Found problems: 1084
We consider a natural number $n$, $n \geq 2$ and the matrices
\begin{tabular}{cc}
$A= \begin{pmatrix} 1 & 2 & 3 & \cdots & n\\ n & 1 & 2 & \cdots & n - 1\\ n - 1 & n & 1 & \cdots & n - 2\\ \cdots & \cdots & \cdots & \cdots & \cdots\\2 & 3 & 4 & \cdots & 1 \end{pmatrix}$
\end{tabular}
Show that$$\epsilon^ndet\left(I_n-A^{2n}\right)+\epsilon^{n-1}det\left(\epsilon I_n-A^{2n}\right)+\epsilon^{n-2}det\left(\epsilon^2 I_n-A^{2n}\right)+\cdots +det\left(\epsilon^n I_n-A^{2n}\right)$$ $$=n(-1)^{n-1}\left[\frac{n^n(n+1)}{2}\right]^{2n^2-4n}\left(1+(n+1)^{2n}\left(2n+(-1)^n{{2n}\choose{n}}\right)\right)$$where $\epsilon \in \mathbb{C}\backslash \mathbb{R}$, $\epsilon^{n+1}=1$
If $S$ is the set of points $z$ in the complex plane such that $(3+4i)z$ is a real number, then $S$ is a
$ \textbf{(A)}\text{ right triangle}\qquad\textbf{(B)}\text{ circle}\qquad\textbf{(C)}\text{ hyperbola}\qquad\textbf{(D)}\text{ line}\qquad\textbf{(E)}\text{ parabola} $
Let $ ABCDE$ be a convex pentagon. If $ P$, $ Q$, $ R$ and $ S$ are the respective centroids of the triangles $ ABE$, $ BCE$, $ CDE$ and $ DAE$, show that $ PQRS$ is a parallelogram and its area is $ 2/9$ of that of $ ABCD$.
Given a regular $n$-gon inscribed in a circle of radius 1, where $n > 3$.
Define $G(n)$ as the average length of the diagonals of this $n$-gon.
Prove that if $n \rightarrow \infty, G(n) \rightarrow \frac{4}{\pi}$.
Let $ f$ be an entire function on $ \mathbb C$ and $ \omega_1,\omega_2$ are complex numbers such that $ \frac {\omega_1}{\omega_2}\in{\mathbb C}\backslash{\mathbb Q}$. Prove that if for each $ z\in \mathbb C$, $ f(z) \equal{} f(z \plus{} \omega_1) \equal{} f(z \plus{} \omega_2)$ then $ f$ is constant.
Let $P(z)=a_d z^d+\dots+ a_1z+a_0$ be a polynomial with complex coefficients. The $reverse$ of $P$ is defined by
$$P^*(z)=\overline{a_0}z^d+\overline{a_1}z^{d-1}+\dots+\overline{a_d}$$
(a) Prove that
$$P^*(z)=z^d \overline{ P\left( \frac{1}{\overline{z}} \right) } $$
(b) Let $m$ be a positive integer and let $q(z)$ be a monic nonconstant polynomial with complex coefficients. Suppose that all roots of $q(z)$ lie inside or on the unit circle. Prove that all roots of the polynomial
$$Q(z)=z^m q(z)+ q^*(z)$$
lie on the unit circle.
Let $ABC$ be an acute triangle with incentre $I$ and $AB \neq AC$. Let lines $BI$ and $CI$ intersect the circumcircle of $ABC$ at $P \neq B$ and $Q \neq C$, respectively. Consider points $R$ and $S$ such that $AQRB$ and $ACSP$ are parallelograms (with $AQ \parallel RB, AB \parallel QR, AC \parallel SP$, and $AP \parallel CS$). Let $T$ be the point of intersection of lines $RB$ and $SC$. Prove that points $R, S, T$, and $I$ are concyclic.
Given a point $P_0$ in the plane of the triangle $A_1A_2A_3$. Define $A_s=A_{s-3}$ for all $s\ge4$. Construct a set of points $P_1,P_2,P_3,\ldots$ such that $P_{k+1}$ is the image of $P_k$ under a rotation center $A_{k+1}$ through an angle $120^o$ clockwise for $k=0,1,2,\ldots$. Prove that if $P_{1986}=P_0$, then the triangle $A_1A_2A_3$ is equilateral.
Let ${AB}$ be a diameter of a circle ${\omega}$ and center ${O}$ , ${OC}$ a radius of ${\omega}$ perpendicular to $AB$,${M}$ be a point of the segment $\left( OC \right)$ . Let ${N}$ be the second intersection point of line ${AM}$ with ${\omega}$ and ${P}$ the intersection point of the tangents of ${\omega}$ at points ${N}$ and ${B.}$ Prove that points ${M,O,P,N}$ are cocyclic.
(Albania)
Every point with integer coordinates in the plane is the center of a disk with radius $1/1000$.
(1) Prove that there exists an equilateral triangle whose vertices lie in different discs.
(2) Prove that every equilateral triangle with vertices in different discs has side-length greater than $96$.
[i]Radu Gologan, Romania[/i]
[hide="Remark"]
The "> 96" in [b](b)[/b] can be strengthened to "> 124". By the way, part [b](a)[/b] of this problem is the place where I used [url=http://mathlinks.ro/viewtopic.php?t=5537]the well-known "Dedekind" theorem[/url].
[/hide]
Let $ABC$ be a fixed acute triangle inscribed in a circle $\omega$ with center $O$. A variable point $X$ is chosen on minor arc $AB$ of $\omega$, and segments $CX$ and $AB$ meet at $D$. Denote by $O_1$ and $O_2$ the circumcenters of triangles $ADX$ and $BDX$, respectively. Determine all points $X$ for which the area of triangle $OO_1O_2$ is minimized.
[i]Proposed by Zuming Feng[/i]
Let $z\ne0$ be a complex number such that $z^8=\overline z$. What are the possible values of $z^{2001}$?
Let $ S_{\nu}\equal{}\sum_{j\equal{}1}^n b_jz_j^{\nu} \;(\nu\equal{}0,\pm 1, \pm 2 ,...) $, where the $ b_j$ are arbitrary and the $ z_j$ are nonzero complex numbers . Prove that \[ |S_0| \leq n \max_{0<|\nu| \leq n} |S_{\nu}|.\]
[i]G. Halasz[/i]
$z$ is a complex number and $|z|=1$ and $z^2\ne1$. Then $\frac z{1-z^2}$ lies on
$\textbf{(A)}~\text{a line not through origin}$
$\textbf{(B)}~\text{|z|=2}$
$\textbf{(C)}~x-\text{axis}$
$\textbf{(D)}~y-\text{axis}$
$ABCD$ is a fixed rhombus. Segment $PQ$ is tangent to the inscribed circle of $ABCD$, where $P$ is on side $AB$, $Q$ is on side $AD$. Show that, when segment $PQ$ is moving, the area of $\Delta CPQ$ is a constant.
In triangle $ABC$, $AD$ and $AE$ trisect $\angle BAC$. The lengths of $BD, DE $ and $EC$ are $1, 3 $ and $5$ respectively. Find the length of $AC$.
A point $M$ is found inside a convex quadrilateral $ABCD$ such that triangles $AMB$ and $CMD$ are isoceles ($AM = MB, CM = MD$) and $\angle AMB= \angle CMD = 120^o$ . Prove that there exists a point N such that triangles$ BNC$ and $DNA$ are equilateral.
(I.Sharygin)
Three non-collinear points $A_1, A_2, A_3$ are given in a plane. For $n = 4, 5, 6, \ldots$, $A_n$ be the centroid of the triangle $A_{n-3}A_{n-2}A_{n-1}$.
[list]
a) Show that there is exactly one point $S$, which lies in the interior of the triangle $A_{n-3}A_{n-2}A_{n-1}$ for all $n\ge 4$.
b) Let $T$ be the intersection of the line $A_1A_2$ with $SA_3$. Determine the two ratios, $A_1T : TA_2$ and $TS : SA_3$.
[/list]
Let $A$, $B$, $C$, $D$ be four points such that no three are collinear and $D$ is not the orthocenter of $ABC$. Let $P$, $Q$, $R$ be the orthocenters of $\triangle BCD$, $\triangle CAD$, $\triangle ABD$, respectively. Suppose that the lines $AP$, $BQ$, $CR$ are pairwise distinct and are concurrent. Show that the four points $A$, $B$, $C$, $D$ lie on a circle.
[i]Andrew Gu[/i]
Let $ABC$ be an acute triangle with $\angle BAC=30^{\circ}$. The internal and external angle bisectors of $\angle ABC$ meet the line $AC$ at $B_1$ and $B_2$, respectively, and the internal and external angle bisectors of $\angle ACB$ meet the line $AB$ at $C_1$ and $C_2$, respectively. Suppose that the circles with diameters $B_1B_2$ and $C_1C_2$ meet inside the triangle $ABC$ at point $P$. Prove that $\angle BPC=90^{\circ}$ .
Let $ ABCD$ be a rectangle with center $ O$, $ AB\neq BC$. The perpendicular from $ O$ to $ BD$ cuts the lines $ AB$ and $ BC$ in $ E$ and $ F$ respectively. Let $ M,N$ be the midpoints of the segments $ CD,AD$ respectively. Prove that $ FM \perp EN$.
Let $m \neq 0 $ be an integer. Find all polynomials $P(x) $ with real coefficients such that
\[ (x^3 - mx^2 +1 ) P(x+1) + (x^3+mx^2+1) P(x-1) =2(x^3 - mx +1 ) P(x) \]
for all real number $x$.
Let $ABC$ be an acute triangle with $\angle BAC=30^{\circ}$. The internal and external angle bisectors of $\angle ABC$ meet the line $AC$ at $B_1$ and $B_2$, respectively, and the internal and external angle bisectors of $\angle ACB$ meet the line $AB$ at $C_1$ and $C_2$, respectively. Suppose that the circles with diameters $B_1B_2$ and $C_1C_2$ meet inside the triangle $ABC$ at point $P$. Prove that $\angle BPC=90^{\circ}$ .
Given that $i^2=-1$, for how many integers $n$ is $(n+i)^4$ an integer?
$\text{(A)} \ \text{none} \qquad \text{(B)} \ 1 \qquad \text{(C)} \ 2 \qquad \text{(D)} \ 3 \qquad \text{(E)} \ 4$
Prove that there exists a convex 1990-gon with the following two properties :
[b]a.)[/b] All angles are equal.
[b]b.)[/b] The lengths of the 1990 sides are the numbers $ 1^2$, $ 2^2$, $ 3^2$, $ \cdots$, $ 1990^2$ in some order.