$\textbf{(Self-Isogonal Cubics)}$ Let $ABC$ be a triangle with $AB = 2$, $AC = 3$, $BC = 4$. The $\emph{isogonal conjugate}$ of a point $P$, denoted $P^\ast$, is the point obtained by intersecting the reflection of lines $PA$, $PB$, $PC$ across the angle bisectors of $\angle A$, $\angle B$, and $\angle C$, respectively. Given a point $Q$, let $\mathfrak K(Q)$ denote the unique cubic plane curve which passes through all points $P$ such that line $PP^\ast$ contains $Q$. Consider: [list] [*] the M'Cay cubic $\mathfrak K(O)$, where $O$ is the circumcenter of $\triangle ABC$, [*] the Thomson cubic $\mathfrak K(G)$, where $G$ is the centroid of $\triangle ABC$, [*] the Napoleon-Feurerbach cubic $\mathfrak K(N)$, where $N$ is the nine-point center of $\triangle ABC$, [*] the Darboux cubic $\mathfrak K(L)$, where $L$ is the de Longchamps point (the reflection of the orthocenter across point $O$), [*] the Neuberg cubic $\mathfrak K(X_{30})$, where $X_{30}$ is the point at infinity along line $OG$, [*] the nine-point circle of $\triangle ABC$, [*] the incircle of $\triangle ABC$, and [*] the circumcircle of $\triangle ABC$. [/list] Estimate $N$, the number of points lying on at least two of these eight curves. An estimate of $E$ earns $\left\lfloor 20 \cdot 2^{-|N-E|/6} \right\rfloor$ points.

Let $(a_1, a_2, ..., a_8)$ be a permutation of $(1, 2, ... , 8)$. Find, with proof, the maximum possible number of elements of the set $$\{a_1, a_1 + a_2, ... , a_1 + a_2 + ... + a_8\}$$ that can be perfect squares.

Let $a > 1$ be a positive integer and $d > 1$ be a positive integer coprime to $a$. Let $x_1=1$, and for $k\geq 1$, define $$x_{k+1} = \begin{cases} x_k + d &\text{if } a \text{ does not divide } x_k \\ x_k/a & \text{if } a \text{ divides } x_k \end{cases}$$ Find, in terms of $a$ and $d$, the greatest positive integer $n$ for which there exists an index $k$ such that $x_k$ is divisible by $a^n$.

Prove that for any integers $a,b$, the equation $2abx^4 - a^2x^2 - b^2 - 1 = 0$ has no integer roots. [i](Dan Schwarz)[/i]

Let $p>2$ be a prime number. For $k=1,2,\ldots,p-1$ we denote by $a_k$ the remainder when $k^p$ is divided by $p^2$. Prove that $$a_1+a_2+\ldots+a_{p-1}=\frac{p^3-p^2}2.$$

How many pairs $(a, b)$ for integers $1 \le a, b \le 2015$ which satisfy that $a$ is divisible by $b + 1$ and $2016 - a$ is divisible by $b$.

Let $ABCD$ and $ABEF$ be two squares situated in two perpendicular planes and let $O$ be the intersection of the lines $AE$ and $BF$. If $AB=4$ compute: a) the distance from $B$ to the line of intersection between the planes $(DOC)$ and $(DAF)$; b) the distance between the lines $AC$ and $BF$.

Polynomials $(P_n(X))_{n\in\mathbb{N}}$ are defined as: $$P_0(X)=0, \quad P_1(X)=X+2,$$ $$P_n(X)=P_{n-1}(X)+3P_{n-1}(X)\cdot P_{n-2}(X)+P_{n-2}(X), \quad (\forall) n\geq2.$$ Show that if $ k $ divides $m$ then $P_k(X)$ divides $P_m(X).$

Let $k$ and $m$, with $k > m$, be positive integers such that the number $km(k^2 - m^2)$ is divisible by $k^3 - m^3$. Prove that $(k - m)^3 > 3km$.

Solve the following system of equations in integers: $\begin{cases} x^2+2xy+8z=4z^2+4y+8\\ x^2+y+2z=156 \\ \end{cases}$

Let $n\geqslant 2$ be an integer and $A{}$ a set of $n$ points in the plane. Find all integers $1\leqslant k\leqslant n-1$ with the following property: any two circles $C_1$ and $C_2$ in the plane such that $A\cap\text{Int}(C_1)\neq A\cap\text{Int}(C_2)$ and $|A\cap\text{Int}(C_1)|=|A\cap\text{Int}(C_2)|=k$ have at least one common point. [i]Cristi Săvescu[/i]

Determine all polynomials $p(x)$ with non-negative integer coefficients such that $p (1) = 7$ and $p (10) = 2014$.

Azar, Carl, Jon, and Sergey are the four players left in a singles tennis tournament. They are randomly assigned opponents in the semifinal matches, and the winners of those matches play each other in the final match to determine the winner of the tournament. When Azar plays Carl, Azar will win the match with probability $\frac23$. When either Azar or Carl plays either Jon or Sergey, Azar or Carl will win the match with probability $\frac34$. Assume that outcomes of different matches are independent. The probability that Carl will win the tournament is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

Let be a complex number $ z $ having the property that $ \Re \left( z^n \right) >\Im \left( z^n \right) , $ for any natural numbers $ n. $ Show that $ z $ is a positive real number. [i]Laurențiu Panaitopol[/i]

We say that a set of positive integers is [i]regular [/i] if, for any selection of numbers from the set, the sum of the chosen numbers is different from $1810$. Divide the set of integers from $452$ to $1809$ (inclusive) into the smallest possible number of regular sets.

A right that passes through the incircle $ I$ of the triangle $ \Delta ABC$ intersects the side $ AB$ and $ CA$ in $ P$, respective $ Q$. We denote $ BC\equal{}a\ , \ AC\equal{}b\ ,\ AB\equal{}c$ and $ \frac{PB}{PA}\equal{}p\ ,\ \frac{QC}{QA}\equal{}q$. i) Prove that: \[ a(1\plus{}p)\cdot \overrightarrow{IP}\equal{}(a\minus{}pb)\overrightarrow{IB}\minus{}pc\overrightarrow{IC}\] ii) Show that $ a\equal{}bp\plus{}cq$. iii) If $ a^2\equal{}4bcpq$, then the rights $ AI\ ,\ BQ$ and $ CP$ are concurrents.

An equilateral triangle $ABC$ has side length $7$. Point $P$ is in the interior of triangle $ABC$, such that $PB=3$ and $PC=5$. The distance between the circumcenters of $ABC$ and $PBC$ can be expressed as $\frac{m\sqrt{n}}{p}$, where $n$ is not divisible by the square of any prime and $m$ and $p$ are relatively prime positive integers. What is $m+n+p$?

\[\int_0^\infty 1+\frac{2}{\sqrt[x]{8}}-\frac{3}{\sqrt[x]{4}}\,\mathrm dx\] [i]Proposed by Connor Gordon[/i]

Let $S$ denote the set of $2$ by $2$ matrices with integer entries and determinant $1$, and let $T$ denote those matrices of $S$ which are congruent to the identity matrix $I\pmod3$ (so $\begin{pmatrix}a&b\\c&d\end{pmatrix}\in T$ means that $a,b,c,d\in\mathbb Z,ad-bc=1,$ and $3$ divides $b,c,a-1,d-1$). (a) Let $f:T\to\mathbb R$ be a function such that for every $X,Y\in T$ with $Y\ne I$, either $f(XY)>f(X)$ or $f(XY^{-1})>f(X)$. Show that given two finite nonempty subsets $A,B$ of $T$, there are matrices $a\in A$ and $b\in B$ such that if $a'\in A$, $b'\in B$ and $a'b'=ab$, then $a'=a$ and $b'=b$. (b) Show that there is no $f:S\to\mathbb R$ such that for every $X,Y\in S$ with $Y\ne\pm I$, either $f(XY)>f(X)$ or $f(XY^{-1})>f(X)$.

The center $O$ of the circle $\omega$ passing through the vertex $C$ of the isosceles triangle $ABC$ ($AB = AC$) is the interior point of the triangle $ABC$. This circle intersects segments $BC$ and $AC$ at points $D \ne C$ and $E \ne C$, respectively, and the circumscribed circle $\Omega$ of the triangle $AEO$ at the point $F \ne E$. Prove that the center of the circumcircle of the triangle $BDF$ lies on the circle $\Omega$.

Let $ ABCD$ be a convex quadrilateral such that the sides $ AB, AD, BC$ satisfy $ AB \equal{} AD \plus{} BC.$ There exists a point $ P$ inside the quadrilateral at a distance $ h$ from the line $ CD$ such that $ AP \equal{} h \plus{} AD$ and $ BP \equal{} h \plus{} BC.$ Show that: \[ \frac {1}{\sqrt {h}} \geq \frac {1}{\sqrt {AD}} \plus{} \frac {1}{\sqrt {BC}} \]

A natural number $k>1$ is given. The sum of some divisor of $k$ and some divisor of $k - 1$ is equal to $a$,where $a>k + 1$. Prove that at least one of the numbers $a - 1$ or $a + 1$ composite.

Mr. Stein is ordering a two-course dessert at a restaurant. For each course, he can choose to eat pie, cake, rødgrød, and crème brûlée, but he doesn't want to have the same dessert twice. In how many ways can Mr. Stein order his meal? (Order matters.)

Let $ n$ be a positive integer. Consider a rectangle $ (90n\plus{}1)\times(90n\plus{}5)$ consisting of unit squares. Let $ S$ be the set of the vertices of these squares. Prove that the number of distinct lines passing through at least two points of $ S$ is divisible by $ 4$.

Stacy has $d$ dollars. She enters a mall with $10$ shops and a lottery stall. First she goes to the lottery and her money is doubled, then she goes into the first shop and spends $1024$ dollars. After that she alternates playing the lottery and getting her money doubled (Stacy always wins) then going into a new shop and spending $\$1024$. When she comes out of the last shop she has no money left. What is the minimum possible value of $d$?