A $\textit{cubic sequence}$ is a sequence of integers given by $a_n =n^3 + bn^2 + cn + d$, where $b, c$ and $d$ are integer constants and $n$ ranges over all integers, including negative integers. $\textbf{(a)}$ Show that there exists a cubic sequence such that the only terms of the sequence which are squares of integers are $a_{2015}$ and $a_{2016}$. $\textbf{(b)}$ Determine the possible values of $a_{2015} \cdot a_{2016}$ for a cubic sequence satisfying the condition in part $\textbf{(a)}$.

Denote by $\mathbb{N}$ the set of all positive integers. Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that for all positive integers $m$ and $n$, the integer $f(m)+f(n)-mn$ is nonzero and divides $mf(m)+nf(n)$. [i]Proposed by Dorlir Ahmeti, Albania[/i]

Find the smallest prime number $p$ that cannot be represented in the form $|3^{a} - 2^{b}|$, where $a$ and $b$ are non-negative integers.

$H$ is the orthocenter of an acute triangle $ABC$, and let $M$ be the midpoint of $BC$. Suppose $(AH)$ meets $AB$ and $AC$ at $D,E$ respectively. $AH$ meets $DE$ at $P$, and the line through $H$ perpendicular to $AH$ meets $DM$ at $Q$. Prove that $P,Q,B$ are collinear.

Let $a,b$ be positive real numbers.Prove that $(1+a)^{8}+(1+b)^{8}\geq 128ab(a+b)^{2}$.

Let $ABC$ be a triangle with circumcircle $\Gamma$ and incenter $I$ and let $M$ be the midpoint of $\overline{BC}$. The points $D$, $E$, $F$ are selected on sides $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ such that $\overline{ID} \perp \overline{BC}$, $\overline{IE}\perp \overline{AI}$, and $\overline{IF}\perp \overline{AI}$. Suppose that the circumcircle of $\triangle AEF$ intersects $\Gamma$ at a point $X$ other than $A$. Prove that lines $XD$ and $AM$ meet on $\Gamma$. [i]Proposed by Evan Chen, Taiwan[/i]

Given a $15\times 15$ chessboard. We draw a closed broken line without self-intersections such that every edge of the broken line is a segment joining the centers of two adjacent cells of the chessboard. If this broken line is symmetric with respect to a diagonal of the chessboard, then show that the length of the broken line is $\leq 200$.

Mary's top book shelf holds five books with the following widths, in centimeters: $ 6$, $ \frac12$, $ 1$, $ 2.5$, and $ 10$. What is the average book width, in centimeters? $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5$

Consider a line $ \ell $ in the plane and let $ B_1, B_2, B_3 $ be different points in $ \ell$. Let $ A $ be a point that is not in $ \ell$. Show that there is $ P, Q $ in $ {B_1, B_2, B_3} $ with $ P \ne Q $ so that the distance from $ A $ to $ \ell$ is greater than the distance from $ P $ to the line that passes through $ A $ and $ Q $.

Given is a convex polygon $ P$ with $ n$ vertices. Triangle whose vertices lie on vertices of $ P$ is called [i]good [/i] if all its sides are unit length. Prove that there are at most $ \frac {2n}{3}$ [i]good[/i] triangles. [i]Author: Vyacheslav Yasinskiy, Ukraine[/i]

Find all pairs of integers $(a,b)$ such that $n|( a^n + b^{n+1})$ for all positive integer $n$

For positive is true $$\frac{3}{abc} \geq a+b+c$$ Prove $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \geq a+b+c$$

Two circles $\omega$ and $\gamma$ have radii $3$ and $4$ respectively, and their centers are $10$ units apart. Let $x$ be the shortest possible distance between a point on $\omega$ and a point on $\gamma$ , and let$ y$ be the longest possible distance between a point on $\omega$ and a point on $\gamma$ . Find the product $xy$.

Find all triplets natural numbers $(a, b, c)$ satisfied \[GCD(a, b) + LCM(a,b) = 2021^c\] with $|a - b|$ and $(a+b)^2 + 4$ are both prime number

In a circle with a radius $R$ a convex hexagon is inscribed. The diagonals $AD$ and $BE$,$BE$ and $CF$,$CF$ and $AD$ of the hexagon intersect at the points $M$,$N$ and$K$, respectively. Let $r_1,r_2,r_3,r_4,r_5,r_6$ be the radii of circles inscribed in triangles $ ABM,BCN,CDK,DEM,EFN,AFK$ respectively. Prove that.$$r_1+r_2+r_3+r_4+r_5+r_6\leq R\sqrt{3}$$ .

The squares of a squared paper are enumerated as shown on the picture. \[\begin{array}{|c|c|c|c|c|c} \ddots &&&&&\\ \hline 10&\ddots&&&&\\ \hline 6&9&\ddots&&&\\ \hline 3&5&8&12&\ddots&\\ \hline 1&2&4&7&11&\ddots\\ \hline \end{array}\] Devise a polynomial $p(m, n)$ in two variables such that for any $m, n \in \mathbb{N}$ the number written in the square with coordinates $(m, n)$ is equal to $p(m, n)$.

Prove that there exists a positive integer $x<5^{2022}$ such that \[\{\varphi\sqrt[3]{x}\}<\varphi^{-2022}.\]

Pablo and Silvia play on a $2010 \times 2010$ board. To start the game, Pablo writes an integer in every cell. After he is done, Silvia repeats the following operation as many times as she wants: she chooses three cells that form an $L$, like in the figure below, and adds $1$ to each of the numbers in these three cells. Silvia wins if, after doing the operation many times, all of the numbers in the board are multiples of $10$. Prove that Silvia can always win. $\begin{array}{|c|c} \cline{1-1} \; & \; \\ \hline \; & \multicolumn{1}{|c|}{\;} \\ \hline \end{array} \qquad \begin{array}{c|c|} \cline{2-2} \; & \; \\ \hline \multicolumn{1}{|c|}{\;} & \; \\ \hline \end{array} \qquad \begin{array}{|c|c} \hline \; & \multicolumn{1}{|c|}{\;} \\ \hline \multicolumn{1}{|c|}{\;} & \; \\ \cline{1-1} \end{array} \qquad \begin{array}{c|c|} \hline \multicolumn{1}{|c|}{\;} & \; \\ \hline \; & \multicolumn{1}{|c|}{\;} \\ \cline{2-2} \end{array}$

Let $\{x_n\}$ be a real sequence defined by: \[x_1=a,x_{n+1}=3x_n^3-7x_n^2+5x_n\] For all $n=1,2,3...$ and a is a real number. Find all $a$ such that $\{x_n\}$ has finite limit when $n\to +\infty$ and find the finite limit in that cases.

Vertices of a square $ABCD$ of side $\frac{25}4$ lie on a sphere. Parallel lines passing through points $A,B,C$ and $D$ intersect the sphere at points $A_1,B_1,C_1$ and $D_1$, respectively. Given that $AA_1=2$, $BB_1=10$, $CC_1=6$, determine the length of the segment $DD_1$.

Find all triples $(x,y,z)\in \mathbb{R}^{3}$ such that $x,y,z>3$ and $\frac{(x+2)^2}{y+z-2}+\frac{(y+4)^2}{z+x-4}+\frac{(z+6)^2}{x+y-6}=36$

Find all positive integers $n$ for which $n^n+1$ and $(2n)^{2n}+1$ are prime numbers.

Define acute $\triangle ABC$ with circumcenter $O$. The circumcircle of $\triangle ABO$ meets segment $BC$ at $D \ne B$, segment $AC$ at $F \ne A$, and the Euler line of $\triangle ABC$ at $P \ne O$. The circumcircle of $\triangle ACO$ meets segment $BC$ at $E \ne C$. Let $\overline{BC}$ and $\overline{FP}$ intersect at $X$, with $C$ between $B$ and $X$. If $BD=13$, $EC=8$, and $CX=27$, find $DE$. $\emph{(The Euler line of a triangle passes through its orthocenter, circumcenter, and centroid.)}$

Let $ABCD$ be a convex cyclic quadrilateral with diagonals $AC$ and $BD$. Each of the four vertixes are reflected across the diagonal on which the do not lie. (a) Investigate when the four points thus obtained lie on a straight line and give as simple an equivalent condition as possible to the cyclic quadrilateral $ABCD$ for it. (b) Show that in all other cases the four points thus obtained lie on one circle. (Theresia Eisenkölbl)

Let $ABC$ be an acute angled triangle with circumcircle $O$. Line $AO$ intersects the circumcircle of triangle $ABC$ again at point $D$. Let $P$ be a point on the side $BC$. Line passing through $P$ perpendicular to $AP$ intersects lines $DB$ and $DC$ at $E$ and $F$ respectively . Line passing through $D$ perpendicular to $BC$ intersects $EF$ at point $Q$. Prove that $EQ = FQ$ if and only if $BP = CP$.