Denote by $ H_n$ the linear space of $ n\times n$ self-adjoint complex matrices, and by $ P_n$ the cone of positive-semidefinite matrices in this space. Let us consider the usual inner product on $ H_n$ \[ \langle A,B\rangle \equal{} {\rm tr} AB\qquad (A,B\in H_n)\] and its derived metric. Show that every $ \phi: P_n\to P_n$ isometry (that is a not necessarily surjective, distance preserving map with respect to the above metric) can be expressed as \[ \phi(A) \equal{} UAU^* \plus{} X\qquad (A\in H_n)\] or \[ \phi(A) \equal{} UA^TU^* \plus{} X\qquad (A\in H_n)\] where $ U$ is an $ n\times n$ unitary matrix, $ X$ is a positive-semidefinite matrix, and $ ^T$ and $ ^*$ denote taking the transpose and the adjoint, respectively.

For $0 \leq d \leq 9$, we define the numbers \[S_{d}=1+d+d^{2}+\cdots+d^{2006}\]Find the last digit of the number \[S_{0}+S_{1}+\cdots+S_{9}.\]

$A,B$, and $C$ are three interior points of a sphere $S$ such that $AB$ and $AC$ are perpendicular to the diameter of $S$ through $A$, and so that two spheres can be constructed through $A$, $B$, and $C$ which are both tangent to $S$. Prove that the sum of their radii is equal to the radius of $S$.

Let $ABCD$ be a parallelogram with $AC=BC.$ A point $P$ is chosen on the extension of ray $AB$ past $B.$ The circumcircle of $ACD$ meets the segment $PD$ again at $Q.$ The circumcircle of triangle $APQ$ meets the segment $PC$ at $R.$ Prove that lines $CD,AQ,BR$ are concurrent.

Consider a triangle $ABC$ and $I$ its incenter. The line $(AI)$ meets the circumcircle of $ABC$ in $D$. Let $E$ and $F$ be the orthogonal projections of $I$ on $(BD)$ and $(CD)$ respectively. Assume that $IE+IF=\frac{1}{2}AD$. Calculate $\angle{BAC}$. [color=red][Moderator edited: Also discussed at http://www.mathlinks.ro/Forum/viewtopic.php?t=5088 .][/color]

Let $ABCD$ be a rhombus such that $m(\widehat{ABC}) = 40^\circ$. Let $E$ be the midpoint of $[BC]$ and $F$ be the foot of the perpendicular from $A$ to $DE$. What is $m(\widehat{DFC})$? $ \textbf{a)}\ 100^\circ \qquad\textbf{b)}\ 110^\circ \qquad\textbf{c)}\ 115^\circ \qquad\textbf{d)}\ 120^\circ \qquad\textbf{e)}\ 135^\circ $

Distinct points $A$ and $B$ are given on the plane. Consider all triangles $ABC$ in this plane on whose sides $BC,CA$ points $D,E$ respectively can be taken so that (i) $\frac{BD}{BC}=\frac{CE}{CA}=\frac{1}{3}$; (ii) points $A,B,D,E$ lie on a circle in this order. Find the locus of the intersection points of lines $AD$ and $BE$.

Given is a trapezoid $ ABCD$ where $ AB$ and $ CD$ are parallel, and $ A,B,C,D$ are clockwise in this order. Let $ \Gamma_1$ be the circle with center $ A$ passing through $ B$, $ \Gamma_2$ be the circle with center $ C$ passing through $ D$. The intersection of line $ BD$ and $ \Gamma_1$ is $ P$ $ ( \ne B,D)$. Denote by $ \Gamma$ the circle with diameter $ PD$, and let $ \Gamma$ and $ \Gamma_1$ meet at $ X$$ ( \ne P)$. $ \Gamma$ and $ \Gamma_2$ meet at $ Y$. If the circumcircle of triangle $ XBY$ and $ \Gamma_2$ meet at $ Q$, prove that $ B,D,Q$ are collinear.

Find the largest possible number of intersection points of the diagonals of a convex $n$-gon.

In a convex hexagon $ABCDEF$, triangles $ACE$ and $BDF$ have the same circumradius $R$. If triangle $ACE$ has inradius $r$, prove that \[ \text{Area}(ABCDEF)\le\frac{R}{r}\cdot\text{Area}(ACE).\]

$\Delta ABC$ and its mirror reflection $\Delta A^{\prime}B^{\prime}C^{\prime}$ is arbitrarily placed on the plane. Prove the midpoints of $AA^{\prime},BB^{\prime},CC^{\prime}$ are collinear.

We define a relation between subsets of $\mathbb R ^n$. $A \sim B\Longleftrightarrow$ we can partition $A,B$ in sets $A_1,\dots,A_n$ and $B_1,\dots,B_n$(i.e $\displaystyle A=\bigcup_{i=1} ^n A_i,\ B=\bigcup_{i=1} ^n B_i, A_i\cap A_j=\emptyset,\ B_i\cap B_j=\emptyset$) and $A_i\simeq B_i$. Say the the following sets have the relation $\sim$ or not ? a) Natural numbers and composite numbers. b) Rational numbers and rational numbers with finite digits in base 10. c) $\{x\in\mathbb Q|x<\sqrt 2\}$ and $\{x\in\mathbb Q|x<\sqrt 3\}$ d) $A=\{(x,y)\in\mathbb R^2|x^2+y^2<1\}$ and $A\setminus \{(0,0)\}$

Consider an acute triangle $ABC$, points $E,F$ are the feet of the perpendiculars from $B$ and $C$ in $\triangle ABC$. Points $I$ and $J$ are the projections of points $F,E$ on the line $BC$, points $K,L$ are on sides $AB,AC$ respectively such that $IK \parallel AC$ and $JL \parallel AB$. Prove that the lines $IE$,$JF$,$KL$ are concurrent.

Let $ABCD$ be a cyclic quadrilateral with circumcenter $O$. Let the internal angle bisectors at $A$ and $B$ meet at $X$, the internal angle bisectors at $B$ and $C$ meet at $Y$, the internal angle bisectors at $C$ and $D$ meet at $Z$, and the internal angle bisectors at $D$ and $A$ meet at $W$. Further, let $AC$ and $BD$ meet at $P$. Suppose that the points $X$, $Y$, $Z$, $W$, $O$, and $P$ are distinct. Prove that $O$, $X$, $Y$, $Z$, $W$ lie on the same circle if and only if $P$, $X$, $Y$, $Z$, and $W$ lie on the same circle.

Given a parallelogram $ABCD$ with $AB<BC$, show that the circumcircles of the triangles $APQ$ share a second common point (apart from $A$) as $P,Q$ move on the sides $BC,CD$ respectively s.t. $CP=CQ$.

Triangles $ABC$ and $DEF$, having a common incircle of radius $R$, intersect at points $X_1, X_2, \ldots , X_6$ and form six triangles (see the figure below). Let $r_1, r_2,\ldots, r_6$ be the radii of the inscribed circles of these triangles, and let $R_1, R_2, \ldots , R_6$ be the radii of the inscribed circles of the triangles $AX_1F, FX_2B, BX_3D, DX_4C, CX_5E$ and $EX_6A$ respectively. [img]https://i.ibb.co/BspgdHB/Image.jpg[/img] Prove that \[ \sum_{i=1}^{6} \frac{1}{r_i} < \frac{6}{R}+\sum_{i=1}^{6} \frac{1}{R_i} \] [i]U. Maksimenkau[/i]

In space, there are \( 13 \) points, no four of which lie in the same plane. Three of the points are colored blue, and the triangle with these points as vertices will be called a [i]blue triangle[/i]. The remaining \( 10 \) points are colored red. We say that a triangle with three red vertices is [i]attached[/i] to the blue triangle if the boundary of the red triangle intersects the blue triangle (either in its interior or on its boundary) at exactly one point. Is it possible for the number of attached triangles to be \( 33 \)?

From a point $C$, tangents $CA$ and $CB$ are drawn to a circle $O$. From an arbitrary point $N$ on the circle, perpendiculars $ND, NE, NF$ are drawn on $AB, CA$ and $CB$, respectively. Prove that the length of $ND$ is the mean proportional of the lengths of $NE$ and $NF$.

In a rectangle $ABCD, E$ is the midpoint of $AB, F$ is a point on $AC$ such that $BF$ is perpendicular to $AC$, and $FE$ perpendicular to $BD$. Suppose $BC = 8\sqrt3$. Find $AB$.

To two circles $r_1$ and $r_2$, intersecting at points $A$ and $B$, their common tangent $CD$ is drawn ($C$ and $D$ are tangency points, respectively, point $B$ is closer to line $CB$ than $A$). Line passing through $A$ , intersects $r_1$ and $r_2$ for second time at points $K$ and $L$, respectively ($A$ lies between $K$ and $L$). Lines $KC$ and $LD$ intersect at point $P$. Prove that $PB$ is the symmedian of triangle $KPL$. (Yu. Blinkov)

Let $ABC$ be acute, non-isosceles triangle, inscribed in the circle $(O)$. Let $D$ be perpendicular projection of $A$ onto $BC$, and $E, F$ be perpendicular projections of $D$ onto $CA,AB$ respectively. (a) Prove that $AO \perp EF$. (b) The line $AO$ intersects $DE,DF$ at $I,J$ respectively. Prove that $\vartriangle DIJ$ and $\vartriangle ABC$ are similar. (c) Prove that circumcenter of $\vartriangle DIJ$ is equidistant from $B$ and $C$

Let $V$ be a point in the exterior of a circle of center $O$, and let $T_1,T_2$ be the points where the tangents from $V$ touch the circle. Let $T$ be an arbitrary point on the small arc $T_1T_2$. The tangent in $T$ at the circle intersects the line $VT_1$ in $A$, and the lines $TT_1$ and $VT_2$ intersect in $B$. We denote by $M$ the intersection of the lines $TT_1$ and $AT_2$. Prove that the lines $OM$ and $AB$ are perpendicular.

In acute triangle $\triangle ABC$, $BC>AC$, $\Gamma$ is its circumcircle, $D$ is a point on segment $AC$ and $E$ is the intersection of the circle with diameter $CD$ and $\Gamma$. $M$ is the midpoint of $AB$ and $CM$ meets $\Gamma$ again at $Q$. The tangents to $\Gamma$ at $A,B$ meet at $P$, and $H$ is the foot of perpendicular from $P$ to $BQ$. $K$ is a point on line $HQ$ such that $Q$ lies between $H$ and $K$. Prove that $\angle HKP=\angle ACE$ if and only if $\frac{KQ}{QH}=\frac{CD}{DA}$.

In a triangle ABC we have $\angle C = \angle A + 90^o$. The point $D$ on the continuation of $BC$ is given such that $AC = AD$. A point $E$ in the side of $BC$ in which $A$ doesn’t lie is chosen such that $\angle EBC = \angle A, \angle EDC = \frac{1}{2} \angle A$ . Prove that $\angle CED = \angle ABC$. by Morteza Saghafian

A hexahedron $ABCDE$ is made of two regular congruent tetrahedra $ABCD$ and $ABCE.$ Prove that there exists only one isometry $\mathbf Z$ that maps points $A, B, C, D, E$ onto $B, C, A, E, D,$ respectively. Find all points $X$ on the surface of hexahedron whose distance from $\mathbf Z(X)$ is minimal.