Found problems: 288
Let $n \geq 2$ be an integer. Prove that if $$\frac{n^2+4^n+7^n}{n}$$ is an integer, then it is divisible by 11.
Let $n$ be a given positive integer. How many solutions are there in ordered positive integer pairs $(x,y)$ to the equation
$$\frac{xy}{x+y}=n?$$
Find all positive integers $n \ge 2$ such that equality $i+j \equiv C_{n}^{i} + C_{n}^{j}$ (mod $2$) is true for arbitrary $0 \le i \le j \le n$.
Are there integers $m, n \geq 2$ such that the following property is always true?
$$``\text{For any real numbers } x, y, \text{ if } x^m + y^m \text{ and } x^n + y^n \text{ are integers, then } x + y \text{ is an integer}".$$
Among $2000$ distinct positive integers, there are equally many even and odd ones. The sum of the numbers is less than $3000000.$ Show that at least one of the numbers is divisible by $3.$
When any $11$ integers are given, prove that you can always choose $6$ integers among them so that the sum of the chosen numbers is a multiple of $6$. The $11$ integers aren't necessarily different.
The sequence $x_{n}$ is de fined by:
$x_{0}=1, x_{1}=0, x_{2}=1,x_{3}=1, x_{n+3}=\frac{(n^2+n+1)(n+1)}{n}x_{n+2}+(n^2+n+1)x_{n+1}-\frac{n+1}{n}x_{n} (n=1,2,3..)$
Prove that all members of the sequence are perfect squares.
Let $(b_n)_{n \ge 0}$ be a sequence of positive integers satisfying $b_n=d\left(\sum_{i=0}^{n-1} b_k\right)$ for all $n \ge 1$. (By $d(m)$ we denote the number of positive divisors of $m$.)
a) Prove that $(b_n)_{n \ge 0}$ is unbounded.
b) Prove that there are infinitely many $n$ such that $b_n>b_{n+1}$.
a. View the second-degree quadratic equation $x^2+? x +? = 0$
Two players successively put an integer each at the location of a question mark. Show that the second player can always ensure that the quadratic gets two integer solutions.
Note: we say that the quadratic also has two integer solutions, even when they are equal (for example if they are both equal to $3$).
b.View the third-degree equation $x^3 +? x^2 +? x +? = 0$
Three players successively put an integer each at the location of a question mark. The equation appears to have three integer (possibly again the same) solutions. It is given that two players each put a $3$ in the place of a question mark. What number did the third player put? Determine that number and the place where it is placed and prove that only one number is possible.
The numbers $a, b$ satisfy the condition $2a + a^2= 2b + b^2$. Prove that if $a$ is an integer, $b$ is also an integer.
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?
A sum of $335$ pairwise distinct positive integers equals $100000$.
a) What is the least number of uneven integers in that sum?
b) What is the greatest number of uneven integers in that sum?
Determine all functions $f\colon\mathbb{Z}_{>0}\to\mathbb{Z}_{>0}$ such that, for all positive integers $a$ and $b$,
\[
f^{bf(a)}(a+1)=(a+1)f(b).
\]
Let $a_1, a_2, ..., a_{2012}$ be pairwise distinct integers. Show that the equation $(x -a_1)(x - a_2)...(x - a_{2012}) = (1006!)^2$ has at most one integral solution.
Let $a, b, c$ be nonzero integers such that $M = \frac{a}{b}+\frac{b}{c}+\frac{c}{a}$ and $N =\frac{a}{c}+\frac{b}{a}+\frac{c}{b}$ are both integers. Find $M$ and $N$.
Determine all positive integers $n < 200$, such that $n^2 + (n+ 1)^2$ is the square of an integer.
Let $a$ be a real number such that $\left(a + \frac{1}{a}\right)^2=11$. For which $n\in N$ is $a^n + \frac{1}{a^n}$ an integer? Does this depend on the exact value of $a$?
Prove that there are infinitely many ordered pairs of positive integers $(m, n)$ such that $\frac{m+1}{n}+\frac{n+1}{m}$ is a positive integer.
Let $n$ be a positive integer. Determine the maximum of the sum $x_1+\cdots+x_n$ where $x_1,\ldots,x_n$ are non-negative integers satisfying the condition \[x_1^3+\cdots+x_n^3\le7n.\]
Given an expression $x^2 + ax + b$ where $a,b$ are integer coefficients. At any step, one can change the expression by adding either $1$ or $-1$ to only one of the two coefficients $a, b$.
a) Suppose that the initial expression has $a =-7$ and $b = 19$. Show your modification steps to obtain a new expression that has zero value at some integer value of $x$.
b) Starting from the initial expression as above, one gets the expression $x^2 - 17x + 9$ after $m$ modification steps. Prove that at a certain step $k$ with $k < m$, the obtained expression has zero value at some integer value of $x$.
Find all real numbers $x$ for which the number $$a =\frac{2x + 1}{x^2 + 2x + 3}$$ is an integer.
Prove that there exist infinitely many positive integers $n$ such that $\sqrt{n}$ is not an integer and $n$ is divisible by $[\sqrt{n}] $.
For a natural number $b$, let $N(b)$ denote the number of natural numbers $a$ for which the equation $x^2 + ax + b = 0$ has integer roots. What is the smallest value of $b$ for which $N(b) = 20$?
$5.$ Let $P(x)$ be a non - zero polynomial with integer coefficients. If $P(n)$ is divisible by $n$ for each integer polynomial $n.$ What is the value of $P(0) ?$
$\textbf{N1.}$ Find all nonnegative integers $a,b,c$ such that
\begin{align*} a^2+b^2+c^2-ab-bc-ca = a+b+c \end{align*}
[i]Proposed by usjl[/i]