Found problems: 788
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 $2014$?
Denote by $S(n)$ the sum of digits of $n$. Given a positive integer $N$, we consider the following process: We take the sum of digits $S(N)$, then take its sum of digits $S(S(N))$, then its sum of digits $S(S(S(N)))$... We continue this until we are left with a one-digit number.
We call the number of times we had to activate $S(\cdot)$ the [b]depth[/b] of $N$.
For example, the depth of 49 is 2, since $S(49)=13\rightarrow S(13)=4$, and the depth of 45 is 1, since $S(45)=9$.
[list=a]
[*] Prove that every positive integer $N$ has a finite depth, that is, at some point of the process we get a one-digit number.
[*] Define $x(n)$ to be the [u]minimal[/u] positive integer with depth $n$. Find the residue of $x(5776)\mod 6$.
[*] Find the residue of $x(5776)-x(5708)\mod 2016$.
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How many three-digit numbers $\underline{abc}$ have the property that when it is added to $\underline{cba}$, the number obtained by reversing its digits, the result is a palindrome? (Note that $\underline{cba}$ is not necessarily a three-digit number since before reversing, $c$ may be equal to $0$.)
A game for two.
One gives a digit and the second substitutes it instead of a star in the following difference:
$$**** - **** = $$
Then the first gives the next digit, and so on $8$ times.
The first wants to obtain the greatest possible difference, the second -- the least. Prove that:
1. The first can operate in such a way that the difference would be not less than $4000$, not depending on the second's behaviour.
2. The second can operate in such a way that the difference would be not greater than $4000$, not depending on the first's behaviour.
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 $56$.
Positive integer $m$ shall be called [i]anagram [/i] of positive $n$ if every digit $a$ appears as many times in the decimal representation of $m$ as it appears in the decimal representation of $n$ also. Is it possible to find $4$ different positive integers such that each of the four to be [i]anagram [/i] of the sum of the other $3$?
(Bulgaria)
Are there $n$-digit numbers M and N such that all digits $M$ are even, all $N$ digits are odd, every digit from $0$ to $9$ occurs in decimal notation M or N at least once, and $M$ is divisible by $N$?
Suppose for some positive integer $n$, the numbers $2^n$ and $5^n$ have equal first digit. What are the possible values of this first digit?
Note: solved [url=https://artofproblemsolving.com/community/c6h312638p1685546]here[/url]
How many 4-digit numbers $ABCD$ are there with the property that $|A-B|= |B-C|= |C-D|$?
Note that the first digit $A$ of a four-digit number cannot be zero.
The teacher writes a $2002$-digit number consisting only of digits $9$ on the blackboard.
The first student factors this number as $ab$ with $a > 1$ and $b > 1$ and replaces it on the blackboard by two numbers $a'$ and $b'$ with $|a-a'| = |b-b'| = 2$. The second student chooses one of the numbers on the blackboard, factors it as $cd$ with $c > 1$ and $d > 1$ and replaces the chosen number by two numbers $c'$ and $d'$ with $|c-c'| = |d-d'| = 2$, etc. Is it possible that after a certain number of students have been to the blackboard all numbers written there are equal to $9$?
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.
The [i]fragments [/i] of a positive integer are the numbers seen when reading one or more of its digits in order. The [i]fragment sum[/i] equals the sum of all the fragments, including the number itself. For example, the fragment sum of $2022$ is $2022+202+022+20+02+22+2+0+2+2 = 2296$.
There is another four-digit number with the same fragment sum. What is it?
As the example shows, if a fragment occurs multiple times, then all its occurrences are added, and the fragments beginning with $0$ also count (for instance, $022$ is worth $22$).
What is the last digit in the number $2007^{2007}$?
Find the smallest possible natural number $n = \overline{a_m ...a_2a_1a_0} $ (in decimal system) such that the number $r = \overline{a_1a_0a_m ..._20} $ equals $2n$.
For a positive integer $n$, let $S(n), P(n)$ denote the sum and the product of all the digits of $n$ respectively.
1) Find all values of n such that $n = P(n)$:
2) Determine all values of n such that $n = S(n) + P(n)$.
Determine all $6$-digit numbers $(abcdef)$ such that $(abcdef) = (def)^2$ where $(x_1x_2...x_n)$ is not a multiplication but a number of $n$ digits.
A three-digit $\overline{abc}$ number is called [i]Ecuadorian [/i] if it meets the following conditions:
$\bullet$ $\overline{abc}$ does not end in $0$.
$\bullet$ $\overline{abc}$ is a multiple of $36$.
$\bullet$ $\overline{abc} - \overline{cba}$ is positive and a multiple of $36$.
Determine all the Ecuadorian numbers.
Find a way to write all the digits of $1$ to $9$ in a sequence and without repetition, so that the numbers determined by any two consecutive digits of the sequence are divisible by $7$ or $13$.
How many consecutive zeros are there at the end of the number $2001! = 2001 \times 2000 \times ... \times 3 \times 2 \times 1$ ?
When $4444^{4444}$ is written in decimal notation, the sum of its digits is $ A.$ Let $B$ be the sum of the digits of $A.$ Find the sum of the digits of $ B.$ ($A$ and $B$ are written in decimal notation.)
Let $N = 2 \: \: \underbrace {99… 9} _{n \,\,\text {times}} \: \: 82 \: \: \underbrace {00… 0} _{n \,\, \text {times} } \: \: 29$. Prove that $N$ can be written as the sum of the squares of $3$ consecutive natural numbers.
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 $n \geq n_0, d_{n+T} = d_n.$
Call a positive integer [i]balanced [/i] if the number of its distinct prime factors is equal to the number of its digits in the decimal representation; for example, the number $385 = 5 \cdot 7 \cdot 11$ is balanced, while $275 = 5^2 \cdot 11$ is not. Prove that there exist only a finite number of balanced numbers.
Prove that for every real number $M$ there exists an infinite arithmetic progression such that:
- each term is a positive integer and the common difference is not divisible by 10
- the sum of the digits of each term (in decimal representation) exceeds $M$.
Find the last three digits in the product $1 \cdot 3\cdot 5\cdot 7 \cdot . . . \cdot 2009 \cdot 2011$.