On the plane are given four distinct points $A,B,C,D$ on a line $g$ in this order, at the mutual distances $AB = a, BC = b, CD = c$. (a) Construct (if possible) a point $P$ outside line $g$ such that $\angle APB =\angle BPC =\angle CPD$. (b) Prove that such a point $P$ exists if and only if $ (a+b)(b+c) < 4ac$

Let be given an obtuse angle $AKS$ in the plane. Construct a triangle $ABC$ such that $S$ is the midpoint of $BC$ and $K$ is the intersection point of $BC$ with the bisector of $\angle BAC$.

Let $ABC$ be an acute-angled triangle with $AC > AB$, let $O$ be its circumcentre, and let $D$ be a point on the segment $BC$. The line through $D$ perpendicular to $BC$ intersects the lines $AO, AC,$ and $AB$ at $W, X,$ and $Y,$ respectively. The circumcircles of triangles $AXY$ and $ABC$ intersect again at $Z \ne A$. Prove that if $W \ne D$ and $OW = OD,$ then $DZ$ is tangent to the circle $AXY.$

Let triangle $ABC$ be an isosceles triangle with $AB = AC$. Suppose that the angle bisector of its angle $\angle B$ meets the side $AC$ at a point $D$ and that $BC = BD+AD$. Determine $\angle A$.

Let $z$ be a complex number and k a positive integer such that $z^k$ is a positive real number other than $1$. Let $f(n)$ denote the real part of the complex number $z^n$. Assume the parabola $p(n) = an^2 +bn+c$ intersects $f(n)$ four times, at $n = 0, 1, 2, 3$. Assuming the smallest possible value of $k$, find the largest possible value of $a$.

Let $\triangle ABC$ be a triangle, and let $P_1,\cdots,P_n$ be points inside where no three given points are collinear. Prove that we can partition $\triangle ABC$ into $2n+1$ triangles such that their vertices are among $A,B,C,P_1,\cdots,P_n$, and at least $n+\sqrt{n}+1$ of them contain at least one of $A,B,C$.

Let $A_1A_2A_3$ be a triangle and, for $1 \leq i \leq 3$, let $B_i$ be an interior point of edge opposite $A_i$. Prove that the perpendicular bisectors of $A_iB_i$ for $1 \leq i \leq 3$ are not concurrent.

Prove that a triangle with angles $\alpha, \beta, \gamma$, circumradius $R$, and area $A$ satisfies \[\tan \frac{ \alpha}{2}+\tan \frac{ \beta}{2}+\tan \frac{ \gamma}{2} \leq \frac{9R^2}{4A}.\] [hide="Remark."]Remark. Can we determine [i]all[/i] of equality cases ?[/hide]

Let $ABCD$ be a convex quadrilateral. The lines parallel to $AD$ and $CD$ through the orthocentre $H$ of $ABC$ intersect $AB$ and $BC$ Crespectively at $P$ and $Q$. prove that the perpendicular through $H$ to th eline $PQ$ passes through th eorthocentre of triangle $ACD$

Given is an acute angled triangle $ \triangle ABC$ with side lengths $ a$, $ b$ and $ c$ (in an usual way) and circumcenter $ O$. Angle bisector of angle $ \angle BAC$ intersects circumcircle at points $ A$ and $ A_{1}$. Let $ D$ be projection of point $ A_{1}$ onto line $ AB$, $ L$ and $ M$ be midpoints of $ AC$ and $ AB$ , respectively. (i) Prove that $ AD\equal{}\frac{1}{2}(b\plus{}c)$ (ii) If triangle $ \triangle ABC$ is an acute angled prove that $ A_{1}D\equal{}OM\plus{}OL$

In the isosceles triangle $ABC$ with the vertex at the point $B$, the altitudes $BH$ and $CL$ are drawn. The point $D$ is such that $BDCH$ is a rectangle. Find the value of the angle $DLH$. (Bogdan Rublev)

Point $P$ is taken interior to a square with side-length $a$ and such that is it equally distant from two consecutive vertices and from the side opposite these vertices. If $d$ represents the common distance, then $d$ equals: $\textbf{(A)}\ \dfrac{3a}{5} \qquad \textbf{(B)}\ \dfrac{5a}{8} \qquad \textbf{(C)}\ \dfrac{3a}{8} \qquad \textbf{(D)}\ \dfrac{a\sqrt{2}}{2} \qquad \textbf{(E)}\ \dfrac{a}{2}$

A triangle with sides $a$, $b$, and $c$ is given. Denote by $s$ the semiperimeter, that is $s = \frac{a + b + c}{2}$. Construct a triangle with sides $s - a$, $s - b$, and $s - c$. This process is repeated until a triangle can no longer be constructed with the side lengths given. For which original triangles can this process be repeated indefinitely?

In a trapezoid $ABCD$ with bases $AD$ and $BC$, the bisector of the angle $\angle DAB$ intersects the bisectors of the angles $\angle ABC$ and $\angle CDA$ at the points $P$ and $S$, respectively, and the bisector of the angle $\angle BCD$ intersects the bisectors of the angles $\angle ABC$ and $\angle CDA$ at the points $Q$ and $R$, respectively. Prove that if $PS\parallel RQ$, then $AB = CD$.

If $x$,$y$ and $z$ are the lengths of a side, a shortest diagonal and a longest diagonal respectively, of a regular nonagon. Write a correct equation consisting of the three lengths

For a given triangle ABC, let X be a variable point on the line BC such that the point C lies between the points B and X. Prove that the radical axis of the incircles of the triangles ABX and ACX passes through a point independent of X. This is a slight extension of the [url=http://www.mathlinks.ro/Forum/viewtopic.php?t=41033]IMO Shortlist 2004 geometry problem 7[/url] and can be found, together with the proposed solution, among the files uploaded at http://www.mathlinks.ro/Forum/viewtopic.php?t=15622 . Note that the problem was proposed by Russia. I could not find the names of the authors, but I have two particular persons under suspicion. Maybe somebody could shade some light on this... Darij

Let $B = (-1, 0)$ and $C = (1, 0)$ be fixed points on the coordinate plane. A nonempty, bounded subset $S$ of the plane is said to be [i]nice[/i] if $\text{(i)}$ there is a point $T$ in $S$ such that for every point $Q$ in $S$, the segment $TQ$ lies entirely in $S$; and $\text{(ii)}$ for any triangle $P_1P_2P_3$, there exists a unique point $A$ in $S$ and a permutation $\sigma$ of the indices $\{1, 2, 3\}$ for which triangles $ABC$ and $P_{\sigma(1)}P_{\sigma(2)}P_{\sigma(3)}$ are similar. Prove that there exist two distinct nice subsets $S$ and $S'$ of the set $\{(x, y) : x \geq 0, y \geq 0\}$ such that if $A \in S$ and $A' \in S'$ are the unique choices of points in $\text{(ii)}$, then the product $BA \cdot BA'$ is a constant independent of the triangle $P_1P_2P_3$.

Given a triangle $ABC$, incircle $(I)$ touches $BC,CA,AB$ at $D,E,F$ respectively. Let $M$ be a point inside $ABC$. Prove that $M$ lie on $(I)$ if and only if one number among $\sqrt{AE\cdot S_{BMC}},\sqrt{BF\cdot S_{CMA}},\sqrt{CD\cdot S_{AMB}}$ is sum of two remaining numbers ($S_{ABC}$ denotes the area of triangle $ABC$)

$(a)$ A plane $\pi$ passes through the vertex $O$ of the regular tetrahedron $OPQR$. We define $p, q, r$ to be the signed distances of $P,Q,R$ from $\pi$ measured along a directed normal to $\pi$. Prove that \[p^2 + q^2 + r^2 + (q - r)^2 + (r - p)^2 + (p - q)^2 = 2a^2,\] where $a$ is the length of an edge of a tetrahedron. $(b)$ Given four parallel planes not all of which are coincident, show that a regular tetrahedron exists with a vertex on each plane. [u]Note:[/u] Part $(b)$ is [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?f=49&t=60825&start=0]IMO 1972 Problem 6[/url]

Given two positive numbers $a, b$, how many non-congruent plane quadrilaterals are there such that $AB = a$, $BC = CD = DA = b$ and $\angle B = 90^o$ ?

Draw a line passing through a point $M$ on the angle bisector of the angle $\angle AOB$, that intersects $OA$ and $OB$ at points $K$ and $L$ respectively. Prove that the valus of the sum $\frac{1}{|OK|}+\frac{1}{|OL|}$ does not depend on the choice of the straight line passing through $M$, i.e. is defined by the size of the angle AOB and the selection of the point $M$ only.

Let $BE$ and $CF$ be the bisectors of a triangle $ABC$. Prove that $2EF \leq BF + CE$.

Does there exist an equilateral triangle (a) on a plane; (b) in a 3-dimensional space; such that all its three vertices have integral coordinates?

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 $\angle BPC = 90^{\circ}$.

(a) Prove that if the six dihedral (i.e. angles between pairs of faces) of a given tetrahedron are congruent, then the tetrahedron is regular. (b) Is a tetrahedron necessarily regular if five dihedral angles are congruent?