Let $\Delta ABC$ be an isosceles triangle with $BC = AC$ and $\angle ACB = 40^{\circ}$. Contruct the circle with diameter $\overline{BC}$, and let $D$ and $E$ be the other intersection points of the circle with the sides $\overline{AC}$ and $\overline{AB}$, respectively. Let $F$ be the intersection of the diagonals of the quadrilateral $BCDE$. What is the degree measure of $\angle BFC ?$ $\textbf{(A) } 90 \qquad\textbf{(B) } 100 \qquad\textbf{(C) } 105 \qquad\textbf{(D) } 110 \qquad\textbf{(E) } 120$

Let $ABC$ be an acute triangle with orthocenter $H$. Let $G$ be the point such that the quadrilateral $ABGH$ is a parallelogram. Let $I$ be the point on the line $GH$ such that $AC$ bisects $HI$. Suppose that the line $AC$ intersects the circumcircle of the triangle $GCI$ at $C$ and $J$. Prove that $IJ = AH$.

A [i]super-integer[/i] triangle is defined to be a triangle whose lengths of all sides and at least one height are positive integers. We will deem certain positive integer numbers to be [i]good[/i] with the condition that if the lengths of two sides of a super-integer triangle are two (not necessarily different) good numbers, then the length of the remaining side is also a good number. Let $5$ be a good number. Prove that all integers larger than $2$ are good numbers.

Let $ABC$ be an acute triangle with circumcenter $O$ and circumradius $R$. $AO$ meets the circumcircle of $BOC$ at $A'$, $BO$ meets the circumcircle of $COA$ at $B'$ and $CO$ meets the circumcircle of $AOB$ at $C'$. Prove that \[OA'\cdot OB'\cdot OC'\geq 8R^{3}.\] Sorry if this has been posted before since this is a very classical problem, but I failed to find it with the search-function.

Eva draws an equilateral triangle and its altitudes. In a first step she draws the center triangle of the equilateral triangle, in a second step the center triangle of this center triangle and so on. After each step Eva counts all triangles whose sides lie completely on drawn lines. What is the minimum number of center triangles she must have drawn so that the figure contains more than 2022 such triangles?

A point $D$ is chosen on the side $AC$ of a triangle $ABC$ with $\angle C < \angle A < 90^\circ$ in such a way that $BD=BA$. The incircle of $ABC$ is tangent to $AB$ and $AC$ at points $K$ and $L$, respectively. Let $J$ be the incenter of triangle $BCD$. Prove that the line $KL$ intersects the line segment $AJ$ at its midpoint.

Let ABC be a triangle and $M$ be an interior point. Prove that \[ \min\{MA,MB,MC\}+MA+MB+MC<AB+AC+BC.\]

Let $\pi$ be the set of points in a plane and $f : \pi \to \pi$ be a mapping such that the image of any triangle (as its polygonal line) is a square. Show that $f(\pi)$ is a square.

Let $ABC$ be a triangle with semiperimeter $s$ and inradius $r$. The semicircles with diameters $BC$, $CA$, $AB$ are drawn on the outside of the triangle $ABC$. The circle tangent to all of these three semicircles has radius $t$. Prove that \[\frac{s}{2}<t\le\frac{s}{2}+\left(1-\frac{\sqrt{3}}{2}\right)r. \] [i]Alternative formulation.[/i] In a triangle $ABC$, construct circles with diameters $BC$, $CA$, and $AB$, respectively. Construct a circle $w$ externally tangent to these three circles. Let the radius of this circle $w$ be $t$. Prove: $\frac{s}{2}<t\le\frac{s}{2}+\frac12\left(2-\sqrt3\right)r$, where $r$ is the inradius and $s$ is the semiperimeter of triangle $ABC$. [i]Proposed by Dirk Laurie, South Africa[/i]

Let $ABC$ be a triangle with semiperimeter $s$ and inradius $r$. The semicircles with diameters $BC$, $CA$, $AB$ are drawn on the outside of the triangle $ABC$. The circle tangent to all of these three semicircles has radius $t$. Prove that \[\frac{s}{2}<t\le\frac{s}{2}+\left(1-\frac{\sqrt{3}}{2}\right)r. \] [i]Alternative formulation.[/i] In a triangle $ABC$, construct circles with diameters $BC$, $CA$, and $AB$, respectively. Construct a circle $w$ externally tangent to these three circles. Let the radius of this circle $w$ be $t$. Prove: $\frac{s}{2}<t\le\frac{s}{2}+\frac12\left(2-\sqrt3\right)r$, where $r$ is the inradius and $s$ is the semiperimeter of triangle $ABC$. [i]Proposed by Dirk Laurie, South Africa[/i]

Let $ABC$ be a triangle and $P$ an exterior point in the plane of the triangle. Suppose the lines $AP$, $BP$, $CP$ meet the sides $BC$, $CA$, $AB$ (or extensions thereof) in $D$, $E$, $F$, respectively. Suppose further that the areas of triangles $PBD$, $PCE$, $PAF$ are all equal. Prove that each of these areas is equal to the area of triangle $ABC$ itself.

Find the locus of centroids of equilateral triangles whose vertices lie on sides of a given square $ABCD$.

Prove that we can put in any arbitrary triangle with sidelengths $a,b,c$ such that $0\le a,b,c \le \sqrt2$ into a unit cube.

Given a triangle $ABC$. The points $A$, $B$, $C$ divide the circumcircle $\Omega$ of the triangle $ABC$ into three arcs $BC$, $CA$, $AB$. Let $X$ be a variable point on the arc $AB$, and let $O_{1}$ and $O_{2}$ be the incenters of the triangles $CAX$ and $CBX$. Prove that the circumcircle of the triangle $XO_{1}O_{2}$ intersects the circle $\Omega$ in a fixed point.

A square $ABCD,AB=s=1$ is given in the plane with its center $S$. Furthermore, points $E,F$ are given on the rays opposite to $CB,DA$, respectively, $CE=a,DF=b$. Determine all triangles $XYZ$ such that $X,Y,Z$ lie in this order on segments $CD,AD,BC$ and $E,S,F$ lie on lines $XY,YZ,ZX$ respectively. Discuss conditions of solvability in terms of $a,b,s$ and unknown $x=CX$.

In the triangle $ABC$, let $A'$ be the intersection of the perpendicular bisector of $AB$ and the angle bisector of $\angle BAC$ and define $B', C'$ analogously. Prove that a) The triangle $ABC$ is equilateral if and only if $A' =B'.$ b) If $A', B'$ and $C'$ are distinct, we have $\angle B' A' C' = 90^{\circ} - \frac{1}{2} \angle BAC.$

Point $A$ lies outside circle $o$ of center $O$. From point $A$ draw two lines tangent to a circle $o$ in points $B$ and $C$. A tangent to a circle $o$ cuts segments $AB$ and $AC$ in points $E$ and $F$, respectively. Lines $OE$ and $OF$ cut segment $BC$ in points $P$ and $Q$, respectively. Prove that from line segments $BP$, $PQ$, $QC$ can construct triangle similar to triangle $AEF$.

Given $100$ coplanar points, no three collinear, prove that at most $70\%$ of the triangles formed by the points have all angles acute.

Let $M$ be a point on the altitude $CD$ of an acute-angled triangle $ABC$, and $K$ and $L$ the orthogonal projections of $M$ on $AC$ and $BC$. Suppose that the incenter and circumcenter of the triangle lie on the segment $KL$. (a) Prove that $CD=R+r$, where $R$ and $r$ are the circumradius and inradius, respectively. (b) Find the minimum value of the ratio $CM:CD$.

Points $A, S$ are given in plane such that $AS = a > 0$ as well as positive numbers $b, c$ satisfying $b < a < c$. Construct an equilateral triangle $ABC$ with the property $BS = b$, $CS = c$. Discuss conditions of solvability.

Let $P$ be a point inside triangle $ABC$. Let $AP$ meet $BC$ at $A_1$, let $BP$ meet $CA$ at $B_1$, and let $CP$ meet $AB$ at $C_1$. Let $A_2$ be the point such that $A_1$ is the midpoint of $PA_2$, let $B_2$ be the point such that $B_1$ is the midpoint of $PB_2$, and let $C_2$ be the point such that $C_1$ is the midpoint of $PC_2$. Prove that points $A_2, B_2$, and $C_2$ cannot all lie strictly inside the circumcircle of triangle $ABC$. (Australia)

Given $100$ coplanar points, no three collinear, prove that at most $70\%$ of the triangles formed by the points have all angles acute.

Let $A_1$ be the center of the square inscribed in acute triangle $ABC$ with two vertices of the square on side $BC$. Thus one of the two remaining vertices of the square is on side $AB$ and the other is on $AC$. Points $B_1,\ C_1$ are defined in a similar way for inscribed squares with two vertices on sides $AC$ and $AB$, respectively. Prove that lines $AA_1,\ BB_1,\ CC_1$ are concurrent.

In all triangles $ABC$ does it hold that: $$\sum\sin^2\frac A2\cos^2A\ge\frac{3\left(s^2-(2R+r)^2\right)}{8R^2}$$ [i]Proposed by Mihály Bencze and Marius Drăgan[/i]

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]