Olympiad problems

2014 Contests, 2

Tags: algebra , sum of cubes , sequence , digit , number theory

The first term of a sequence is $2014$. Each succeeding term is the sum of the cubes of the digits of the previous term. What is the $2014$ th term of the sequence?

1974 All Soviet Union Mathematical Olympiad, 197

Tags: number theory , digit

Find all the natural $n$ and $k$ such that $n^n$ has $k$ digits and $k^k$ has $n$ digits.

1994 Bundeswettbewerb Mathematik, 1

Tags: pigeonhole , decimal representation , real number , digit , algebra

Given eleven real numbers, prove that there exist two of them such that their decimal representations agree infinitely often.

2001 All-Russian Olympiad Regional Round, 9.8

Tags: number theory , perfect square , digit

Sasha wrote a non-zero number on the board and added it to it on the right, one non-zero digit at a time, until he writes out a million digits. Prove that an exact square has been written on the board no more than $100$ times.

2020 Colombia National Olympiad, 1

Tags: number theory , digit , perfect square

A positive integer is called [i]sabroso [/i]if when it is added to the number obtained when its digits are interchanged from one side of its written form to the other, the result is a perfect square. For example, $143$ is sabroso, since $143 + 341 =484 = 22^2$. Find all two-digit sabroso numbers.

2010 Contests, 1

Tags: number theory , sum of squares , algebra , digit

Nine positive integers $a_1,a_2,...,a_9$ have their last $2$-digit part equal to $11,12,13,14,15,16,17,18$ and $19$ respectively. Find the last $2$-digit part of the sum of their squares.

2022 Durer Math Competition Finals, 7

Tags: number theory , digit

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 $22$).

2018 Estonia Team Selection Test, 6

Tags: number theory , digit

We call a positive integer $n$ whose all digits are distinct [i]bright[/i], if either $n$ is a one-digit number or there exists a divisor of $n$ which can be obtained by omitting one digit of $n$ and which is bright itself. Find the largest bright positive integer. (We assume that numbers do not start with zero.)

2014 Czech-Polish-Slovak Junior Match, 4

Tags: number theory , divide , divisible , digit

The number $a_n$ is formed by writing in succession, without spaces, the numbers $1, 2, ..., n$ (for example, $a_{11} = 1234567891011$). Find the smallest number t such that $11 | a_t$.

2021 Malaysia IMONST 1, 17

Tags: number theory , digit

Determine the sum of all positive integers $n$ that satisfy the following condition: when $6n + 1$ is written in base $10$, all its digits are equal.

2012 Bosnia and Herzegovina Junior BMO TST, 2

Tags: combinatorics , digit , transformation

Let $\overline{abcd}$ be $4$ digit number, such that we can do transformations on it. If some two neighboring digits are different than $0$, then we can decrease both digits by $1$ (we can transform $9870$ to $8770$ or $9760$). If some two neighboring digits are different than $9$, then we can increase both digits by $1$ (we can transform $9870$ to $9980$ or $9881$). Can we transform number $1220$ to: $a)$ $2012$ $b)$ $2021$

2006 VTRMC, Problem 1

Tags: number theory , digit

Find, with proof, all positive integers $n$ such that neither $n$ nor $n^2$ contain a $1$ when written in base $3$.

2003 Cuba MO, 1

Tags: number theory , digit , prime

Given the following list of numbers: $$1990, 1991, 1992, ..., 2002, 2003, 2003, 2003, ..., 2003$$ where the number $2003$ appears $12$ times. Is it possible to write these numbers in some order so that the $100$-digit number that we get is prime?

2014 India PRMO, 2

Tags: algebra , sum of cubes , sequence , digit , number theory

The first term of a sequence is $2014$. Each succeeding term is the sum of the cubes of the digits of the previous term. What is the $2014$ th term of the sequence?

1999 Greece JBMO TST, 3

Tags: number theory , decimal representation , digit

Find digits $a,b,c,x$ ($a>0$) such that $\overline{abc}+\overline{acb}=\overline{199x}$

2015 Junior Regional Olympiad - FBH, 4

Tags: digit , fraction

Which number we need to substract from numerator and add to denominator of $\frac{\overline{28a3}}{7276}$ such that we get fraction equal to $\frac{2}{7}$

1991 Tournament Of Towns, (287) 3

Tags: number theory , digit

We are looking for numbers ending with the digit $5$ such that in their decimal expansion each digit beginning with the second digit is no less than the previous one. Moreover the squares of these numbers must also possess the same property. (a) Find four such numbers. (b) Prove that there are infinitely many. (A. Andjans, Riga)

1970 IMO Shortlist, 7

Tags: modular arithmetic , number theory , digit , equation

For which digits $a$ do exist integers $n \geq 4$ such that each digit of $\frac{n(n+1)}{2}$ equals $a \ ?$

2021 Durer Math Competition Finals, 4

Tags: digit , number theory

What is the number of $4$-digit numbers that contains exactly $3$ different digits that have consecutive value? Such numbers are for instance $5464$ or $2001$. Two digits in base $10$ are consecutive if their difference is $1$.

1989 Chile National Olympiad, 1

Tags: number theory , sum of digits , digit

Writing $1989$ in base $b$, we obtain a three-digit number: $xyz$. It is known that the sum of the digits is the same in base $10$ and in base $b$, that is, $1 + 9 + 8 + 9 = x + y + z$. Determine $x,y,z,b.$

2018 Singapore Junior Math Olympiad, 1

Tags: multiple , number theory , divide , digit

Consider the integer $30x070y03$ where $x, y$ are unknown digits. Find all possible values of $x, y$ so that the given integer is a multiple of $37$.

2013 Chile National Olympiad, 1

Tags: combinatorics , digit , number theory

Find the sum of all $5$-digit positive integers that they have only the digits $1, 2$, and $5$, none repeated more than three consecutive times.

1978 IMO Longlists, 7

Tags: modular arithmetic , number theory , digit , decimal representation

Let $ m$ and $ n$ be positive integers such that $ 1 \le m < n$. In their decimal representations, the last three digits of $ 1978^m$ are equal, respectively, to the last three digits of $ 1978^n$. Find $ m$ and $ n$ such that $ m \plus{} n$ has its least value.

2020 Regional Competition For Advanced Students, 2

Tags: digit , number theory

The set $M$ consists of all $7$-digit positive integer numbers that contain (in decimal notation) each of the digits $1, 3, 4, 6, 7, 8$ and $9$ exactly once. (a) Find the smallest positive difference $d$ of two numbers from $M$. (b) How many pairs $(x, y)$ with $x$ and $y$ from M are there for which $x - y = d$? (Gerhard Kirchner)

1980 Bulgaria National Olympiad, Problem 1

Tags: algebra , digit , number theory

Show that there exists a unique sequence of decimal digits $p_0=5,p_1,p_2,\ldots$ such that, for any $k$, the square of any positive integer ending with $\overline{p_kp_{k-1}\cdots p_0}$ ends with the same digits.