Found problems: 546
I Soros Olympiad 1994-95 (Rus + Ukr), 9.10
For which natural $n$ there exists a natural number multiple of $n$, whose decimal notation consists only of the digits $8$ and $9$ (possibly only from numbers $8$ or only from numbers $9$)?
2021 Malaysia IMONST 1, 20
The cells of a $2021\times 2021$ table are filled with numbers using the following rule. The bottom left cell, which we label with coordinate $(1, 1)$, contains the number $0$. For every other cell $C$, we consider a route from $(1, 1)$ to $C$, where at each step we can only go one cell to the right or one cell up (not diagonally). If we take the number of steps in the route and add the numbers from the cells along the route, we obtain the number in cell $C$. For example, the cell with coordinate $(2, 1)$ contains $1 = 1 + 0$, the cell with coordinate $(3, 1)$ contains $3 = 2 + 0 + 1$, and the cell with coordinate $(3, 2)$ contains $7 = 3 + 0 + 1 + 3$. What is the last digit of the number in the cell $(2021, 2021)$?
2020-IMOC, C3
Sunny wants to send some secret message to usjl. The secret message is a three digit number, where each digit is one digit from $0$ to $9$ (so $000$ is also possibly the secret message). However, when Sunny sends the message to usjl, at most one digit might be altered. Therefore, Sunny decides to send usjl a longer message so that usjl can decipher the message to get the original secret message Sunny wants to send. Sunny and usjl can communicate the strategy beforehand. Show that sending a $4$-digit message does not suffice. Also show that sending a $6$-digit message suffices. If it is deduced that sending a $c$-digit message suffices for some $c>6$, then partial credits may be awarded.
1999 Denmark MO - Mohr Contest, 5
Is there a number whose digits are only $1$'s and which is divided by $1999$?
2012 Ukraine Team Selection Test, 6
For the positive integer $k$ we denote by the $a_n$ , the $k$ from the left digit in the decimal notation of the number $2^n$ ($a_n = 0$ if in the notation of the number $2^n$ less than the digits). Consider the infinite decimal fraction $a = \overline{0, a_1a_2a_3...}$. Prove that the number $a$ is irrational.
1991 Swedish Mathematical Competition, 5
Show that there are infinitely many odd positive integers $n$ such that in binary $n$ has more $1$s than $n^2$.
2004 All-Russian Olympiad Regional Round, 11.6
Let us call the [i]distance [/i] between the numbers $\overline{a_1a_2a_3a_4a_5}$ and $\overline{b_1b_2b_3b_4b_5}$ the maximum $i$ for which $a_i \ne b_i$. All five-digit numbers are written out one after another in some order. What is the minimum possible sum of distances between adjacent numbers?
1977 Germany Team Selection Test, 4
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.)
2018 Brazil EGMO TST, 1
(a) Let $m$ and $n$ be positive integers and $p$ a positive rational number, with $m > n$, such that $\sqrt{m} -\sqrt{n}= p$. Prove that $m$ and $n$ are perfect squares.
(b) Find all four-digit numbers $\overline{abcd}$, where each letter $a, b, c$ and $d$ represents a digit, such that $\sqrt{\overline{abcd}} -\sqrt{\overline{acd}}= \overline{bb}$.
2001 Switzerland Team Selection Test, 5
Let $a_1 < a_2 < ... < a_n$ be a sequence of natural numbers such that for $i < j$ the decimal representation of $a_i$ does not occur as the leftmost digits of the decimal representation of $a_j$ . (For example, $137$ and $13729$ cannot both occur in the sequence.) Prove that $\sum_{i=1}^n \frac{1}{a_i} \le 1+\frac12 +\frac13 +...+\frac19$
.
2020 Flanders Math Olympiad, 2
Every officially published book used to have an ISBN code (International Standard Book Number) which consisted of $10$ symbols. Such code looked like this: $$a_1a_2 . . . a_9a_{10}$$ with $a_1, . . . , a_9 \in \{0, 1, . . . , 9\}$ and $a_{10} \in \{0, 1, . . . , 9, X\}$. The symbol $X$ stood for the number $10$. With a valid ISBN code was
$$a_1 + 2a2 + . . . + 9a_9 + 10a_{10}$$ a multiple of $11$. Prove the following statements.
(a) If one symbol is changed in a valid ISBN code, the result is no valid ISBN code.
(b) When two different symbols swap places in a valid ISBN code then the result is not a valid ISBN.
2017 Latvia Baltic Way TST, 15
Let's call the number string $D = d_{n-1}d_{n-2}...d_0$ a [i]stable ending[/i] of a number , if for any natural number $m$ that ends in $D$, any of its natural powers $m^k$ also ends in $D$. Prove that for every natural number $n$ there are exactly four stable endings of a number of length $n$.
[hide=original wording]Ciparu virkni $D = d_{n-1}d_{n-2}...d_0$ sauksim par stabilu skaitļa nobeigumu, ja jebkuram naturālam skaitlim m, kas beidzas ar D, arī jebkura tā naturāla pakāpe $m^k$ beidzas ar D. Pierādīt, ka katram naturālam n ir tieši četri stabili skaitļa nobeigumi, kuru garums ir n.[/hide]
2020 Polish Junior MO First Round, 1.
Determine all natural numbers $n$, such that it's possible to insert one digit at the right side of $n$ to obtain $13n$.
2022 Durer Math Competition Finals, 7
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 $22$).
2009 Singapore Junior Math Olympiad, 3
Suppose $\overline{a_1a_2...a_{2009}}$ is a $2009$-digit integer such that for each $i = 1,2,...,2007$, the $2$-digit integer $\overline{a_ia_{i+1}}$ contains $3$ distinct prime factors. Find $a_{2008}$
(Note: $\overline{xyz...}$ denotes an integer whose digits are $x, y,z,...$.)
1969 All Soviet Union Mathematical Olympiad, 117
Given a finite sequence of zeros and ones, which has two properties:
a) if in some arbitrary place in the sequence we select five digits in a row and also select five digits in any other place in a row, then these fives will be different (they may overlap);
b) if you add any digit to the right of the sequence, then property (a) will no longer hold true.
Prove that the first four digits of our sequence coincide with the last four
1991 Tournament Of Towns, (319) 6
An arithmetical progression (whose difference is not equal to zero) consists of natural numbers without any nines in its decimal notation.
(a) Prove that the number of its terms is less than $100$.
(b) Give an example of such a progression with $72$ terms.
(c) Prove that the number of terms in any such progression does not exceed $72$.
(V. Bugaenko and Tarasov, Moscow)
2015 Saudi Arabia JBMO TST, 1
A $2015$- digit natural number $A$ has the property that any $5$ of it's consecutive digits form a number divisible by $32$. Prove that $A$ is divisible by $2^{2015}$
1927 Eotvos Mathematical Competition, 2
Find the sum of all distinct four-digit numbers that contain only the digits $1, 2, 3, 4,5$, each at most once.
1991 Tournament Of Towns, (307) 4
A sequence $a_n$ is determined by the rules $a_0 = 9$ and for any nonnegative $k$,
$$a_{k+1}=3a_k^4+4a_k^3.$$
Prove that $a_{10}$ contains more than $1000$ nines in decimal notation.
(Yao)
2014 Junior Regional Olympiad - FBH, 5
From digits $0$, $1$, $3$, $4$, $7$ and $9$ were written $5$ digit numbers which all digits are different. How many numbers from them are divisible with $5$