Olympiad problems

2006 Switzerland - Final Round, 10

Decide whether there is an integer $n > 1$ with the following properties: (a) $n$ is not a prime number. (b) For all integers $a$, $a^n - a$ is divisible by $n$

2001 Paraguay Mathematical Olympiad, 3

Tags: number theory , divisible , sum of digits

Find a $10$-digit number, in which no digit is zero, that is divisible by the sum of their digits.

2015 Caucasus Mathematical Olympiad, 1

Tags: digit , sum , divisible , number theory

Does there exist a four-digit positive integer with different non-zero digits, which has the following property: if we add the same number written in the reverse order, then we get a number divisible by $101$?

2010 QEDMO 7th, 9

Tags: number theory , divisible

Let $p$ be an odd prime number and $c$ an integer for which $2c -1$ is divisible by $p$. Prove that $$(-1)^{\frac{p+1}{2}}+\sum_{n=0}^{\frac{p-1}{2}} {2n \choose n}c^n$$ is divisible by $p$.

2016 Costa Rica - Final Round, N2

Tags: divide , divisible , number theory

Let $x, y, z$ be positive integers and $p$ a prime such that $x <y <z <p$. Also $x^3, y^3, z^3$ leave the same remainder when divided by $p$. Prove that $x + y + z$ divides $x^2 + y^2 + z^2$.

2019 Durer Math Competition Finals, 11

Tags: sum , divide , divisible , number theory

What is the smallest $N$ for which $\sum_{k=1}^{N} k^{2018}$ is divisible by $2018$?

2008 Postal Coaching, 3

Tags: product , divide , divisible , number theory

Prove that there exists an innite sequence $<a_n>$ of positive integers such that for each $k \ge 1$ $(a_1 - 1)(a_2 - 1)(a_3 -1)...(a_k - 1)$ divides $a_1a_2a_3 ...a_k + 1$.

2010 All-Russian Olympiad Regional Round, 10.4

Tags: number theory , divisible

We call a natural number $b$ [i]lucky [/i] if for any natural number $a$ such that $a^5$ is divisible by $b^2$, the number $a^2$ is divisible by $b$. Find the number of [i]lucky [/i] natural numbers less than $2010$.

1999 Israel Grosman Mathematical Olympiad, 1

Tags: divisible , number theory

For any $16$ positive integers $n,a_1,a_2,...,a_{15}$ we define $T(n,a_1,a_2,...,a_{15}) = (a_1^n+a_2^n+ ...+a_{15}^n)a_1a_2...a_{15}$. Find the smallest $n$ such that $T(n,a_1,a_2,...,a_{15})$ is divisible by $15$ for any choice of $a_1,a_2,...,a_{15}$.

2018 Swedish Mathematical Competition, 4

Tags: consecutive , divisible , number theory , sum of digits

Find the least positive integer $n$ with the property: Among arbitrarily $n$ selected consecutive positive integers, all smaller than $2018$, there is at least one that is divisible by its sum of digits .

VMEO I 2004, 2

Tags: fibonacci number , fibonacci sequence , fibonacci , divisible , perfect power , number theory

The Fibonacci numbers $(F_n)_{n=1}^{\infty}$ are defined as follows: $$F_1 = F_2 = 1, F_n = F_{n-2} + F_{n-1}, n = 3, 4, ...$$ Assume $p$ is a prime greater than $3$. With $m$ being a natural number greater than $3$, find all $n$ numbers such that $F_n$ is divisible by $p^m$.

2007 Dutch Mathematical Olympiad, 3

Tags: digit , number theory , divisible

Does there exist an integer having the form $444...4443$ (all fours, and ending with a three) that is divisible by $13$? If so, give an integer having that form that is divisible by $13$, if not, prove that such an integer cannot exist.

2018 Estonia Team Selection Test, 12

Tags: divisible , polynomial

We call the polynomial $P (x)$ simple if the coefficient of each of its members belongs to the set $\{-1, 0, 1\}$. Let $n$ be a positive integer, $n> 1$. Find the smallest possible number of terms with a non-zero coefficient in a simple $n$-th degree polynomial with all values at integer places are divisible by $n$.

2019 Saudi Arabia IMO TST, 1

Tags: divide , number theory , divisible

Let $a_0$ be an arbitrary positive integer. Let $(a_n)$ be infinite sequence of positive integers such that for every positive integer $n$, the term $a_n$ is the smallest positive integer such that $a_0 + a_1 +... + a_n$ is divisible by $n$. Prove that there exist $N$ such that $a_{n+1} = a_n$ for all $n \ge N$

2016 NZMOC Camp Selection Problems, 4

Tags: divisible , number theory

A quadruple $(p, a, b, c)$ of positive integers is a[i] karaka quadruple[/i] if $\bullet$ $p$ is an odd prime number $\bullet$ $a, b$ and $c$ are distinct, and $\bullet$ $ab + 1$, $bc + 1$ and $ca + 1$ are divisible by $p$. (a) Prove that for every karaka quadruple $(p, a, b, c)$ we have $p + 2 \le\frac{a + b + c}{3}$. (b) Determine all numbers $p$ for which a karaka quadruple $(p, a, b, c)$ exists with $p + 2 =\frac{a + b + c}{3}$

VMEO IV 2015, 11.3

Tags: floor function , divisible , number theory

How many natural number $n$ less than $2015$ that is divisible by $\lfloor\sqrt[3]{n}\rfloor$ ?

1925 Eotvos Mathematical Competition, 1

Tags: divisible , divide , number theory

Let $a,b, c,d$ be four integers. Prove that the product of the six differences $$b - a,c - a,d - a,d - c,d - b, c - b$$ is divisible by $12$.

2018 Istmo Centroamericano MO, 1

Tags: number theory , divide , recurrence relation , divisible

A sequence of positive integers $g_1$, $g_2$, $g_3$, $. . . $ is defined as follows: $g_1 = 1$ and for every positive integer $n$, $$g_{n + 1} = g^2_n + g_n + 1.$$ Show that $g^2_{n} + 1$ divides $g^2_{n + 1}+1$ for every positive integer $n$.

2008 Korea Junior Math Olympiad, 6

Tags: sum of powers , sum , divisible , number theory , divisor

If $d_1,d_2,...,d_k$ are all distinct positive divisors of $n$, we define $f_s(n) = d_1^s+d_2^s+..+d_k^s$. For example, we have $f_1(3) = 1 + 3 = 4, f_2(4) = 1 + 2^2 + 4^2 = 21$. Prove that for all positive integers $n$, $n^3f_1(n) - 2nf_9(n) + n^2f_3(n)$ is divisible by $8$.