Practice (113)

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The sets $A = \{z : z^{18} = 1\}$ and $B = \{w : w^{48} = 1\}$ are both sets of complex roots of unity. The set $C = \{zw : z \in A ~ \mbox{and} ~ w \in B\}$ is also a set of complex roots of unity. How many distinct elements are in $C_{}^{}$?

Given a positive integer $n$, it can be shown that every complex number of the form $r+si$, where $r$ and $s$ are integers, can be uniquely expressed in the base $-n+i$ using the integers $1,2,\ldots,n^2$ as digits. That is, the equation $r+si=a_m(-n+i)^m+a_{m-1}(-n+i)^{m-1}+\cdots +a_1(-n+i)+a_0$ is true for a unique choice of non-negative integer $m$ and digits $a_0,a_1,\ldots,a_m$ chosen from the set $\{0,1,2,\ldots,n^2\}$, with $a_m e 0$. We write $r+si=(a_ma_{m-1}\ldots a_1a_0)_{-n+i}$ to denote the base $-n+i$ expansion of $r+si$. There are only finitely many integers $k+0i$ that have four-digit expansions $k=(a_3a_2a_1a_0)_{-3+i}~~$ $~~a_3 e 0.$ Find the sum of all such $k$.

Let $w_1, w_2, \dots, w_n$ be complex numbers. A line $L$ in the complex plane is called a mean line for the points $w_1, w_2, \dots, w_n$ if $L$ contains points (complex numbers) $z_1, z_2, \dots, z_n$ such that \[\sum_{k = 1}^n (z_k - w_k) = 0.\]For the numbers $w_1 = 32 + 170i$, $w_2 = - 7 + 64i$, $w_3 = - 9 + 200i$, $w_4 = 1 + 27i$, and $w_5 = - 14 + 43i$, there is a unique mean line with $y$-intercept 3. Find the slope of this mean line.

Find all pairs $(a,b)$ of nonnegative reals such that $(a-bi)^n = a^n - b^n i$ for some positive integer $n>1$.

Let $ABCD$ be an cyclic quadrilateral and let $HA, HB, HC, HD$ be the orthocentres of triangles $BCD, CDA, DAB$, and ABC respectively. Prove that the quadrilaterals $ABCD$ and $H_AH_BH_CH_D$ are congruent.

Consider triangle ABC and its circumcircle S. Reflect the circle with respect to AB, AC, BC to get three new circles SAB, SBC, and SBC. Show that these three circles intersect at a common point and identify this point.

The line $T$ is tangent to the circumcircle of acute triangle $ABC$ at $B$. Let $K$ be the projection of the orthocenter of triangle $ABC$ onto line $T$ ($K$ is the root of the perpendicular from the orthocenter to $S$). Let $L$ be the midpoint of side $AC$. Show that the triangle $BKL$ is isosceles.

The squares $BCDE$, $CAFG$, and $ABHI$ are constructed outside the triangle $ABC$. Let $GCDQ$ and $EBHP$ be parallelograms. Prove that $APQ$ is isosceles and $\angle PAQ =\frac{\pi}{2} $.

Given triangle $ABC$, construct equilateral triangle $ABC_1$, $BCA_1$, $CAB_1$ on the outside of $ABC$. Let $P, Q$ denote the midpoints of $C_1A_1$ and $C_1B_1$ respectively. Let $R$ be the midpoint of $AB$. Prove that triangle $PQR$ is isosceles.

(Napolean's Triangle) Given triangle $ABC$, construct an equilateral triangle on the outside of each of the sides. Let $P, Q, R$ be the centroids of these equilateral triangles, prove that triangle $PQR$ is equilateral.

(Simson Line) Let $Z$ be a point on the circumcircle of triangle $ABC$ and $P, Q, R$ be the feet of perpendiclars from $Z$ to $BC, AC, AB$ respectively. Prove that $P, Q, R$ are collinear. (This line is called the Simson line of triangle $ABC$ from $Z$.)

Given cyclic quadrilateral $ABCD$, let $P$ and $Q$ be the reflection of $C$ across lies $AB$ and $AD$ respectively. Prove that PQ passes through the orthocentre of triangle $ABD$.

Let $W_1W_2W_3$ be a triangle with circumcircle $S$, and let $A_1, A_2, A_3$ be the midpoints of $W_2W_3, W_1W_3, W_1W_2$ respectively. From Ai drop a perpendicular to the line tangent to $S$ at $W_i$. Prove that these perpendicular lines are concurrent and identify this point of concurrency.

Let $A_0A_1A_2A_3A_4A_5A_6$ be a regular 7-gon. Prove that $$\frac{1}{A_0A_1} = \frac{1}{A_0A_2}+\frac{1}{A_0A_3}$$

Given point $P_0$ in the plane of triangle $A_1A_2A_3$. Denote $A_s = A_{s-3}$, for $s > 3$. Construct points $P_1; P_2; \cdots$ sequentially such that point $P_{k+1}$ is $P_k$ rotated $120^\circ$ counter-clockwise around $A_{k+1}$. Prove that if $P_{1986} = P_0$ then triangle $A_1A_2A_3$ is isosceles.

Point $H$ is the orthocenter of triangle $ABC$. Points $D, E$ and $F$ lie on the circumcircle of triangle $ABC$ such that $AD\parallel BE\parallel CF$. Points $S, T,$ and $U$ are the respective reflections of $D, E, F$ across the lines $BC, CA$ and $AB$. Prove that $S, T, U, H$ are cyclic.

Let $ABCD$ be a cyclic quadrilateral. Let $P, Q, R$ be the feet of the perpendiculars from $D$ to the lines $BC, CA$ and $AB$ respectively. Show that $PQ = QR$ iff the bisectors of $\angle ABC$ and $\angle ADC$ meet on $AC$.

Let $O$ be the circumcentre of triangle $ABC$. A line through $O$ intersects sides $AB$ and $AC$ at $M$ and $N$ respectively. Let $S$ and $R$ be the midpoints of $BN$ and $CM$, respectively. Prove that $\angle ROS = \angle BAC$.

Let $ABCD$ be a convex quadrilateral for which $AC = BD$. Equilateral triangles are constructed on the sides of the quadrilateral and pointing outward. Let $O_1, O_2, O_3, O_4$ be the centres of the triangles constructed on $AB, BC, CD,$ and $DA$ respectively. Prove that lines $O_1O_3$ and $O_2O_4$ are perpendicular.

Let $ABC$ be a triangle. Triangles $PAB$ and $QAC$ are constructed outside of $ABC$ such that $AP = AB$ and $AQ = AC$ and $\angle BAP = \angle CAQ$. Segments $BQ$ and $CP$ meet at $R$. Let $O$ be the circumcentre of triangle $BCR$. Prove that $AO \perp PQ$.

Calculate $(1+i)^{20}$.

Let complex number $z$ satisfy $|z|=1$. If $f(z)=|z+1+i|$ reaches its maximum and minimal values when $z=z_1$ and $z=z_2$, respectively. Compute $z_1-z_2$.

Let complex number $z$ satisfy $|z|=1$, $w = z^4-z^3-3z^2i-z+1$. Find the minimal value of $|w|$.

Let complex numbers $a$, $b$, and $c$ satisfy $a|bc| + b|ca| + c|ab| = 0$. Show that $$|(a-b)(b-c)(c-a)|\ge 3\sqrt{3}|abc|$$

Let $x, y \in \big(0, \frac{\pi}{2}\big)$. Show that if the equation $(\cos x + i \sin y)^n = \cos nx + i \sin ny$ holds for two consecutive positive integers, then it will hold for all positive integers.