Olympiad problems

1996 Tournament Of Towns, (512) 5

Does there exist a $6$-digit number $A$ such that none of its $500 000$ multiples $A$, $2A$, $3A$, ..., $500 000A$ ends in $6$ identical digits? (S Tokarev)

1961 Poland - Second Round, 4

Tags: number theory , last digits

Find the last four digits of $5^{5555}$.

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 $a_9$.

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 $x < 10^{2015}$?

1996 Denmark MO - Mohr Contest, 4

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

Regarding a natural number $n$, it is stated that the number $n^2$ has $7$ as the second to last digit. What is the last digit of $n^2$?

2019 Polish Junior MO First Round, 1

Tags: number theory , last digits , Digit , Digits

The natural number $n$ was multiplied by $3$, resulting in the number $999^{1000}$. Find the unity digit of $n$.

2010 Flanders Math Olympiad, 1

Tags: number theory , last digits , Digits

How many zeros does $101^{100} - 1$ end with?

1949-56 Chisinau City MO, 2

Tags: number theory , last digits

What is the last digit of $777^{777}$?