Olympiad problems

2004 Switzerland - Final Round, 6

Determine all $k$ for which there exists a natural number n such that $1^n + 2^n + 3^n + 4^n$ with exactly $k$ zeros at the end.

2019 Hanoi Open Mathematics Competitions, 2

Tags: number theory , last digits , Digit

What is the last digit of $4^{3^{2019}}$? [b]A.[/b] $0$ [b]B.[/b] $2$ [b]C.[/b] $4$ [b]D.[/b] $6$ [b]E.[/b] $8$

2009 Puerto Rico Team Selection Test, 2

Tags: number theory , Digits , Perfect Square , last digits

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

1964 Czech and Slovak Olympiad III A, 1

Tags: number theory , divisible , last digits

Show that the number $11^{100}-1$ is both divisible by $6000$ and its last four decimal digits are $6000$.

2011 Denmark MO - Mohr Contest, 5

Tags: number theory , Digits , last digits , Perfect Square

Determine all sets $(a, b, c)$ of positive integers where one obtains $b^2$ by removing the last digit in $c^2$ and one obtains $a^2$ by removing the last digit in $b^2$. .

1979 Dutch Mathematical Olympiad, 3

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

Define $a_1 = 1979$ and $a_{n+1} = 9^{a_n}$ for $n = 1,2,3,...$. Determine the last two digits of $a_{1979}$.

1964 German National Olympiad, 4

Tags: number theory , last digits , Digit , periodic

Denote by $a_n$ the last digit of the number $n^{(n^n)}$ (let $n\ne 0$ be a natural number ). Prove that the numbers $a_n$ form a periodic sequence and state this period!

2019 Durer Math Competition Finals, 14

Tags: number theory , last digits , Digits

Let $S$ be the set of all positive integers less than $10,000$ whose last four digits in base $2$ are the same as its last four digits in base $5$. What remainder do we get if we divide the sum of all elements of $S$ by $10000$?

1987 Tournament Of Towns, (150) 1

Tags: number theory , Digits , last digits , power of 3

Prove that the second last digit of each power of three is even . (V . I . Plachkos)

2001 All-Russian Olympiad Regional Round, 9.6

Tags: number theory , last digits , Digits , Divisors

Is there such a natural number that the product of all its natural divisors (including $1$ and the number itself) ends exactly in $2001$ zeros?

2019 Durer Math Competition Finals, 8

Tags: number theory , Digits , last digits

Let $N$ be a positive integer such that $N$ and $N^2$ both end in the same four digits $\overline{abcd}$, where $a \ne 0$. What is the four-digit number $\overline{abcd}$?

2015 Puerto Rico Team Selection Test, 4

Tags: number theory , last digits

Let $n$ be a positive integer. Find as many as possible zeros as last digits the following expression: $1^n + 2^n + 3^n + 4^n$.

2013 Saudi Arabia Pre-TST, 2.3

Tags: number theory , Power , last digits

The positive integer $a$ is relatively prime with $10$. Prove that for any positive integer $n$, there exists a power of $a$ whose last $n$ digits are $\underbrace{0...0}_\text{n-1}1$.

1974 Chisinau City MO, 72

Tags: number theory , last digits

Find the last two digits of each of the numbers $3^{1974}$ and $7^{1974}$.

1983 All Soviet Union Mathematical Olympiad, 356

Tags: Sequence , number theory , periodical , last digits , Digits , floor function

The sequences $a_n$ and $b_n$ members are the last digits of $[\sqrt{10}^n]$ and $[\sqrt{2}^n]$ respectively (here $[ ...]$ denotes the whole part of a number). Are those sequences periodical?

1949-56 Chisinau City MO, 5

Tags: number theory , Perfect Square , last digits

Prove that the square of any integer cannot end with two fives.

1990 Greece National Olympiad, 4

Tags: number theory , last digits , Digits

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

2020 Malaysia IMONST 1, 5

Tags: last digits , number theory , modulo

Determine the last digit of $5^5+6^6+7^7+8^8+9^9$.

1962 Dutch Mathematical Olympiad, 4

Tags: floor function , number theory , Digits , last digits

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

2007 Denmark MO - Mohr Contest, 2

Tags: number theory , last digits

What is the last digit in the number $2007^{2007}$?

2019 Durer Math Competition Finals, 12

Tags: number theory , polynomial , last digits , Digits

$P$ 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 $Q(P(-1))$?

1925 Eotvos Mathematical Competition, 2

Tags: number theory , Digits , last digits

How maay zeros are there at the end of the number $$1000! = 1 \cdot 2 \cdot 3 \cdot ... \cdot 999 \cdot 1000?$$

2014 Saudi Arabia Pre-TST, 3.4

Tags: number theory , Power , last digits

Prove that there exists a positive integer $n$ such that the last digits of $n^3$ are $...201320132013$.

1952 Polish MO Finals, 5

Tags: number theory , last digits

Prove that none of the digits $2$, $4$, $7$, $9$ can be the last digit of a number $$ 1 + 2 + 3 + \ldots + n,$$ where $n$ is a natural number.

2001 Denmark MO - Mohr Contest, 2

Tags: number theory , factorial , Digits , last digits

If there is a natural number $n$ such that the number $n!$ has exactly $11$ zeros at the end? (With $n!$ is denoted the number $1\cdot 2\cdot 3 \cdot ... (n - )1 \cdot n$).