Found problems: 1084
A quadrilateral $ABCD$ without parallel sides is circumscribed around a circle with centre $O$. Prove that $O$ is a point of intersection of middle lines of quadrilateral $ABCD$ (i.e. barycentre of points $A,\,B,\,C,\,D$) iff $OA\cdot OC=OB\cdot OD$.
In a triangle $ABC$, with $\widehat{A} > 90^\circ$, let $O$ and $H$ denote its circumcenter and orthocenter, respectively. Let $K$ be the reflection of $H$ with respect to $A$. Prove that $K, O$ and $C$ are collinear if and only if $\widehat{A} - \widehat{B} = 90^\circ$.
If $ a$ and $ b$ are two of the roots of $ x^4\plus{}x^3\minus{}1\equal{}0$, prove that $ ab$ is a root of $ x^6\plus{}x^4\plus{}x^3\minus{}x^2\minus{}1\equal{}0$.
Consider a fixed circle $\Gamma$ with three fixed points $A, B,$ and $C$ on it. Also, let us fix a real number $\lambda \in(0,1)$. For a variable point $P \not\in\{A, B, C\}$ on $\Gamma$, let $M$ be the point on the segment $CP$ such that $CM =\lambda\cdot CP$ . Let $Q$ be the second point of intersection of the circumcircles of the triangles $AMP$ and $BMC$. Prove that as $P$ varies, the point $Q$ lies on a fixed circle.
[i]Proposed by Jack Edward Smith, UK[/i]
Let $A_1, B_1$ and $C_1$ be points on sides $BC$, $CA$ and $AB$ of an acute triangle $ABC$ respectively, such that $AA_1$, $BB_1$ and $CC_1$ are the internal angle bisectors of triangle $ABC$. Let $I$ be the incentre of triangle $ABC$, and $H$ be the orthocentre of triangle $A_1B_1C_1$. Show that $$AH + BH + CH \geq AI + BI + CI.$$
Let $\omega$ be the circumcircle of $\triangle{ABC}$. $P$ is an interior point of $\triangle{ABC}$. $A_{1}, B_{1}, C_{1}$ are the intersections of $AP, BP, CP$ respectively and $A_{2}, B_{2}, C_{2}$ are the symmetrical points of $A_{1}, B_{1}, C_{1}$ with respect to the midpoints of side $BC, CA, AB$.
Show that the circumcircle of $\triangle{A_{2}B_{2}C_{2}}$ passes through the orthocentre of $\triangle{ABC}$.
Fix a circle $\Gamma$, a line $\ell$ to tangent $\Gamma$, and another circle $\Omega$ disjoint from $\ell$ such that $\Gamma$ and $\Omega$ lie on opposite sides of $\ell$. The tangents to $\Gamma$ from a variable point $X$ on $\Omega$ meet $\ell$ at $Y$ and $Z$. Prove that, as $X$ varies over $\Omega$, the circumcircle of $XYZ$ is tangent to two fixed circles.
A regular $2008$-gon is located in the Cartesian plane such that $(x_1,y_1)=(p,0)$ and $(x_{1005},y_{1005})=(p+2,0)$, where $p$ is prime and the vertices, \[(x_1,y_1),(x_2,y_2),(x_3,y_3),\cdots,(x_{2008},y_{2008}),\]
are arranged in counterclockwise order. Let \begin{align*}S&=(x_1+y_1i)(x_3+y_3i)(x_5+y_5i)\cdots(x_{2007}+y_{2007}i),\\T&=(y_2+x_2i)(y_4+x_4i)(y_6+x_6i)\cdots(y_{2008}+x_{2008}i).\end{align*} Find the minimum possible value of $|S-T|$.
Let $a$, $b$ and $c$ be complex numbers such that $abc = 1$. Find the value of the cubic root of
\begin{tabular}{|ccc|}
$b + n^3c$ & $n(c - b)$ & $n^2(b - c)$\\
$n^2(c - a)$ & $c + n^3a$ & $n(a - c)$\\
$n(b - a)$ & $n^2(a - b)$ & $a + n^3b$
\end{tabular}
Given $f(z) = z^2-19z$, there are complex numbers $z$ with the property that $z$, $f(z)$, and $f(f(z))$ are the vertices of a right triangle in the complex plane with a right angle at $f(z)$. There are positive integers $m$ and $n$ such that one such value of $z$ is $m+\sqrt{n}+11i$. Find $m+n$.
Suppose that $4^{n}+2^{n}+1$ is prime for some positive integer $n$. Show that $n$ must be a power of $3$.
Given an integer $n\geq 2$ and a closed unit disc, evaluate the maximum of the product of the lengths of all $\frac{n(n-1)}{2}$ segments determined by $n$ points in that disc.
a) Show that $\forall n \in \mathbb{N}_0, \exists A \in \mathbb{R}^{n\times n}: A^3=A+I$.
b) Show that $\det(A)>0, \forall A$ fulfilling the above condition.
Let $Z,W,\lambda$ be complex numbers, $|\lambda|\neq1$. Which statements are correct about the equation $\overline{Z}-\lambda Z=W$?
I. $Z=\frac{\overline{\lambda}W+\overline{W}}{1-|\lambda|^2}$ is a solution to the equation.
II. The equation has only one solution.
III. The equation has two solutions.
IV. The equation has infinitely many solutions.
$\text{(A)}$ Only I and II.
$\text{(B)}$ Only I and III.
$\text{(C)}$ Only I and IV.
$\text{(D)}$ None of $\text{(A)(B)(C)}$.
Answer the following questions:
(1) For complex numbers $\alpha ,\ \beta$, if $\alpha \beta =0$, then prove that $\alpha =0$ or $\beta =0$.
(2) For complex number $\alpha$, if $\alpha^2$ is a positive real number, then prove that $\alpha$ is a real number.
(3) For complex numbers $\alpha_1,\ \alpha_2,\ \cdots,\ \alpha_{2n+1}\ (n=1,\ 2,\ \cdots)$, assume that $\alpha_1\alpha_2,\ \cdots ,\ \alpha_k\alpha_{k+1},\ \cdots,\ \alpha_{2n}\alpha_{2n+1}$ and $\alpha_{2n+1}\alpha_1$ are all positive real numbers. Prove that $\alpha_1,\ \alpha_2,\ \cdots,\ \alpha_{2n+1}$ are all real numbers.
There are 24 different complex numbers $z$ such that $z^{24} = 1$. For how many of these is $z^6$ a real number?
$\textbf{(A) }1\qquad\textbf{(B) }3\qquad\textbf{(C) }6\qquad\textbf{(D) }12\qquad\textbf{(E) }24$
Let $n \ge 3$ be a fixed integer. A game is played by $n$ players sitting in a circle. Initially, each player draws three cards from a shuffled deck of $3n$ cards numbered $1, 2, \dots, 3n$. Then, on each turn, every player simultaneously passes the smallest-numbered card in their hand one place clockwise and the largest-numbered card in their hand one place counterclockwise, while keeping the middle card.
Let $T_r$ denote the configuration after $r$ turns (so $T_0$ is the initial configuration). Show that $T_r$ is eventually periodic with period $n$, and find the smallest integer $m$ for which, regardless of the initial configuration, $T_m=T_{m+n}$.
[i]Proposed by Carl Schildkraut and Colin Tang[/i]
Let $n$ be a positive integer. There are $n$ soldiers stationed on the $n$th root of unity in the complex plane. Each round, you pick a point, and all the soldiers shoot in a straight line towards that point; if their shot hits another soldier, the hit soldier dies and no longer shoots during the next round. What is the minimum number of rounds, in terms of $n$, required to eliminate all the soldiers?
[i]David Yang.[/i]
Let $\Gamma$ and $\Gamma'$ be two concentric circles. Let $ABC$ and $A'B'C'$ be any two equilateral triangles inscribed in $\Gamma$ and $\Gamma'$ respectively. If $P$ and $P'$ are any two points on $\Gamma$ and $\Gamma'$ respectively, show that \[ P'A^2 + P'B^2 + P'C^2 = A'P^2 + B'P^2 + C'P^2. \]
Prove that whatever the complex number $z$ is, it is true that
$$(1 + z^{2^n})(1-z^{2^n})= 1- z^{2^{n+1}}.$$
Writing the equalities that result from giving $n$ the values $0, 1, 2, . . .$ and multiplying them, show that for $|z| < 1$ holds
$$\frac{1}{1-z}= \lim_{k\to \infty}(1 + z)(1 + z^2)(1 + z^{2^2})...(1 + z^{2^k}).$$
Find all polynomials $P$ with real coefficients satisfying $P(x^2)=P(x)\cdot P(x+2)$ for all real numbers $x.$
Prove that for any polynomial $P$ with real coefficients, and for any positive integer $n$, there exists a polynomial $Q$ with real coefficients such that $P(x)^2 +Q(x)^2$ is divisible by $(1+x^2)^n$.
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]
Let $ABCDEF$ be a convex hexagon with $AB=DE$, $BC=EF$, $CD=FA$, and $\angle A-\angle D = \angle C -\angle F = \angle E -\angle B$. Prove that the diagonals $AD$, $BE$, and $CF$ are concurrent.
Let $ a,b,c\in\mathbb{C}^* $ pairwise distinct, having the same absolute value, and satisfying:
$$ a^2+b^2+c^2-ab-bc-ca=0. $$
Prove that $ a,b,c $ represents the affixes of the vertices of a right or equilateral triangle.