Found problems: 159
We say that a natural number is called special if all of its digits are non-zero and any two adjacent digits in its decimal representation are consecutive (not necessarily in ascending order).
a) Determine the largest special number $m$ whose sum of digits is equal to $2023$.
b) Determine the smallest special number $n$ whose sum of digits is equal to $2022$.
Given a polynomial $P(x)$ with
a) natural coefficients;
b) integer coefficients;
Let us denote with $a_n$ the sum of the digits of $P(n)$ value. Prove that there is a number encountered in the sequence $a_1, a_2, ... , a_n, ...$ infinite times.
Let $n,k$ be positive integers such that n is not divisible by 3 and $k \geq n$. Prove that there exists a positive integer $m$ which is divisible by $n$ and the sum of its digits in decimal representation is $k$.
Prove that among $18$ consecutive three digit numbers there must be at least one which is divisible by the sum of its digits.
When the natural numbers are written one after another in an increasing way, you get an infinite succession of digits $123456789101112 ....$ Denote $A_k$ the number formed by the first $k$ digits of this sequence . Prove that for all positive integer $n$ there is a positive integer $m$ which simultaneously verifies the following three conditions:
(i) $n$ divides $A_m$,
(ii) $n$ divides $m$,
(iii) $n$ divides the sum of the digits of $A_m$.
With the digits $1, 2, 3,. . . . . . , 9$ three-digit numbers are written such that the sum of the three digits is $17$. How many numbers can be written?
For each positive integer $n$, let $S(n)$ be the sum of the digits of $n^2 +1$. A sequence $\{a_n\}$ is defined, with $a_0$ an arbitrary positive integer and $a_{n+1} = S(a_n)$. Prove that the sequence $\{a_n\}$ is eventually periodic with period three.
How many integers from $1$ to $2008$ have the sum of their digits divisible by $5$ ?
The product of the digits of the natural number $N$ is denoted by $P(N)$ whereas the sum of these digits is denoted by $S(N)$. How many solutions does the equation $P(P(N)) + P(S(N)) + S(P(N)) + S(S(N)) = 1984$ have?
For each positive integer $n$ we define $f (n)$ as the product of the sum of the digits of $n$ with $n$ itself.
Examples: $f (19) = (1 + 9) \times 19 = 190$, $f (97) = (9 + 7) \times 97 = 1552$.
Show that there is no number $n$ with $f (n) = 19091997$.
The function $f : N \to N$ is defined by $f(n) = n + S(n)$, where $S(n)$ is the sum of digits in the decimal representation of positive integer $n$.
a) Prove that there are infinitely many numbers $a \in N$ for which the equation $f(x) = a$ has no natural roots.
b) Prove that there are infinitely many numbers $a \in N$ for which the equation $f(x) = a$ has at least two distinct natural roots.
(I. Voronovich)
Let $q(n)$ denote the sum of the digits of a natural number $n$. Determine $q(q(q(2000^{2000})))$.
The numbers $d_1, d_2,..., d_6$ are distinct digits of the decimal number system other than $6$.
Prove that $d_1+d_2+...+d_6= 36$ if and only if $(d_1-6) (d_2-6) ... (d_6 -6) = -36$.
Calculate the sum of digit of the number
$$9 \times 99 \times 9999 \times ... \times \underbrace{ 99...99}_{2^n}$$
where the number of nines doubles in each factor.
For each positive integer $k$, let $S(k)$ the sum of digits of $k$ in decimal system.
Show that there is an integer $k$, with no $9$ in it's decimal representation, such that:
$$S(2^{24^{2017}}k)=S(k)$$
Does there exist an increasing arithmetic progression of
(a) $11$
(b) $10000$
(c) infinitely many
positive integers such that the sums of their digits in base $10$ also form an increasing arithmetic progression?
(A Shapovalov)
For any positive integer $k$, denote the sum of digits of $k$ in its decimal representation by $S(k)$. Find all polynomials $P(x)$ with integer coefficients such that for any positive integer $n \geq 2016$, the integer $P(n)$ is positive and $$S(P(n)) = P(S(n)).$$
[i]Proposed by Warut Suksompong, Thailand[/i]
Let $s (n)$ denote the sum of digits of a positive integer $n$. Using six different digits, we formed three 2-digits $p, q, r$ such that $$p \cdot q \cdot s(r) = p\cdot s(q) \cdot r = s (p) \cdot q \cdot r.$$ Find all such numbers $p, q, r$.
Find all two-digit numbers $x$ the sum of whose digits is the same as that of $2x$, $3x$, ... , $9x$.
Let us denote by $S(m)$ the sum of the digits of the natural number $m$. Prove that there are infinitely many positive integers $n$ such that $$S(3^n) \ge S(3^{n+1}).$$
For every positive integer $n$, $s(n)$ denotes the sum of the digits in the decimal representation of $n$. Prove that for every integer $n \ge 5$, we have $$S(1)S(3)...S(2n-1) \ge S(2)S(4)...S(2n)$$
Let $s(n)$ denote the sum of digits of a positive integer $n$. Prove that $s(9^n) > 9$ for all $n\geq 3$.
We have a positive integer $n$, whose sum of digits is $100$ . If the sum of digits of $44n$ is $800$ then what is the sum of digits of $3n$?
A natural number $x$ is written on the board. In one move, we can take the number on the board and between any two of its digits in its decimal notation we can we put a sign $+$, or we may not put it, then we calculate the obtained result and we write it on the board in place of $x$. For example, from the number $819$. we can get $18$ by $8 + 1 + 9$, $90$ by $81 + 9$, and $27$ by $8 + 19$. Prove that no matter what $x$ is, we can reach a single digit number with at most $4$ moves.
A pair of positive integers $(a,b)$ is called an [b]average couple[/b] if there exist positive integers $k$ and $c_1, \dots, c_k$ for which
\[\frac{c_1+c_2+\cdots+c_k}{k}=a\qquad \text{and} \qquad \frac{s(c_1)+s(c_2)+\cdots+s(c_k)}{k}=b\]
where $s(n)$ denotes the sum of digits of $n$ in decimal representation.
Find the number of average couples $(a,b)$ for which $a,b<10^{10}$.