Found problems: 387
Find a subset of $\{1,2, ...,2022\}$ with maximum number of elements such that it does not have two elements $a$ and $b$ such that $a = b + d$ for some divisor $d$ of $b$.
We define $N !!$ to be $N(N - 2)(N -4)...5 \cdot 3 \cdot 1$ if $N$ is odd and $N(N -2)(N -4)... 6\cdot 4\cdot 2$ if $N$ is even .
For example, $8 !! = 8 \cdot 6\cdot 4\cdot 2$ , and $9 !! = 9v 7 \cdot 5\cdot 3 \cdot 1$ .
Prove that $1986 !! + 1985 !!$ i s divisible by $1987$.
(V.V . Proizvolov , Moscow)
a) Suppose that $n$ is an odd integer. Prove that $k(n-k)$ is divisible by $2$ for all positive integers $k$.
b) Find an integer $k$ such that $k(100-k)$ is not divisible by $11$.
c) Suppose that $p$ is an odd prime, and $n$ is an integer.
Prove that there is an integer $k$ such that $k(n-k)$ is not divisible by $p$.
d) Suppose that $p,q$ are two different odd primes, and $n$ is an integer.
Prove that there is an integer $k$ such that $k(n-k)$ is not divisible by any of $p,q$.
Find the smallest positive integer $n$ that satisfies that for any $n$ different integers, the product of all the positive differences of these numbers is divisible by $2014$.
Prove that $(n!)!$ is a multiple of $(n!)^{(n-1)!}$
Find a positive integer $x$, with $x> 1$ such that all numbers in the sequence $$x + 1,x^x + 1,x^{x^x}+1,...$$ are divisible by $2009.$
Let $p$ be a prime, and let $a_1, a_2, a_3, . . .$ be a sequence of positive integers so that $a_na_{n+2} = a^2_{n+1} + p$ for all positive integers $n$. Show that $a_{n+1}$ divides $a_n + a_{n+2}$ for all positive integers $n$.
Let coefficients of the polynomial$ P (x) = a_dx^d + ... + a_2x^2 + a_0$ where $d \ge 2$, are positive integers. The sequences $(b_n)$ is defined by $b_1 = a_0$ and $b_{n+1} = P (b_n)$ for $n \ge 1$. Prove that for any $n \ge 2$, there exists a prime number $p$ such that $p|b_n$ but it does not divide $b_1, b_2, ..., b_{n-1}$.
Let $n$ be a positive integer. Prove that there exist integers $a_1, a_2,..., a_n$ such that for any integer $x$, the number $(... (((x^2 + a_1)^2 + a_2)^2 + ...)^2 + a_{n-1})^2 + a_n$ is divisible by $2n - 1$.
Find all the positive integers $n , n\ge 3$ such that $n\mid (n-2)!$
a) Prove that when you divide any prime number by $30$, the remainder is either $1$ or is a prime number!
b) Does this also apply when dividing a prime number by $60$? Justify your answer!
Find all pairs of integers $(x,y)$ satisfying the following condition:
[i]each of the numbers $x^3 + y$ and $x + y^3$ is divisible by $x^2 + y^2$
[/i]
Tournament of Towns
For every positive integer $n$, show that there is a positive integer $k$ such that $2k^2 + 2001k + 3 \equiv 0$ (mod $2^n$).
Let $a_1, a_2,..., a_9$ be integers. Prove that if $19$ divides $a_1^9+a_2^9+...+a_9^9$ then $19$ divides the product $a_1a_2...a_9$.
Let $(F_n )_{n\ge o}$ be the sequence of Fibonacci numbers: $F_0 = 0$, $F_1 = 1$ and $F_{n+2} = F_{n+1}+F_n$ , for every $n \ge 0$.
Prove that for any prime $p \ge 3$, $p$ divides $F_{2p} - F_p$ .
Determine all positive integers$ n$ such that $f_n(x,y,z) = x^{2n} + y^{2n} + z^{2n} - xy - yz - zx$ divides $g_n(x,y, z) = (x - y)^{5n} + (y -z)^{5n} + (z - x)^{5n}$, as polynomials in $x, y, z$ with integer coefficients.
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 $n \ge N$
Let $n$ be a natural number such that $n!$ is a multiple of $2023$ and is not divisible by $37$. Find the largest power of $11$ that divides $n!$.
Prove that the number $n^n-n$ is divisible by $24$ for any odd integer $n$.
Find all functions $f : Z_{>0} \to Z_{>0}$ with $a - f(b) | af(a) - bf(b)$ for all $a, b \in Z_{>0}$.
[i](Theresia Eisenkoelbl)[/i]
Prove that the product of three consecutive natural numbers, the middle of which is the cube of a natural number, is divisible by $ 504 $ .
Determine, for each positive integer $n$, the largest positive integer $k$ such that $2^k$ is a divisor of $3^n+1$.
Determine the highest power of $2$ that divides $2^n!$.
How many integers in the set $\{1, 2 ,..., 2010\}$ divide $5^{2010!}- 3^{2010!}$?
The sequence of positive integers $\{a_n, n\ge 1\}$ is such that $a_n\le a_{n+1}\le a_n+5$ and $a_n$ is divisible by $n$ for all $n \ge 1$. What are the possible values of $a_1$?