combinatorics , digit

Let $\overline{abcd}$ be $4$ digit number, such that we can do transformations on it. If some two neighboring digits are different than $0$, then we can decrease both digits by $1$ (we can transform $9870$ to $8770$ or $9760$). If some two neighboring digits are different than $9$, then we can increase both digits by $1$ (we can transform $9870$ to $9980$ or $9881$). Can we transform number $1220$ to: $a)$ $2012$ $b)$ $2021$

2017 Irish Math Olympiad, 1

Tags: multiple , digit , number theory

Determine, with proof, the smallest positive multiple of $99$ all of whose digits are either $1$ or $2$.

2013 Saudi Arabia IMO TST, 3

Tags: digit , divisor , number theory

For a positive integer $n$, we consider all its divisors (including $1$ and itself). Suppose that $p\%$ of these divisors have their unit digit equal to $3$. (For example $n = 117$, has six divisors, namely $1,3,9,13,39,117$. Two of these divisors namely $3$ and $13$, have unit digits equal to $3$. Hence for $n = 117$, $p =33.33...$). Find, when $n$ is any positive integer, the maximum possible value of $p$.

2018 Pan-African Shortlist, N2

Tags: number theory , divisibility , digit

A positive integer is called special if its digits can be arranged to form an integer divisible by $4$. How many of the integers from $1$ to $2018$ are special?