Found problems: 387
Prove that for any integer $a$ the polynomial $3x^{2n}+ax^n+2$ cannot be divided by $2x^{2m}+ax^m+3$ without a remainder.
Let $a,b,c$ be positive integers such that $a|b^3, b|c^3$ and $c|a^3$. Prove that $abc|(a+b+c)^{13}$
Find the smallest positive integer $x$ such that $2^{254}$ divides $x^{2005} + 1$.
Prove that the integer $1^1 + 3^3 + 5^5 + .. + (2^n - 1)^{2^n-1}$ is a multiple of $2^n$ but not a multiple of $2^{n+1}$.
Prove that for every nonnegative integer $ k$ there is a product
$$(k + 1)(k + 2)...(k + 1980)$$
divisible by $ 1980^{197}$.
(1) Find the smallest positive integer $a$ such that $221|3^a -2^a$,
(2) Let $A=\{n\in N^*: 211|1+2^n+3^n+4^n\}$.
Are there infinitely many numbers $n$ such that both $n$ and $n+1$ belong to set $A$?
What is the biggest positive integer which divides $p^4 - q^4$ for all primes $p$ and $q$ greater than $10$?
Let $a, b,c,d$ be positive integers such that $ac+bd$ is divisible by $a^2 +b^2$. Prove that $gcd(c^2 + d^2, a^2 + b^2) > 1$.
Trần Nam Dũng
Let $f \in Z[X]$, $f = X^2 + aX + b$, be a quadratic polynomial. Prove that $f$ has integer zeros if and only if for each positive integer $n$ there is an integer $u_n$ such that $n | f(u_n)$.
Suppose that we have a set $S$ of $756$ arbitrary integers between $1$ and $2008$ ($1$ and $2008$ included).
Prove that there are two distinct integers $a$ and $b$ in $S$ such that their sum $a + b$ is divisible by $8$.
Given $N = 2^ap_1p_2...p_m$, $m \ge 1$, $a \in N$ with $p_1, p_2,..., p_m$ are different primes. It is known that $\sigma (N) = 3N $ where $\sigma (N)$ is the sum of all positive integers which are factors of $N$. Show that there exists a prime number $p$ such that $2^p- 1$ is also a prime, and $2^p - 1|N$.
Suppose $S= \{x|x=a^2+ab+b^2,a,b \in Z\}$. Prove that:
(1) If $m \in S$, $3|m$ , then $\frac{m}{3} \in S$
(2) If $m,n \in S$ , then $mn\in S$.
Prove that if $ n $ is a non-negative integer, then number $$
2^{n+2} + 3^{2n+1}$$
is divisible by $7$.
For each real number $x$, $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$. For example $\lfloor 2.8 \rfloor = 2$. Let $r \ge 0$ be a real number such that for all integers $m, n, m|n$ implies $\lfloor mr \rfloor| \lfloor nr \rfloor$. Prove that $r$ is an integer.
The natural numbers $n$ and $z$ are relatively prime and greater than $1$. For $k = 0, 1, 2,..., n - 1$ let $s(k) = 1 + z + z^2 + ...+ z^k.$ Prove that:
a) At least one of the numbers $s(k)$ is divisible by $n$.
b) If $n$ and $z - 1$ are also coprime, then already one of the numbers $s(k)$ with $k = 0,1, 2,..., n- 2$ is divisible by $n$.
Find all pairs $(x,n)$ of positive integers for which $x^n + 2^n + 1$ divides $x^{n+1} +2^{n+1} +1$.
Let $p> 3$ be a prime number and let $q = \frac{4^p-1}{3}$. Show that $q$ is a composite integer as well is a divisor of $2^{q-1}- 1$.
Given a sequence of numbers $ 13, 25, 43, \ldots $ whose $ n $-th term is defined by the formula
$$a_n =3(n^2 + n) + 7$$
Prove that this sequence has the following properties:
1) Of every five consecutive terms of the sequence, exactly one is divisible by $ 5 $,
2( No term of the sequence is the cube of an integer.
Let $a, b, c, d$ and $u$ be integers such that each of the numbers
$$ac\ \ , \ \ bc + ad \ \ , \ \ bd$$
is a multiple of $u$. Show that $bc$ and $ad$ are multiples of $u$.
Prove that $46^{2n+1} + 296 \cdot 13^{2n+1}$ is divisible by $1947$.
Prove that there is a unique positive integer formed only by the digits $2$ and $5$, which has $ 2007$ digits and is divisible by $2^{2007}$.
Show that each number is of the form $$2^{5^{2^{5^{...}}}}+ 4^{5^{4^{5^{...}}}}$$
is divisible by $2008$, where the exponential towers can be any independent ones have height $\ge 3$.
Consider the $2015$ integers $n$, from $ 1$ to $2015$. Determine for how many values of $n$ it is verified that the number $n^3 + 3^n$ is a multiple of $5$.
Let $N$ be the set of positive integers.
Find all the functions $f: N\to N$ with $f (1) = 2$ and such that $max \{f(m)+f(n), m+n\}$ divides $min\{2m+2n,f (m+ n)+1\}$ for all $m, n$ positive integers
Which positive integers satisfy the following three conditions?
a) The number consists of at least two digits.
b) The last digit is not zero.
c) Inserting a zero between the last two digits yields a number divisible by the original number.