Found problems: 41
Let $n$ be a positive integer.
(a) Prove that $1^5 +3^5 +5^5 +...+(2n-1)^5$ is divisible by $n$.
(b) Prove that $1^3 +3^3 +5^3 +...+(2n-1)^3$ is divisible by $n^2$.
The number $2019$ has the following nice properties:
(a) It is the sum of the fourth powers of fuve distinct positive integers.
(b) It is the sum of six consecutive positive integers.
In fact,
$2019 = 1^4 + 2^4 + 3^4 + 5^4 + 6^4$ (1)
$2019 = 334 + 335 + 336 + 337 + 338 + 339$ (2)
Prove that $2019$ is the smallest number that satis es [b]both [/b] (a) and (b).
(You may assume that (1) and (2) are correct!)
The sum of several integers (not necessarily distinct) equals $1492$. Decide whether the sum of their seventh powers can equal (a) $1996$; (b) $1998$.
Let $a, b, c, d$ be four positive real numbers. Prove that
$$a^{10} + b^{10}+c^{10} + d^{10} \ge (0.1a + 0.2b + 0.3c + 0.4d)^{10} + (0.4a + 0.3b + 0.2c + 0.ld)^{10} + (0.2a + 0.4b + 0.1c + 0.3d)^{10} + (0.3a + 0.1b + 0.4c + 0.2d)^{10}$$
Determine all prime numbers $p_1, p_2,..., p_{12}, p_{13}, p_1 \le p_2 \le ... \le p_{12} \le p_{13}$, such
that $p_1^2+ p_2^2+ ... + p_{12}^2 = p_{13}^2$ and one of them is equal to $2p_1 + p_9$.
Compute the remainder when $1^{2547} + 2^{2547} +...+ 2547^{2547}$ is divided by $25$.
Let $n$ be a natural number and let $P (t) = 1 + t + t^2 + ... + t^{2n}$.
If $x \in R$ such that $P (x)$ and $P (x^2)$ are rational numbers, prove that $x$ is rational number.
Prove that, for each odd integer $n \ge 5$, the number $1^n+2^n+...+15^n$ is divisible by $480$.
Let $a, b$, and $n$ be integers such that $a + b$ is divisible by $n$ and $a^2 + b^2$ is divisible by $n^2$. Prove that $a^m + b^m$ is divisible by $n^m$ for all positive integers $m$.
Let $p$ be a prime number. Prove that there exist infinitely many positive integers $n$ such that $p$ divides $1^n + 2^n +... + (p + 1)^n.$
Let $a, b, c$ be real numbers, and define $S_n = a^n + b^n + c^n$ for positive integers $n$. Suppose that $S_1, S_2, S_3$ are integers satisfying $6 | 5S_1 - 3S_2 - 2S_3$. Show that $S_n$ is an integer for all positive integers $n$.
The sequence ($a_n$) of natural numbers is defined by $a_1 = 2$ and $a_{n+1}$ equals the sum of tenth powers of the decimal digits of $a_n$ for all $n \ge 1$. Are there numbers which appear twice in the sequence ($a_n$)?
Find all numbers that can be expressed in exactly $2010$ different ways as the sum of powers of two with non-negative exponents, each power appearing as a summand at most three times. A sum can also be made from just one summand.
Let $p$ be a prime and $k$ be a positive integer. Set $S$ contains all positive integers $a$ satisfying $1\le a \le p-1$, and there exists positive integer $x$ such that $x^k\equiv a \pmod p$.
Suppose that $3\le |S| \le p-2$. Prove that the elements of $S$, when arranged in increasing order, does not form an arithmetic progression.
Let $d = a^{1999} + b^{1999} + c^{1999}$ , where $a, b$ and $c$ are integers such that $a + b + c = 0$.
(a) May it happen that $d = 2$?
(b) May it happen that $d$ is prime?
(V Senderov)
Let $p$ be an odd prime number. Prove that
$$1^{p-1} + 2^{p-1} +...+ (p-1)^{p-1} \equiv p + (p-1)! \mod p^2$$