Olympiad problems

2022 CHMMC Winter (2022-23), 2

Tags: number theory

Select a number $X$ from the set of all $3$-digit natural numbers uniformly at random. Let $A \in [0,1]$ be the probability that $X$ is divisible by $11$, given that it is palindromic. Let $B \in [0,1]$ be the probability that X is palindromic, given that it is divisible by $11$. Compute $B-A$. Recall that a $3$-digit number is a palindrome if it reads the same left to right as right to left. For instance, $484$ is a palindrome, but $603$ is not a palindrome.

2010 Kazakhstan National Olympiad, 3

Tags: number theory

Call $A \in \mathbb{N}^0$ be $number of year$ if all digits of $A$ equals $0$, $1$ or $2$ (in decimal representation). Prove that exist infinity $N \in \mathbb{N}$, such that $N$ can't presented as $A^2+B$ where $A \in \mathbb{N}^0 ; B$- $number of year$.

2019 Middle European Mathematical Olympiad, 8

Tags: pell equation , number theory

Let $N$ be a positive integer such that the sum of the squares of all positive divisors of $N$ is equal to the product $N(N+3)$. Prove that there exist two indices $i$ and $j$ such that $N=F_iF_j$ where $(F_i)_{n=1}^{\infty}$ is the Fibonacci sequence defined as $F_1=F_2=1$ and $F_n=F_{n-1}+F_{n-2}$ for $n\geq 3$. [i]Proposed by Alain Rossier, Switzerland[/i]

2005 AMC 8, 20

Tags: least common multiple , number theory , modular arithmetic

Alice and Bob play a game involving a circle whose circumference is divided by 12 equally-spaced points. The points are numbered clockwise, from 1 to 12. Both start on point 12. Alice moves clockwise and Bob, counterclockwise. In a turn of the game, Alice moves 5 points clockwise and Bob moves 9 points counterclockwise. The game ends when they stop on the same point. How many turns will this take? $ \textbf{(A)}\ 6\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 14\qquad\textbf{(E)}\ 24 $

2023 Indonesia MO, 4

Tags: digit , nt construction , number theory

Determine whether or not there exists a natural number $N$ which satisfies the following three criteria: 1. $N$ is divisible by $2^{2023}$, but not by $2^{2024}$, 2. $N$ only has three different digits, and none of them are zero, 3. Exactly 99.9% of the digits of $N$ are odd.

1962 Dutch Mathematical Olympiad, 3

Tags: recurrence relation , digit , number theory

Consider the positive integers written in the decimal system with $n$ digits, the start of which is not zero and where there are no two sevens next to each other. The number of these numbers is called $u_n$. Derive a relation that expresses $u_{n+2}$ in terms of $u_{n+1}$ and $u_n$.

2015 Postal Coaching, Problem 4

Tags: number theory , divisor

For $ n \in \mathbb{N}$, let $s(n)$ denote the sum of all positive divisors of $n$. Show that for any $n > 1$, the product $s(n - 1)s(n)s(n + 1)$ is an even number.

2005 China Team Selection Test, 3

Tags: floor function , euler , number theory

Let $a_1,a_2 \dots a_n$ and $x_1, x_2 \dots x_n$ be integers and $r\geq 2$ be an integer. It is known that \[\sum_{j=0}^{n} a_j x_j^k =0 \qquad \text{for} \quad k=1,2, \dots r.\] Prove that \[\sum_{j=0}^{n} a_j x_j^m \equiv 0 \pmod m, \qquad \text{for all}\quad m \in \{ r+1, r+2, \cdots, 2r+1 \}.\]

2011 ITAMO, 5

Tags: number theory

Determine all solutions $(p,n)$ of the equation \[n^3=p^2-p-1\] where $p$ is a prime number and $n$ is an integer

2014 National Olympiad First Round, 26

Tags: number theory , modular arithmetic

Let $f(n)$ be the smallest prime which divides $n^4+1$. What is the remainder when the sum $f(1)+f(2)+\cdots+f(2014)$ is divided by $8$? $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 7 \qquad\textbf{(E)}\ \text{None of the preceding} $

2011 Armenian Republican Olympiads, Problem 4

Tags: number theory

What is the maximal number of elements we can choose form the set $\{1, 2, \ldots, 31\}$, such that the sum of any two of them is not a perfect square.

2020 Baltic Way, 18

Tags: number theory

Let $n\geq 1$ be a positive integer. We say that an integer $k$ is a [i]fan [/i]of $n$ if $0\leq k\leq n-1$ and there exist integers $x,y,z\in\mathbb{Z}$ such that \begin{align*} x^2+y^2+z^2 &\equiv 0 \pmod n;\\ xyz &\equiv k \pmod n. \end{align*} Let $f(n)$ be the number of fans of $n$. Determine $f(2020)$.

2012 AIME Problems, 12

Tags: trigonometry , ratio , relatively prime , number theory

Let $\triangle ABC$ be a right triangle with right angle at $C$. Let $D$ and $E$ be points on $\overline{AB}$ with $D$ between $A$ and $E$ such that $\overline{CD}$ and $\overline{CE}$ trisect $\angle C$. If $\frac{DE}{BE} = \frac{8}{15}$, then $\tan B$ can be written as $\frac{m\sqrt{p}}{n}$, where $m$ and $n$ are relatively prime positive integers, and $p$ is a positive integer not divisible by the square of any prime. Find $m+n+p$.

2022 Germany Team Selection Test, 2

Tags: construction , number theory , combinatorics

Let $r>1$ be a rational number. Alice plays a solitaire game on a number line. Initially there is a red bead at $0$ and a blue bead at $1$. In a move, Alice chooses one of the beads and an integer $k \in \mathbb{Z}$. If the chosen bead is at $x$, and the other bead is at $y$, then the bead at $x$ is moved to the point $x'$ satisfying $x'-y=r^k(x-y)$. Find all $r$ for which Alice can move the red bead to $1$ in at most $2021$ moves.

2003 Junior Balkan Team Selection Tests - Romania, 2

Tags: prime , consecutive , divide , sum of powers , number theory

Consider the prime numbers $n_1< n_2 <...< n_{31}$. Prove that if $30$ divides $n_1^4 + n_2^4+...+n_{31}^4$, then among these numbers one can find three consecutive primes.

2005 Iran Team Selection Test, 1

Tags: function , number theory

Find all $f : N \longmapsto N$ that there exist $k \in N$ and a prime $p$ that: $\forall n \geq k \ f(n+p)=f(n)$ and also if $m \mid n$ then $f(m+1) \mid f(n)+1$

2005 iTest, 31

Tags: digit , number theory

Let $X = 123456789$. Find the sum of the tens digits of all integral multiples of $11$ that can be obtained by interchanging two digits of $X$.

2005 iTest, 39

Tags: sum of powers , number theory

What is the smallest positive integer that when raised to the $6^{th}$ power, it can be represented by a sum of the $6^{th}$ powers of distinct smaller positive integers?

2001 Federal Math Competition of S&M, Problem 1

Tags: number theory

Let $S=\{x^2+2y^2\mid x,y\in\mathbb Z\}$. If $a$ is an integer with the property that $3a$ belongs to $S$, prove that then $a$ belongs to $S$ as well.

2019 Peru IMO TST, 4

Tags: sequence , floor function , number theory

Let $k\geq 0$ an integer. The sequence $a_0,\ a_1,\ a_2, \ a_3, \ldots$ is defined as follows: [LIST] [*] $a_0=k$ [/*] [*] For $n\geq 1$, we have that $a_n$ is the smallest integer greater than $a_{n-1}$ so that $a_n+a_{n-1}$ is a perfect square. [/*] [/LIST] Prove that there are exactly $\left \lfloor{\sqrt{2k}} \right \rfloor$ positive integers that cannot be written as the difference of two elements of such a sequence. [i]Note.[/i] If $x$ is a real number, $\left \lfloor{x} \right \rfloor$ denotes the greatest integer smaller or equal than $x$.

2009 Junior Balkan Team Selection Test, 1

Tags: number theory

Find all two digit numbers $ \overline{AB}$ such that $ \overline{AB}$ divides $ \overline{A0B}$.

2007 Bosnia and Herzegovina Junior BMO TST, 2

Tags: divisible , divide , number theory

Find all pairs of relatively prime numbers ($x, y$) such that $x^2(x + y)$ is divisible by $y^2(y - x)^2$. .

2010 China Team Selection Test, 3

Tags: number theory

For integers $n>1$, define $f(n)$ to be the sum of all postive divisors of $n$ that are less than $n$. Prove that for any positive integer $k$, there exists a positive integer $n>1$ such that $n<f(n)<f^2(n)<\cdots<f^k(n)$, where $f^i(n)=f(f^{i-1}(n))$ for $i>1$ and $f^1(n)=f(n)$.

1992 IMO Longlists, 16

Tags: equation , number theory

Find all triples $(x, y, z)$ of integers such that \[\frac{1}{x^2}+\frac{2}{y^2}+\frac{3}{z^2} =\frac 23\]

1998 IMO Shortlist, 4

Tags: integer sequence , calculate , number theory

A sequence of integers $ a_{1},a_{2},a_{3},\ldots$ is defined as follows: $ a_{1} \equal{} 1$ and for $ n\geq 1$, $ a_{n \plus{} 1}$ is the smallest integer greater than $ a_{n}$ such that $ a_{i} \plus{} a_{j}\neq 3a_{k}$ for any $ i,j$ and $ k$ in $ \{1,2,3,\ldots ,n \plus{} 1\}$, not necessarily distinct. Determine $ a_{1998}$.