Let $AB$ be the diameter of a semicircle. $C$, $D$, and $E$ are points on $AB$, in that order, such that $AC=1$, $CD=3$, $DE=4$, and $EB=2$. From $C$ and $E$, draw perpendicular lines of $AB$, intersecting the semicircle at $F$ and $G$, respectively. Find the measurement of $\angle{FDG}$.
A bug stands at point $A$ of $\triangle{ABC}$. In each move, it craws randomly to one of the other two vertices. Let the probability of it returns on point $A$ in 2015 steps be $\frac{m}{n}$, where $m$ and $n$ are co-prime. Find the last three digits of $m+n$.
Find the smallest positive integer $n$ such that the last $3$ digits of $n^3$ is $888$.
Given that $z$ is a complex number such that $z+\frac 1z=2\cos 3^\circ$, find the least integer that is greater than $z^{2000}+\frac 1{z^{2000}}$.
A function $f$ is defined on the complex numbers by $f(z)=(a+bi)z,$ where $a_{}$ and $b_{}$ are positive numbers. This function has the property that the image of each point in the complex plane is equidistant from that point and the origin. Given that $|a+bi|=8$ and that $b^2=m/n,$ where $m_{}$ and $n_{}$ are relatively prime positive integers, find $m+n.$
If $\{a_1,a_2,a_3,\ldots,a_n\}$ is a set of real numbers, indexed so that $a_1 < a_2 < a_3 < \cdots < a_n,$ its complex power sum is defined to be $a_1i + a_2i^2+ a_3i^3 + \cdots + a_ni^n,$ where $i^2 = - 1.$ Let $S_n$ be the sum of the complex power sums of all nonempty subsets of $\{1,2,\ldots,n\}.$ Given that $S_8 = - 176 - 64i$ and $S_9 = p + qi,$ where $p$ and $q$ are integers, find $|p| + |q|.$
Let $v$ and $w$ be distinct, randomly chosen roots of the equation $z^{1997}-1=0$. Let $\frac{m}{n}$ be the probability that $\sqrt{2+\sqrt{3}}\le\left|v+w\right|$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
Let $\mathrm {P}$ be the product of the roots of $z^6+z^4+z^3+z^2+1=0$ that have a positive imaginary part, and suppose that $\mathrm {P}=r(\cos{\theta^{\circ}}+i\sin{\theta^{\circ}})$, where $0 < r$ and $0\leq \theta <360$. Find $\theta$.
For certain real values of $a, b, c,$ and $d_{},$ the equation $x^4+ax^3+bx^2+cx+d=0$ has four non-real roots. The product of two of these roots is $13+i$ and the sum of the other two roots is $3+4i,$ where $i=\sqrt{-1}.$ Find $b.$
The equation
$x^{10}+(13x-1)^{10}=0\,$
has 10 complex roots $r_1, \overline{r_1}, r_2, \overline{r_2}, r_3, \overline{r_3}, r_4, \overline{r_4}, r_5, \overline{r_5},\,$ where the bar denotes complex conjugation. Find the value of
$\frac 1{r_1\overline{r_1}}+\frac 1{r_2\overline{r_2}}+\frac 1{r_3\overline{r_3}}+\frac 1{r_4\overline{r_4}}+\frac 1{r_5\overline{r_5}}.$
Consider the region $A$ in the complex plane that consists of all points $z$ such that both $\frac{z}{40}$ and $\frac{40}{\overline{z}}$ have real and imaginary parts between $0$ and $1$, inclusive. What is the integer that is nearest the area of $A$?
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.
How many positive integers less than $1998$ are relatively prime to $1547$?
For a positive integer m, we define $m$ as a $\textit{factorial}$ number if and only if there exists a positive integer $k$ for which $m = k\cdot(k - 1)\cdots 2\cdot 1$. We define a positive integer $n$ as a $\textit{Thai}$ number if and only if $n$ can be written as both the sum of two factorial numbers and the product of two factorial numbers.
What is the sum of the five smallest $\textit{Thai}$ numbers?
Let $a_1, a_2, a_3, \cdots a_n$ be a randomly ordered sequence of 1, 2, 3, . . . , $n$. Prove the following product is an even number if $n$ is an odd integer: $$(a_1 -1)(a_2-2)(a_3-3)\cdots(a_n-n)$$
If integer d is not equal to 2,5 or 13. Prove: there must exist two different elements $a$ and $b$ in set {2, 5, 3, d} such that $ab -
Find all 3-digit integer $\overline{abc}$ that satisfy $\overline{abc} = (a + b + c)^3$.
Find all positive integer $1\le n \le 100$ such that sum of all its digits divides $n$ itself.
Let point $O$ be the centroid of $\triangle{ABC}$, prove the areas of $\triangle{ABO}, \triangle{BOC}$, and $\triangle{COA}$ are equal.
If $a_0, a_1,\cdots, a_n \in \{0, 1, 2,\cdots, 9\}, n\ge 1, a_0\ge 1$, then the zeros of $f(x)=a_0 x^n + a_1x^{n-1} +\cdots +a_n$ have real parts less then 4.
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.