Let $ABC$ be a non-isosceles triangle with circumcircle $k$, incenter $I$ and $C$-excenter $I_C$. Let $M$ be the midpoint of $AB$ and $N$ be the midpoint of arc $\widehat{ACB}$ on $k$. Prove that $\angle IMI_C + \angle INI_C = 180^{\circ}$.

A scalene acute triangle $ABC$ is drawn on the plane, in which $BC$ is the longest side. Points $P$ and $D$ are constructed, the first inside $ABC$ and the second outside, so that $\angle ABC = \angle CBD$, $\angle ACP = \angle BCD$ and that the area of triangle $ABC$ is equal to the area of quadrilateral $BPCD$. Prove that triangles $BCD$ and $ACP$ are similar.

Let $ABC$ be a triangle and $D, E, F$ be the foots of altitudes drawn from $A,B,C$ respectively. Let $H$ be the orthocenter of $ABC$. Lines $EF$ and $AD$ intersect at $G$. Let $K$ the point on circumcircle of $ABC$ such that $AK$ is a diameter of this circle. $AK$ cuts $BC$ in $M$. Prove that $GM$ and $HK$ are parallel.

Three pairwise intersecting rays are given. At some point in time not on every ray from its beginning a point begins to move with speed. It is known that these three points form a triangle at any time, and the center of the circumscribed circle of this the triangle also moves uniformly and on a straight line. Is it true, that all these triangles are similar to each other?

Given a regular pentagon, consider the convex pentagon limited by its diagonals. You are asked to calculate: a) The similarity relation between the two convex pentagons. b) The relationship of their areas. c) The ratio of the homothety that transforms the first into the second.

Let $E$ be an interior point of the convex quadrilateral $ABCD$. Construct triangles $\triangle ABF,\triangle BCG,\triangle CDH$ and $\triangle DAI$ on the outside of the quadrilateral such that the similarities $\triangle ABF\sim\triangle DCE,\triangle BCG\sim \triangle ADE,\triangle CDH\sim\triangle BAE$ and $ \triangle DAI\sim\triangle CBE$ hold. Let $P,Q,R$ and $S$ be the projections of $E$ on the lines $AB,BC,CD$ and $DA$, respectively. Prove that if the quadrilateral $PQRS$ is cyclic, then \[EF\cdot CD=EG\cdot DA=EH\cdot AB=EI\cdot BC.\]

The diagonals of a quadrilateral $ABCD$ are perpendicular: $AC\perp BD$. Four squares, $ABEF,BCGH,CDIJ,DAKL$, are erected externally on its sides. The intersection points of the pairs of straight lines $CL,DF; DF,AH; AH,BJ; BJ,CL$ are denoted by $P_1,Q_1,R_1, S_1$, respectively, and the intersection points of the pairs of straight lines $AI,BK; BK,CE;$ $ CE,DG; DG,AI$ are denoted by $P_2,Q_2,R_2, S_2$, respectively. Prove that $P_1Q_1R_1S_1 \cong P_2Q_2R_2S_2.$

Four circles are drawn with the sides of quadrilateral $ABCD$ as diameters. The two circles passing through $A$ meet again at $A'$, two circles through $B$ at $B'$ , two circles at $C$ at $C'$ and the two circles at $D$ at $D'$. Suppose the points $A',B',C'$ and $D'$ are distinct. Prove quadrilateral $A'B'C'D'$ is similar to $ABCD$.

Two tetrahedrons are given. Each two faces of the same tetrahedron are not similar, but each face of the first tetrahedron is similar to some face of the second one. Does this yield that these tetrahedrons are similar?

Let $ABC$ be a triangle, $I$ its incenter, and $\Gamma$ its circumcircle. Let $D$ be the second point of intersection of $AI$ with $\Gamma$. The line parallel to $BC$ through $I$ intersects $AB$ and $AC$ at $P$ and $Q$, respectively. The lines $PD$ and $QD$ intersect $BC$ at $E$ and $F$, respectively. Prove that triangles $IEF$ and $ABC$ are similar.

We assign to any quadrilateral $ ABCD $ the centers of the circles circumscribed in the triangles $ BCD $, $ CDA $, $ DAB $, $ ABC $. Prove that if the vertices of a convex quadrilateral $ Q $ do not lie on the circle, then a) the four points assigned to quadrilateral Q in the above manner are the vertices of the convex quadrilateral. Let us denote this quadrilateral by $ t(Q) $, b) the vertices of the quadrilateral $ t(Q) $ do not lie on the circle, c) quadrilaterals $ Q $ and $ t(t(Q) $ are similar.

For a triangle $T = ABC$ we take the point $X$ on the side $(AB)$ such that $AX/AB=4/5$, the point $Y$ on the segment $(CX)$ such that $CY = 2YX$ and, if possible, the point $Z$ on the ray ($CA$ such that $\widehat{CXZ} = 180 - \widehat{ABC}$. We denote by $\Sigma$ the set of all triangles $T$ for which $\widehat{XYZ} = 45$. Prove that all triangles from $\Sigma$ are similar and find the measure of their smallest angle.

Given a positive integer $n > 1$ and an angle $\alpha < 90^o$, Jaime draws a spiral $OP_0P_1...P_n$ of the following form (the figure shows the first steps): $\bullet$ First draw a triangle $OP_0P_1$ with $OP_0 = 1$, $\angle P_1OP_0 = \alpha$ and $P_1P_0O = 90^o$ $\bullet$ then for every integer $1 \le i \le n$ draw the point $P_{i+1}$ so that $\angle P_{i+1}OP_i = \alpha$, $\angle P_{i+1}P_iO = 90^o$ and $P_{i-1}$ and $P_{i+1}$ are in different half-planes with respect to the line $OP_i$ [img]https://cdn.artofproblemsolving.com/attachments/f/2/aa3913989dac1cf04f2b42b5d630b2e096dcb6.png[/img] a) If $n = 6$ and $\alpha = 30^o$, find the length of $P_0P_n$. b) If $n = 2018$ and $\alpha= 45^o$, find the length of $P_0P_n$.

$AB,AC$ are tangent to $\Omega$ at $B$ and $C$, respectively. $D,E,F$ lie on segments $BC,CA,AB$ such that $AF<AE$ and $\angle FDB= \angle EDC$. The circumcircle of $\triangle FEC$ intersects $\Omega$ at $G$ and $C$. Show that $ \angle AEF= \angle BGD$

Let $ABCD$ be cyclic quadrilateral. Let $AC$ and $BD$ intersect at $R$, and let $AB$ and $CD$ intersect at $K$. Let $M$ and $N$ are points on $AB$ and $CD$ such that $\frac{AM}{MB}=\frac{CN}{ND}$. Let $P$ and $Q$ be the intersections of $MN$ with the diagonals of $ABCD$. Prove that circumcircles of triangles $KMN$ and $PQR$ are tangent at a fixed point.

Let $ABCD$ be a convex plane quadrilateral and let $A_1$ denote the circumcenter of $\triangle BCD$. Define $B_1, C_1,D_1$ in a corresponding way. (a) Prove that either all of $A_1,B_1, C_1,D_1$ coincide in one point, or they are all distinct. Assuming the latter case, show that $A_1$, C1 are on opposite sides of the line $B_1D_1$, and similarly,$ B_1,D_1$ are on opposite sides of the line $A_1C_1$. (This establishes the convexity of the quadrilateral $A_1B_1C_1D_1$.) (b) Denote by $A_2$ the circumcenter of $B_1C_1D_1$, and define $B_2, C_2,D_2$ in an analogous way. Show that the quadrilateral $A_2B_2C_2D_2$ is similar to the quadrilateral $ABCD.$

Let $ABC$ be an acute scalene triangle inscribed in a circle \( \Gamma \). The angle bisector of \( \angle BAC \) intersects \( BC \) at \( L \) and \( \Gamma \) at \( S \). The point \( M \) is the midpoint of \( AL \). Let \( AD \) be the altitude in \( \triangle ABC \), and the circumcircle of \( \triangle DSL \) intersects \( \Gamma \) again at \( P \). Let \( N \) be the midpoint of \( BC \), and let \( K \) be the reflection of \( D \) with respect to \( N \). Prove that the triangles \( \triangle MPS \) and \( \triangle ADK \) are similar.

Let $D$ be a point in the interior of $\triangle{ABC}$ such that $AB=ab$, $AC=ac$, $BC=bc$, $AD=ad$, $BD=bd$, $CD=cd$. Show that $\angle{ABD}+\angle{ACD}=60^{\circ}$. Source: 2016 Singapore Mathematical Olympiad (Open) Round 2, Problem 1

Show that a convex polygon with more than four sides cannot be decomposed into two others, both similar to the first (directly or inversely), by means of a single rectilinear cut. Reasonably specify which are the quadrilaterals and triangles that admit a decomposition of this type.

Let $C_1$ and $C_2$ be two circles with centers $O_1$ and $O_2$, respectively, intersecting at $A$ and $B$. Let $l_1$ be the line tangent to $C_1$ passing trough $A$, and $l_2$ the line tangent to $C_2$ passing through $B$. Suppose that $l_1$ and $l_2$ intersect at $P$ and $l_1$ intersects $C_2$ again at $Q$. Show that $PO_1B$ and $PO_2Q$ are similar triangles. [i]Proposed by Pablo Valeriano[/i]

Let $U$ be the center of the circumcircle of the acute-angled triangle $ABC$. Let $M_A, M_B$ and $M_C$ be the circumcenters of triangles $UBC, UAC$ and $UAB$ respecrively. For which triangles $ABC$ is the triangle $M_AM_BM_C$ similar to the starting triangle (with a suitable order of the vertices)?

Let $ABC$ be an acute triangle and $D$ be the foot of the altitude from $A$ onto $BC$. A semicircle with diameter $BC$ intersects segments $AB, AC$ and $AD$ in the points $F, E$ and $X$, respectively. The circumcircles of the triangles $DEX$ and $DXF$ intersect $BC$ in $L$ and $N$, respectively, other than $D$. Prove that $BN = LC$.

Decide, whether exist tetrahedrons $T$, $T'$ with walls $S_1$, $S_2$, $S_3$, $S_4$ and $S_1'$, $S_2'$, $S_3'$, $S_4'$, respectively, such that for $i = 1, 2, 3, 4$ triangle $S_i$ is similar to triangle $S_i'$, but despite this, tetrahedron $T$ is not similar to tetrahedron $T'$.

Let \( M \) be the midpoint of side \( BC \) of triangle \( ABC \), and let \( D \) be an arbitrary point on the arc \( BC \) of the circumcircle that does not contain \( A \). Let \( N \) be the midpoint of \( AD \). A circle passing through points \( A \), \( N \), and tangent to \( AB \) intersects side \( AC \) at point \( E \). Prove that points \( C \), \( D \), \( E \), and \( M \) are concyclic. [i]Proposed by Matthew Kurskyi[/i]