Found problems: 288
Find all integers $n \ge 2$ for which $\sqrt[n]{3^n+ 4^n+5^n+8^n+10^n}$ is an integer.
Let $a_1, a_2, \ldots a_{2n+1}$ be a set of integers such that, if any one of them is removed, the remaining ones can be divided into two sets of $n$ integers with equal sums. Prove $a_{1}=a_2 =\cdots=a_{2n+1}.$
Arithmetical progression $a_1, a_2, a_3, a_4,...$ contains $a_1^2 , a_2^2$ and $a_3^2$ at some positions. Prove that all terms of this progression are integers.
The positive integers are arranged in a row in some order, each occuring exactly once. Does there always exist an adjacent block of at least two numbers somewhere in this row such that the sum of the numbers in the block is a prime number?
(a) Determine all pairs $(x, y)$ of (real) numbers with $0 < x < 1$ and $0 <y < 1$ for which $x + 3y$ and $3x + y$ are both integer. An example is $(x,y) =( \frac{8}{3}, \frac{7}{8}) $, because $ x+3y =\frac38 +\frac{21}{8} =\frac{24}{8} = 3$ and $ 3x+y = \frac98 + \frac78 =\frac{16}{8} = 2$.
(b) Determine the integer $m > 2$ for which there are exactly $119$ pairs $(x,y)$ with $0 < x < 1$ and $0 < y < 1$ such that $x + my$ and $mx + y$ are integers.
Remark: if $u \ne v,$ the pairs $(u, v)$ and $(v, u)$ are different.
Suppose $a, b$ are integers and $a+b$ is a root of $x^2 +ax+b = 0$. What is the maximum possible value of $b^2$?
(a) Let $a$ and $b$ be rational numbers such that line $y = ax + b$ intersects the circle $x^2 + y^2 = 5$ at two different points. Show that if one of the intersections has two rational coordinates, so does the other intersection.
(b) Show that there are infinitely many triples ($k, n, m$) that are such that $k^2 + n^2 = 5m^2$, where $k, n$ and $m$ are integers, and not all three have any in common prime factor.
Prove that $\frac{1}{1\times2013}+\frac{1}{2\times2012}+\frac{1}{3\times2011}+...+\frac{1}{2012\times2}+\frac{1}{2013\times1}<1$
Each integer is colored with exactly one of the following colors: red, blue, or orange, and all three colors are used in the coloring. The coloring also satisfies the following properties:
1. The sum of a red number and an orange number results in a blue-colored number,
2. The sum of an orange and blue number results in an orange-colored number;
3. The sum of a blue number and a red number results in a red-colored number.
(a) Prove that $0$ and $1$ must have distinct colors.
(b) Determine all possible colorings of the integers which also satisfy the properties stated above.
Prove that there are infinitely many positive integers $n$ such that $\frac{5^n-1}{n+2}$ is an integer.
You are given $n$ different primes $p_1, p_2,..., p_n$. Consider the polynomial $$x^n + a_1x^{n -1} + a_2x^{n - 2} + ...+ a_{n - 1}x + a_n$$, where $a_i$ is the product of the first $i$ given prime numbers. For what $n$ can it have an integer root?
Find all real numbers $a$ such that $3 < a < 4$ and $a(a-3\{a\})$ is an integer.
(Here $\{a\}$ denotes the fractional part of $a$.)
The first two terms of a sequence are $2$ and $3$. Each next term thereafter is the sum of the nearestly previous two terms if their sum is not greather than $10, 0$ otherwise. The $2014$th term is:
(A): $0$, (B): $8$, (C): $6$, (D): $4$, (E) None of the above.
Prove that the polynomial $P (x)$ with integer coefficients, taking odd values for $x = 0$ and $x= 1$, has no integer roots.
In how many ways can one select eight integers $a_1,a_2, ... ,a_8$, not necesarily distinct, such that $1 \le a_1 \le ... \le a_8 \le 8$?
Let $x$ and $y$ be positive integers such that $\frac{x^2-1}{y+1}+\frac{y^2-1}{x+1}$ is integer. Prove that numbers $\frac{x^2-1}{y+1}$ and $\frac{y^2-1}{x+1}$ are integers
Find all non-negative integers $a, b, c$ such that the roots of equations: $\begin{cases}x^2 - 2ax + b = 0 \\
x^2- 2bx + c = 0 \\
x^2 - 2cx + a = 0 \end{cases}$ are non-negative integers.
Find all integers $x$ and $y$ such that $2^x+1=y^2$
A Pythagorean triangle is a right triangle whose three sides are integers.
The best known example is the triangle with rectangular sides $3$ and $4$ and hypotenuse $5$.
Determine all Pythagorean triangles whose area is twice the perimeter.
Define a sequence $<x_n>$ by $x_0 = 0$ and $$\large x_n = \left\{
\begin{array}{ll}
x_{n-1} + \frac{3^r-1}{2} & if \,\,n = 3^{r-1}(3k + 1)\\
& \\
x_{n-1} - \frac{3^r+1}{2} & if \,\, n = 3^{r-1}(3k + 2)\\
\end{array}
\right. $$
where $k, r$ are integers. Prove that every integer occurs exactly once in the sequence.
$a, b$ are odd numbers that satisfy $(a-b)^2 \le 8\sqrt {ab}$. For $n=ab$, show that the equation
$$x^2-2([\sqrt n]+1)x+n=0$$ has two integral solutions. $[r]$ denotes the biggest integer, not strictly bigger than $r$.
Let $a$, $b$ and $c$ are integers such that $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}=3$. Prove that $abc$ is a perfect cube of an integer.
What is the greatest number of integers that can be selected from a set of $2015$ consecutive numbers so that no sum of any two selected numbers is divisible by their difference?
For each natural number $n$, let $f(n)$ denote the number of ordered integer pairs $(x,y)$ satisfying the following equation:
\[ x^2 - xy + y^2 = n. \]
a) Determine $f(2022)$.
b) Determine the largest natural number $m$ such that $m$ divides $f(n)$ for every natural number $n$.
In a right-angled triangle in which all side lengths are integers, one has a cathetus length $1994$. Determine the length of the hypotenuse.