Found problems: 288
Let $a$ and $b$ be positive integers such that $\frac{5a^4 + a^2}{b^4 + 3b^2 + 4}$ is an integer. Show that $a$ is not prime.
Let $a, b, c$ be positive integers. Show that if $\frac{a}{b} +\frac{b}{c} +\frac{c}{a}$ is an integer then $abc$ is a perfect cube.
$11.$ Let $a,$ $b,$ and $c$ be real numbers such that $a-7b+8c=4.$ and $8a+4b-c=7.$ What is the value of $a^2-b^2+c^2 ?$
Determine all functions $f\colon\mathbb{Z}_{>0}\to\mathbb{Z}_{>0}$ such that, for all positive integers $a$ and $b$,
\[
f^{bf(a)}(a+1)=(a+1)f(b).
\]
Every term of an infinite geometric progression is also a term of a given infinite arithmetic progression. Prove that the common ratio of the geometric progression is an integer. (4)
Find all polynomials $p(x)$ with integer coefficients such that for each positive integer $n$, the number $2^n - 1$ is divisible by $p(n)$.
Let $m$ be a given positive integer. Define $a_k=\frac{(2km)!}{3^{(k-1)m}},k=1,2,\cdots.$ Prove that there are infinite many integers and infinite many non-integers in the sequence $\{a_k\}$.
Let $x$, $y$, $z$ be positive integers satisfying $x<y<z$ and $x+xy+xyz=37$. Find the greatest possible value of $x+y+z$.
A positive integer $n$ is called [i]naughty[/i] if it can be written in the form $n=a^b+b$ with integers $a,b \geq 2$.
Is there a sequence of $102$ consecutive positive integers such that exactly $100$ of those numbers are naughty?
Find all positive integers $m, n$ such that $\frac{2m-1}{n}$ and $\frac{2n-1}{m}$ are both integers.
Every positive integer $n>2$ can be written as a sum of distinct positive integers. Let $A(n)$ be the maximal number of summands in such a representation. Find a formula for $A(n).$
Give a triangle $ABC$ with heights $h_a = 3$ cm, $h_b = 7$ cm and $h_c = d$ cm, where $d$ is an integer. Determine $d$.
$a, b, c$ are three distinct integers and $n$ is a positive integer. Show that $$\frac{a^n}{(a - b)(a - c)}+\frac{ b^n}{(b - a)(b - c)} +\frac{ c^n}{(c - a)(c - b)}$$ is an integer.
Prove that the sum of all numbers representable as $\frac{1}{mn}$, where $m,n$ -- natural numbers, $1 \le m < n \le1986$, is not an integer.
Determine all pairs $(a, b)$ of positive integers for which $\frac{a^2b+b}{ab^2+9}$ is an integer number.
The $12$ numbers on a clock face are rearranged. Show that we can still find three adjacent numbers whose sum is $21$ or more.
Find all fractions which can be written simultaneously in the forms $\frac{7k- 5}{5k - 3}$ and $\frac{6l - 1}{4l - 3}$
, for some integers $k, l$.
Show that there exist integers $j,k,l,m,n$ greater than $100$ such that $j^2 +k^2 +l^2 +m^2 +n^2 = jklmn-12$.
Given $k \in \mathbb{N}^+$. A sequence of subset of the integer set $\mathbb{Z} \supseteq I_1 \supseteq I_2 \supseteq \cdots \supseteq I_k$ is called a $k-chain$ if for each $1 \le i \le k$ we have
(i) $168 \in I_i$;
(ii) $\forall x, y \in I_i$, we have $x-y \in I_i$.
Determine the number of $k-chain$ in total.
What is the maximum integer $n$ such that $\frac{50!}{2^n}$ is an integer?
Show that if $2 + 2\sqrt{28n^2 + 1}$ is an integer, then it is a square (for $n$ an integer).
Let $x, y$ be distinct positive integers. Prove that the number $$\frac{(x+y)^2}{ x^3 + xy^2 - x^2y - y^3}$$ is not an integer
Show that for any non-negative integer $n$, the number $2^{2n+1}$ cannot be expressed as a sum of four non-zero square numbers.
Find all triples of integers $(x, y, z)$ not zero and relative primes in pairs such that $\frac{(y+z-x)^2}{4x}$, $\frac{(z+x-y)^2}{4y}$ and $\frac{(x+y-z)^2}{4z}$ are all integers.
Find the least positive integer $n$, such that there is a polynomial \[ P(x) = a_{2n}x^{2n}+a_{2n-1}x^{2n-1}+\dots+a_1x+a_0 \] with real coefficients that satisfies both of the following properties:
- For $i=0,1,\dots,2n$ it is $2014 \leq a_i \leq 2015$.
- There is a real number $\xi$ with $P(\xi)=0$.