Found problems: 1084
Given $t\in\mathbb C$. Complex numbers $x,y,z$ satisfy that $|x|=|y|=|z|=1$ and $\frac{t}{y}=\frac{1}{x}+\frac{1}{z}$. Calculate
$$\left|\frac{2xy+2yz+3zx}{x+y+z}\right|.$$
Let $n\ge 2$ be an integer. Let $z_1, z_2,..., z_n$ be complex numbers in such a way that for all integers $k$ with $1\le k\le n$: $$\Pi_{i = 1,i\ne k}^{n} (z_k- z_i) = \Pi_{i = 1,i\ne k}^{n} (z_k+ z_i).$$ Show that two of them are the same.
While there do not exist pairwise distinct real numbers $a,b,c$ satisfying $a^2+b^2+c^2 = ab+bc+ca$, there do exist complex numbers with that property. Let $a,b,c$ be complex numbers such that $a^2+b^2+c^2 = ab+bc+ca$ and $|a+b+c| = 21$. Given that $|a-b| = 2\sqrt{3}$, $|a| = 3\sqrt{3}$, compute $|b|^2+|c|^2$.
[hide="Clarifications"]
[list]
[*] The problem should read $|a+b+c| = 21$. An earlier version of the test read $|a+b+c| = 7$; that value is incorrect.
[*] $|b|^2+|c|^2$ should be a positive integer, not a fraction; an earlier version of the test read ``... for relatively prime positive integers $m$ and $n$. Find $m+n$.''[/list][/hide]
[i]Ray Li[/i]
A real number $\alpha \geq 0$ is given. Find the smallest $\lambda = \lambda (\alpha ) > 0$, such that for any complex numbers ${z_1},{z_2}$ and $0 \leq x \leq 1$, if $\left| {{z_1}} \right| \leq \alpha \left| {{z_1} - {z_2}} \right|$, then $\left| {{z_1} - x{z_2}} \right| \leq \lambda \left| {{z_1} - {z_2}} \right|$.
Find the smallest real number $M$ such that there exist four complex numbers $a,b,c,d$ with $|a|=|b|=|c|=|d|=1$, and for any complex number $z$, if $|z| = 1$, then\[|az^3+bz^2+cz+d|\le M.\]
Let $v$ and $w$ be distinct, randomly chosen roots of the equation $z^{1997}-1=0.$ Let $m/n$ be the probability that $\sqrt{2+\sqrt{3}}\le |v+w|,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
Let $a,b$ complex numbers. Show that if the roots of the equation $z^2+az+b=0$ and 0 form a triangle with the right angle at the origin, then $a^2=2b\neq0.$ Also determine whether the opposite implication holds.
Let $ABC$ be a triangle with orthocenter $H,$ and let $M$ be the midpoint of $\overline{BC}.$ Suppose that $P$ and $Q$ are distinct points on the circle with diameter $\overline{AH},$ different from $A,$ such that $M$ lies on line $PQ.$ Prove that the orthocenter of $\triangle APQ$ lies on the circumcircle of $\triangle ABC.$
[i]Proposed by Michael Ren[/i]
Let $ z_1,z_2,z_3 $ be nonzero complex numbers and pairwise distinct, having the property that $\left( z_1+z_2\right)^3 =\left( z_2+z_3\right)^3 =\left( z_3+z_1\right)^3. $ Show that $ \left| z_1-z_2\right| =\left| z_2-z_3\right| =\left| z_3-z_1\right| . $
Let a constant $\alpha$ as $0<\alpha<1$, prove that:
$(1)$ There exist a constant $C(\alpha)$ which is only depend on $\alpha$ such that for every $x\ge 0$, $\ln(1+x)\le C(\alpha)x^\alpha$.
$(2)$ For every two complex numbers $z_1,z_2$, $|\ln|\frac{z_1}{z_2}||\le C(\alpha)\left(|\frac{z_1-z_2}{z_2}|^\alpha+|\frac{z_2-z_1}{z_1}|^\alpha\right)$.
Consider a $1$-indexed array that initially contains the integers $1$ to $10$ in increasing order.
The following action is performed repeatedly (any number of times):
[code]
def action():
Choose an integer n between 1 and 10 inclusive
Reverse the array between indices 1 and n inclusive
Reverse the array between indices n+1 and 10 inclusive (If n = 10, we do nothing)
[/code]
How many possible orders can the array have after we are done with this process?
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]
In $\triangle ABC$, let its incircle touch $\overline{AC}$ and $\overline{AB}$ at $E$ and $F$ respectively. Let its $A$-excircle have center $I_A$ and touch $\overline{BC}$ at $K$. Let $P$ and $Q$ be points so that $BFPI_A$ and $CEQI_A$ are parallelograms. If $\overline{AI_A}$ and $\overline{PQ}$ intersect at $X$, prove $\overline{KX} \perp \overline{PQ}$.
Let $ABC$ be an acute triangle with circumcenter $O$ and orthocenter $H$. Let $\Gamma$ denote the circumcircle of triangle $ABC$, and $N$ the midpoint of $OH$. The tangents to $\Gamma$ at $B$ and $C$, and the line through $H$ perpendicular to line $AN$, determine a triangle whose circumcircle we denote by $\omega_A$. Define $\omega_B$ and $\omega_C$ similarly.
Prove that the common chords of $\omega_A$,$\omega_B$ and $\omega_C$ are concurrent on line $OH$.
Proposed by Anant Mudgal
Prove that if non-zero complex numbers $\alpha_1,\alpha_2,\alpha_3$ are distinct and noncollinear on the plane, and satisfy $\alpha_1+\alpha_2+\alpha_3=0$, then there holds
\[\sum_{i=1}^{3}\left(\frac{|\alpha_{i+1}-\alpha_{i+2}|}{\sqrt{|\alpha_i|}}\left(\frac{1}{\sqrt{|\alpha_{i+1}|}}+\frac{1}{\sqrt{|\alpha_{i+2}|}}-\frac{2}{\sqrt{|\alpha_{i}|}}\right)\right)\leq 0......(*)\]
where $\alpha_4=\alpha_1, \alpha_5=\alpha_2$. Verify further the sufficient and necessary condition for the equality holding in $(*)$.
Up to similarity, there is a unique nondegenerate convex equilateral 13-gon whose internal angles have measures that are multiples of 20 degrees. Find it. Give your answer by listing the degree measures of its 13 [i]external[/i] angles in clockwise or counterclockwise order. Start your list with the biggest external angle. You don't need to write the degree symbol $^\circ$.
Let $O$ be the circumcentre, and $\Omega$ be the circumcircle of an acute-angled triangle $ABC$. Let $P$ be an arbitrary point on $\Omega$, distinct from $A$, $B$, $C$, and their antipodes in $\Omega$. Denote the circumcentres of the triangles $AOP$, $BOP$, and $COP$ by $O_A$, $O_B$, and $O_C$, respectively. The lines $\ell_A$, $\ell_B$, $\ell_C$ perpendicular to $BC$, $CA$, and $AB$ pass through $O_A$, $O_B$, and $O_C$, respectively. Prove that the circumcircle of triangle formed by $\ell_A$, $\ell_B$, and $\ell_C$ is tangent to the line $OP$.
Let be four distinct complex numbers $ a,b,c,d $ chosen such that
$$ |a|=|b|=|c|=|d|=|b-c|=\frac{|c-d|}{2}=1, $$
and
$$ \min_{\lambda\in\mathbb{C}} |a-\lambda d -(1-\lambda )c| =\min_{\lambda\in\mathbb{C}} |b-\lambda d -(1-\lambda )c| . $$
Calculate $ |a-c| $ and $ |a-d|. $
[i]Carmen Botea[/i]
Let $ABC$ be a scalene triangle, and points $O$ and $H$ be its circumcenter and orthocenter, respectively. Point $P$ lies inside triangle $AHO$ and satisfies $\angle AHP = \angle POA$. Let $M$ be the midpoint of segment $\overline{OP}$. Suppose that $BM$ and $CM$ intersect with the circumcircle of triangle $ABC$ again at $X$ and $Y$, respectively.
Prove that line $XY$ passes through the circumcenter of triangle $APO$.
[i]Proposed by Li4[/i]
Let $\{ z_n \}_{n \ge 1}$ be a sequence of complex numbers, whose odd terms are real, even terms are purely imaginary, and for every positive integer $k$, $|z_k z_{k+1}|=2^k$. Denote $f_n=|z_1+z_2+\cdots+z_n|,$ for $n=1,2,\cdots$
(1) Find the minimum of $f_{2020}$.
(2) Find the minimum of $f_{2020} \cdot f_{2021}$.
Determine all real numbers $x$ for which the following statement is true: the field $\mathbb C$ of complex numbers contains a proper subfield $F$ such that adjoining $x$ to $F$ we get $\mathbb C$. [M. Laczkovich]
Let $ABCDE$ be a convex pentagon such that $\angle ABC = \angle AED = 90^\circ$. Suppose that the midpoint of $CD$ is the circumcenter of triangle $ABE$. Let $O$ be the circumcenter of triangle $ACD$.
Prove that line $AO$ passes through the midpoint of segment $BE$.
Let $z=x+iy$ be a complex number with $x$ and $y$ rational and with $|z| = 1.$
(a) Find two such complex numbers.
(b) Show that $|z^{2n}-1|=2|\sin n\theta|,$ where $z=e^{i\theta}.$
(c) Show that $|z^2n -1|$ is rational for every $n.$
Given a nondegenerate triangle $ABC$, with circumcentre $O$, orthocentre $H$, and circumradius $R$, prove that $|OH| < 3R$.
Given a cyclic quadrilateral $ABCD$, let the diagonals $AC$ and $BD$ meet at $E$ and the lines $AD$ and $BC$ meet at $F$. The midpoints of $AB$ and $CD$ are $G$ and $H$, respectively. Show that $EF$ is tangent at $E$ to the circle through the points $E$, $G$ and $H$.
[i]Proposed by David Monk, United Kingdom[/i]