Found problems: 52
2009 Puerto Rico Team Selection Test, 2
The last three digits of $ N$ are $ x25$. For how many values of $ x$ can $ N$ be the square of an integer?
2006 Cuba MO, 8
Prove that for any integer $k$ ($k \ge 2$) there exists a power of $2$ that among its last $k$ digits, the nines constitute no less than half. For example, for $k = 2$ and $k = 3$ we have the powers $2^{12} = ... 96$ and $2^{53} = ... 992$.
[hide=original wording]
Probar que para cualquier k entero existe una potencia de 2 que entre sus ultimos k dıgitos, los nueves constituyen no menos de la mitad. [/hide]
1925 Eotvos Mathematical Competition, 2
How maay zeros are there at the end of the number $$1000! = 1 \cdot 2 \cdot 3 \cdot ... \cdot 999 \cdot 1000?$$
1966 IMO Shortlist, 54
We take $100$ consecutive natural numbers $a_{1},$ $a_{2},$ $...,$ $a_{100}.$ Determine the last two digits of the number $a_{1}^{8}+a_{2}^{8}+...+a_{100}^{8}.$
1964 Czech and Slovak Olympiad III A, 1
Show that the number $11^{100}-1$ is both divisible by $6000$ and its last four decimal digits are $6000$.
2019 Durer Math Competition Finals, 12
$P$ 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 $Q(P(-1))$?
1987 Tournament Of Towns, (150) 1
Prove that the second last digit of each power of three is even .
(V . I . Plachkos)
1949-56 Chisinau City MO, 2
What is the last digit of $777^{777}$?
1990 Greece National Olympiad, 4
Since this is the $6$th Greek Math Olympiad and the year is $1989$, can you find the last two digits of $6^{1989}$?
2014 Saudi Arabia Pre-TST, 3.4
Prove that there exists a positive integer $n$ such that the last digits of $n^3$ are $...201320132013$.
2000 Rioplatense Mathematical Olympiad, Level 3, 1
Let $a$ and $b$ be positive integers such that the number $b^2 + (b +1)^2 +...+ (b + a)^2-3$ is multiple of $5$ and $a + b$ is odd. Calculate the digit of the units of the number $a + b$ written in decimal notation.
1966 IMO Longlists, 54
We take $100$ consecutive natural numbers $a_{1},$ $a_{2},$ $...,$ $a_{100}.$ Determine the last two digits of the number $a_{1}^{8}+a_{2}^{8}+...+a_{100}^{8}.$
1996 Denmark MO - Mohr Contest, 4
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 $n^2$?
2007 Denmark MO - Mohr Contest, 2
What is the last digit in the number $2007^{2007}$?
1999 Greece Junior Math Olympiad, 4
Defi ne alternate sum of a set of real numbers $A =\{a_1,a_2,...,a_k\}$ with $a_1 < a_2 <...< a_k$, the number
$S(A) = a_k - a_{k-1} + a_{k-2} - ... + (-1)^{k-1}a_1$ (for example if $A = \{1,2,5, 7\}$ then $S(A) = 7 - 5 + 2 - 1$)
Consider the alternate sums, of every subsets of $A = \{1, 2, 3, 4, 5, 6, 7, 8,9, 10\}$ and sum them.
What is the last digit of the sum obtained?
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.
1988 Nordic, 1
The positive integer $ n$ has the following property:
if the three last digits of $n$ are removed, the number $\sqrt[3]{n}$ remains.
Find $n$.
2015 Belarus Team Selection Test, 2
In the sequence of digits $2,0,2,9,3,...$ any digit it equal to the last digit in the decimal representation of the sum of four previous digits. Do the four numbers $2,0,1,5$ in that order occur in the sequence?
Folklore
2021 Chile National Olympiad, 1
Consider the sequence of numbers defined by $a_1 = 7$, $a_2 = 7^7$ , $ ...$ , $a_n = 7^{a_{n-1}}$ for $n \ge 2$. Determine the last digit of the decimal representation of $a_{2021}$.
1979 Dutch Mathematical Olympiad, 3
Define $a_1 = 1979$ and $a_{n+1} = 9^{a_n}$ for $n = 1,2,3,...$. Determine the last two digits of $a_{1979}$.
2010 Flanders Math Olympiad, 1
How many zeros does $101^{100} - 1$ end with?
2009 Kyiv Mathematical Festival, 1
Let $X$ be the sum of all divisors of the number $(3\cdot 2009)^{((2\cdot 2009)^{2009}-1)}$ . Find the last digit of $X$.
1991 Nordic, 1
Determine the last two digits of the number $2^5 + 2^{5^{2}} + 2^{5^{3}} +... + 2^{5^{1991}}$ , written in decimal notation.
2018 Junior Regional Olympiad - FBH, 4
Determine the last digit of number $18^1+18^2+...+18^{19}+18^{20}$
1952 Polish MO Finals, 5
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.