$\textbf{N1.}$ 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]

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 $0 \le i \le j \le n$.

The sum of two integers is $8$. The sum of the squares of those two integers is $34$. What is the product of the two integers?

Given $\lambda^3 - 2\lambda^2- 1 = 0$ for some real $\lambda$ prove that $[\lambda[\lambda[\lambda n]]] - n$ is odd for any positive integer $n$ . (I Voronovich)

Find the smallest real number $A$, such that there are two different triangles, with integer sidelengths and so that the area of each be $A$.

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.

Show that for any integer $n$ the number \[a_n=\frac{\bigl(2+\sqrt3\bigr)^n-\bigl(2-\sqrt3\bigr)^n}{2\sqrt3}\] is also integer. Determine all integers $n$ such that $a_n$ is divisible by 3.

Determine all positive integers $n$ such that $\frac{2^n}{n^2}$ is an integer.

To [i]dissect [/i] a polygon means to divide it into several regions by cutting along finitely many line segments. For example, the diagram below shows a dissection of a hexagon into two triangles and two quadrilaterals: [img]https://cdn.artofproblemsolving.com/attachments/0/a/378e477bcbcec26fc90412c3eada855ae52b45.png[/img] An [i]integer-ratio[/i] right triangle is a right triangle whose side lengths are in an integer ratio. For example, a triangle with sides $3,4,5$ is an[i] integer-ratio[/i] right triangle, and so is a triangle with sides $\frac52 \sqrt3 ,6\sqrt3, \frac{13}{2} \sqrt3$. On the other hand, the right triangle with sides$ \sqrt2 ,\sqrt5, \sqrt7$ is not an [i]integer-ratio[/i] right triangle. Determine, with proof, all integers $n$ for which it is possible to completely [i]dissect [/i] a regular $n$-sided polygon into [i]integer-ratio[/i] right triangles.

Prove that if a triangle has integral side lengths and its circumradius is a prime number then the triangle is right-angled.

Several rational numbers were written on the blackboard. Dima wrote off their fractional parts on paper. Then all the numbers on the board squared, and Dima wrote off another paper with fractional parts of the resulting numbers. It turned out that on Dima's papers were written the same sets of numbers (maybe in different order). Prove that the original numbers on the board were integers. (The fractional part of a number $x$ is such a number $\{x\}, 0 \le \{x\} <1$, that $x-\{x\}$ is an integer.)

A lattice point is a point $(x, y)$ with both $x$ and $y$ integers. Find, with proof, the smallest $n$ such that every set of $n$ lattice points contains three points that are the vertices of a triangle with integer area. (The triangle may be degenerate, in other words, the three points may lie on a straight line and hence form a triangle with area zero.)

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 $3.$

Let $P$ be a $2016$ sided polygon with all its adjacent sides perpendicular to each other, i.e., all its internal angles are either $90^o$ or $270^o$. If the lengths of its sides are odd integers, prove that its area is an even integer.

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 $b_n>b_{n+1}$.

Let $n$ be a given positive integer. How many solutions are there in ordered positive integer pairs $(x,y)$ to the equation $$\frac{xy}{x+y}=n?$$

a) Prove that for every positive integer n, there exist $a, b \in R - Z$ such that the set $A_n = \{a - b, a^2 - b^2, a^3 - b^3,...,a^n - b^n\}$ contains only positive integers. b) Let $a$ and $b$ be two real numbers such that the set $A = \{a^k - b^k | k \in N*\}$ contains only positive integers. Prove that $a$ and $b$ are integers.

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?

Determine all positive integers that cannot be written as $\frac{a}{b} + \frac{a+1}{b+1}$ where $a$ and $b$ are positive integers.

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 $N$.

Let $k$ be a fixed positive integer. Consider the sequence definited by \[a_{0}=1 \;\; , a_{n+1}=a_{n}+\left\lfloor\root k \of{a_{n}}\right\rfloor \;\; , n=0,1,\cdots\] where $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$. For each $k$ find the set $A_{k}$ containing all integer values of the sequence $(\sqrt[k]{a_{n}})_{n\geq 0}$.

Find all integers that can be written in the form $\frac{1}{a_1}+\frac{2}{a_2}+...+\frac{9}{a_9}$ where $a_1,a_2, ...,a_9$ are nonzero digits, not necessarily different.

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 $a$.)

Consider the integral $Z(0)=\int^{\infty}_{-\infty} dx e^{-x^2}= \sqrt{\pi}$. [b](a)[/b] Show that the integral $Z(j)=\int^{\infty}_{-\infty} dx e^{-x^{2}+jx}$, where $j$ is not a function of $x$, is $Z(j)=e^{j^{2}/4a} Z(0)$. [b](b)[/b] Show that $$\dfrac 1 {Z(0)}=\int x^{2n} e^{-x^2}= \dfrac {(2n-1)!!}{2^n},$$ where $(2n-1)!!$ is defined as $(2n-1)(2n-3)\times\cdots\times3\times 1$. [b](c)[/b] What is the number of ways to form $n$ pairs from $2n$ distinct objects? Interpret the previous part of the problem in term of this answer.

Determine all positive integers that cannot be written as $\frac{a}{b} + \frac{a+1}{b+1}$ where $a$ and $b$ are positive integers.