Find all positive integers $n$ for which the number $\frac{n^{3n-2}-3n+1}{3n-2}$ is whole. [hide=EDIT:] In the original problem instead of whole we search for integers, so with this change $n=1$ will be a solution. [/hide]

At a meeting, there are \( N \) people who do not know each other. Prove that it is possible to introduce them in such a way that no three of them have the same number of acquaintances.

Show that the $n$ digits after the decimal point in $(5 +\sqrt{26})^n$ are all equal.

Let $n\ge 2$ be a positive integer. Fill up a $n\times n$ table with the numbers $1,2,...,n^2$ exactly once each. Two cells are termed adjacent if they have a common edge. It is known that for any two adjacent cells, the numbers they contain differ by at most $n$. Show that there exist a $2\times 2$ square of adjacent cells such that the diagonally opposite pairs sum to the same number.

Let $n$ be a natural number. Prove that if $n^5+n^4+1$ has $6$ divisors then $n^3-n+1$ is a square of an integer.

For distinct complex numbers $z_1,z_2,\dots,z_{673}$, the polynomial \[ (x-z_1)^3(x-z_2)^3 \cdots (x-z_{673})^3 \] can be expressed as $x^{2019} + 20x^{2018} + 19x^{2017}+g(x)$, where $g(x)$ is a polynomial with complex coefficients and with degree at most $2016$. The value of \[ \left| \sum_{1 \le j <k \le 673} z_jz_k \right| \] can be expressed in the form $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

We say that a four-digit number $\overline{abcd}$ ($a \ne 0$) is [i]pora [/i] if the following terms are true : $\bullet$ $a\ge b$ $\bullet$ $ab - cd = cd -ba$. For example, $2011$ is pora because $20-11 = 11-02$ Find all the numbers around.

Expanding $(1+0.2)^{1000}$ by the binomial theorem and doing no further manipulation gives \begin{eqnarray*} &\ & \binom{1000}{0}(0.2)^0+\binom{1000}{1}(0.2)^1+\binom{1000}{2}(0.2)^2+\cdots+\binom{1000}{1000}(0.2)^{1000}\\ &\ & = A_0 + A_1 + A_2 + \cdots + A_{1000}, \end{eqnarray*} where $A_k = \binom{1000}{k}(0.2)^k$ for $k = 0,1,2,\ldots,1000$. For which $k$ is $A_k$ the largest?

Compute the number of subsets $S$ with at least two elements of $\{2^2, 3^3, \dots, 216^{216}\}$ such that the product of the elements of $S$ has exactly $216$ positive divisors. [i]Proposed by Sean Li[/i]

Let the interior point $P$ of the convex quadrilateral $ABCD$ be such that $$|\angle PAD| = |\angle ADP| = |\angle CBP| = |\angle PCB| = |\angle CPD|.$$ Let $O$ be the center of the circumcircle of the triangle $CPD$. Prove that $|OA| = |OB|$.

a) Evaluate \[\lim_{n\to \infty} \underbrace{\sqrt{a+\sqrt{a+\ldots+\sqrt{a+\sqrt{b}}}}}_{n\ \text{square roots}}\] with $a,b>0$. b)Let $(a_n)_{n\ge 1}$ and $(x_n)_{n\ge 1}$ such that $a_n>0$ and \[x_n=\sqrt{a_n+\sqrt{a_{n-1}+\ldots+\sqrt{a_2+\sqrt{a_1}}}},\ \forall n\in \mathbb{N}^*\] Prove that: 1) $(x_n)_{n\ge 1}$ is bounded if and only if $(a_n)_{n\ge 1}$ is bounded. 2) $(x_n)_{n\ge 1}$ is convergent if and only if $(a_n)_{n\ge 1}$ is convergent. [i]Valentin Matrosenco[/i]

Let $ ABC$ be a triangle. Point $ D$ lies on its sideline $ BC$ such that $ \angle CAD \equal{} \angle CBA.$ Circle $ (O)$ passing through $ B,D$ intersects $ AB,AD$ at $ E,F$, respectively. $ BF$ meets $ DE$ at $ G$.Denote by$ M$ the midpoint of $ AG.$ Show that $ CM\perp AO.$

Eight card players are seated around a table. One remarks that at some moment, any player and his two neighbours have altogether an odd number of winning cards. Show that any player has at that moment at least one winning card.

Determine all odd natural numbers of the form $$\frac{p + q}{p - q},$$ where $p > q$ are prime numbers.

$7501$ points in a $96 \times 96$ square is marked. We call the $4 \times 4$ square without its central $2 \times 2$ square a [i]frame[/i]. Prove that there exist a frame with sides parallel to the $96 \times 96$ square (not necessarily from the grid lines) that contains at least $10$ marked points. [i]Proposed by Negar Babashah, Shima Amirbeygie[/i] [b]Rated 5[/b]

Positive real numbers $a,b,c$ are given such that $abc=1$. Prove that$$a+b+c+\frac{3}{ab+bc+ca}\geq4.$$

Let $ S \equal{} \{x_1, x_2, \ldots, x_{k \plus{} l}\}$ be a $ (k \plus{} l)$-element set of real numbers contained in the interval $ [0, 1]$; $ k$ and $ l$ are positive integers. A $ k$-element subset $ A\subset S$ is called [i]nice[/i] if \[ \left |\frac {1}{k}\sum_{x_i\in A} x_i \minus{} \frac {1}{l}\sum_{x_j\in S\setminus A} x_j\right |\le \frac {k \plus{} l}{2kl}\] Prove that the number of nice subsets is at least $ \dfrac{2}{k \plus{} l}\dbinom{k \plus{} l}{k}$. [i]Proposed by Andrey Badzyan, Russia[/i]

Find all non constant polynomials $P(x),Q(x)$ with real coefficients such that: $P((Q(x))^3)=xP(x)(Q(x))^3$

In $\triangle A B C$, $A B=A C$ and $D$ is foot of the perpendicular from $C$ to $A B$ and $E$ the foot of the perpendicular from $B$ to $A C,$ then [list=1] [*] $BC^3>BD^3+BE^3$ [*] $BC^3 <BD^3+BE^3$ [*] $BC^3=BD^3+BE^3$ [*] None of these [/list]

As shown in figure, from a point $P$ exterior of circle $\odot O$, we draw tangent $PA$ and the secant $PBC$. Let $AD \perp PO$ Prove that $AC$ is tangent to the circumcircle of $\vartriangle ABD$. [img]https://cdn.artofproblemsolving.com/attachments/a/f/32da6d4626bb3592cec19a4cf0202121ba64db.png[/img]

In a school there $b$ teachers and $c$ students. Suppose that a) each teacher teaches exactly $k$ students, and b)for any two (distinct) students , exactly $h$ teachers teach both of them. Prove that $\frac{b}{h}=\frac{c(c-1)}{k(k-1)}$.

Let $ABC$ be a triangle and let $P$ denote the midpoint of the side $BC$. Suppose that there exist two points $M$ and $N$ interior to the side $AB$ and $AC$ respectively, such that $$|AD|=|DM|=2|DN|,$$ where $D$ is the intersection point of the lines $MN$ and $AP$. Show that $|AC|=|BC|$.

If $a,b>0$ then prove: $$\left(\frac{a+b}2-\frac{2ab}{a+b}\right)\operatorname{arctan}\left(\frac{\sqrt{2ab}-\sqrt{a^2+b^2}}{\sqrt2+\sqrt{ab}\left(a^2+b^2\right)}\right)+\left(\sqrt{\frac{a^2+b^2}2}-\sqrt{ab}\right)\arctan\left(\frac{(a-b)^2}{2+2ab}\right)\ge0$$ [i]Proposed by Daniel Sitaru[/i]

Find all pairs of positive integers $(a,b)$ such that $a^2 + b^2$ divides both $a^3 + 1$ and $b^3 + 1$.

Butch and Sundance need to get out of Dodge. To travel as quickly as possible, each alternates walking and riding their only horse, Sparky, as follows. Butch begins walking as Sundance rides. When Sundance reaches the first of their hitching posts that are conveniently located at one-mile intervals along their route, he ties Sparky to the post and begins walking. When Butch reaches Sparky, he rides until he passes Sundance, then leaves Sparky at the next hitching post and resumes walking, and they continue in this manner. Sparky, Butch, and Sundance walk at 6, 4, and 2.5 miles per hour, respectively. The first time Butch and Sundance meet at a milepost, they are $n$ miles from Dodge, and have been traveling for $t$ minutes. Find $n + t$.