Brian starts at the point $\left(1,0\right)$ in the plane. Every second, he performs one of two moves: he can move from $\left(a,b\right)$ to $\left(a-b,a+b\right)$ or from $\left(a,b\right)$ to $\left(2a-b,a+2b\right)$. How many different paths can he take to end up at $\left(28,-96\right)$? [i]2018 CCA Math Bonanza Individual Round #14[/i]

Find $c$ if $a$, $b$, and $c$ are positive integers which satisfy $c=(a + bi)^3 - 107i$, where $i^2 = -1$.

A set $A_1 , A_2 , A_3 , A_4$ of 4 points in the plane is said to be [i]Athenian[/i] set if there is a point $P$ of the plane satsifying (*) $P$ does not lie on any of the lines $A_i A_j$ for $1 \leq i < j \leq 4$; (**) the line joining $P$ to the mid-point of the line $A_i A_j$ is perpendicular to the line joining $P$ to the mid-point of $A_k A_l$, $i,j,k,l$ being distinct. (a) Find all [i]Athenian[/i] sets in the plane. (b) For a given [i]Athenian[/i] set, find the set of all points $P$ in the plane satisfying (*) and (**)

Each point in the plane is labeled with a real number. Show that there exist two distinct points $P$ and $Q$ whose labels differ by less than the distance from $P$ to $Q$. [i]Holden Mui[/i]

Let $a$, $b$, $c$, $d$ be complex numbers satisfying \begin{align*} 5 &= a+b+c+d \\ 125 &= (5-a)^4 + (5-b)^4 + (5-c)^4 + (5-d)^4 \\ 1205 &= (a+b)^4 + (b+c)^4 + (c+d)^4 + (d+a)^4 + (a+c)^4 + (b+d)^4 \\ 25 &= a^4+b^4+c^4+d^4 \end{align*} Compute $abcd$. [i]Proposed by Evan Chen[/i]

Let $ABC$ be an acute triangle inscribed in a circle $\omega$ with center $O$. Points $E$, $F$ lie on its side $AC$, $AB$, respectively, such that $O$ lies on $EF$ and $BCEF$ is cyclic. Let $R$, $S$ be the intersections of $EF$ with the shorter arcs $AB$, $AC$ of $\omega$, respectively. Suppose $K$, $L$ are the reflection of $R$ about $C$ and the reflection of $S$ about $B$, respectively. Suppose that points $P$ and $Q$ lie on the lines $BS$ and $RC$, respectively, such that $PK$ and $QL$ are perpendicular to $BC$. Prove that the circle with center $P$ and radius $PK$ is tangent to the circumcircle of $RCE$ if and only if the circle with center $Q$ and radius $QL$ is tangent to the circumcircle of $BFS$. [i]Proposed by Mehran Talaei[/i]

Suppose $2016$ points of the circumference of a circle are colored red and the remaining points are colored blue . Given any natural number $n\ge 3$, prove that there is a regular $n$-sided polygon all of whose vertices are blue.

Let $ABC$ be an arbitrary triangle. A regular $n$-gon is constructed outward on the three sides of $\triangle ABC$. Find all $n$ such that the triangle formed by the three centres of the $n$-gons is equilateral.

Let $ ABCD$ be a convex quadrilateral. The perpendicular bisectors of its sides $ AB$ and $ CD$ meet at $ Y$. Denote by $ X$ a point inside the quadrilateral $ ABCD$ such that $ \measuredangle ADX \equal{} \measuredangle BCX < 90^{\circ}$ and $ \measuredangle DAX \equal{} \measuredangle CBX < 90^{\circ}$. Show that $ \measuredangle AYB \equal{} 2\cdot\measuredangle ADX$.

A point $T$ is chosen inside a triangle $ABC$. Let $A_1$, $B_1$, and $C_1$ be the reflections of $T$ in $BC$, $CA$, and $AB$, respectively. Let $\Omega$ be the circumcircle of the triangle $A_1B_1C_1$. The lines $A_1T$, $B_1T$, and $C_1T$ meet $\Omega$ again at $A_2$, $B_2$, and $C_2$, respectively. Prove that the lines $AA_2$, $BB_2$, and $CC_2$ are concurrent on $\Omega$. [i]Proposed by Mongolia[/i]

There is a complex number $ z$ with imaginary part $ 164$ and a positive integer $ n$ such that \[ \frac {z}{z \plus{} n} \equal{} 4i. \]Find $ n$.

For complex number constant $c$, and real number constants $p$ and $q$, there exist three distinct complex values of $x$ that satisfy $x^3 + cx + p(1 + qi) = 0$. Suppose $c$, $p$, and $q$ were chosen so that all three complex roots $x$ satisfy $\tfrac{5}{6} \leq \tfrac{\mathrm{Im}(x)}{\mathrm{Re}(x)} \leq \tfrac{6}{5}$, where $\mathrm{Im}(x)$ and $\mathrm{Re}(x)$ are the imaginary and real part of $x$, respectively. The largest possible value of $|q|$ can be expressed as a common fraction $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.

Let $\triangle{ABC}$ be a scalene triangle with incentre $I$ and circumcircle $\omega$. Lines $AI, BI, CI$ intersect $\omega$ for the second time at points $D, E, F$, respectively. The parallel lines from $I$ to the sides $BC, AC, AB$ intersect $EF, DF, DE$ at points $K, L, M$, respectively. Prove that the points $K, L, M$ are collinear. [i](Cyprus)[/i]

Let $ABC$ be a triangle. Let $A_1$ and $A_2$ be points on side $BC, B_1$ and $B_2$ be points on side $CA$ and $C_1$ and $C_2$ be points on side $AB$ such that $A_1A_2B_1B_2C_1C_2$ is a convex hexagon and that $B,A_1,A_2$ and $C$ are located in that order on side $BC$. We say that triangles $AB_2C_1, BA_1C_2$ and $CA_2B_1$ are glueable if there exists a triangle $PQR$ and there exist $X,Y$ and $Z$ on sides $QR, RP$ and $PQ$ respectively, such that triangle $AB_2C_1$ is congruent in that order to triangle $PYZ$, triangle $BA_1C_2$ is congruent in that order to triangle $QXZ$ and triangle $CA_2B_1$ is congruent in that order to triangle $RXY$. Prove that triangles $AB_2C_1, BA_1C_2$ and $CA_2B_1$ are glueable if and only if the centroids of triangles $A_1B_1C_1$ and $A_2B_2C_2$ coincide.

A function $ f$ is defined by $ f(z) \equal{} i\bar z$, where $ i \equal{}\sqrt{\minus{}\!1}$ and $ \bar z$ is the complex conjugate of $ z$. How many values of $ z$ satisfy both $ |z| \equal{} 5$ and $ f (z) \equal{} z$? $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 8$