### Oral Moscow Geometry 2012-9 102p

geometry problems from Oral Moscow* Geometry Olympiad
with aops links in the names

mistakenly it was called by me Oral Sharygin Geometry Olympiad,
but it is a Qualification / Regional Round that happens in Moscow

2012 - 2019
the previous years are under construction

In trapezoid $ABCD$, the sides $AD$ and $BC$ are parallel, and $AB = BC = BD$. The height $BK$ intersects the diagonal $AC$ at $M$. Find $\angle CDM$.

Two equal polygons $F$ and $F'$ are given on the plane. It is known that the vertices of the polygon $F$ belong to $F'$ (may lie inside it or on the border). Is it true that all the vertices of these polygons coincide?

Given an equilateral triangle $ABC$ and a straight line $\ell$, passing through its center. Intersection points of this line with sides $AB$ and $BC$ are reflected wrt to the midpoints of these sides respectively. Prove that the line passing through the resulting points, touches the inscribed circle triangle $ABC$.

In triangle $ABC$, point $I$ is the center of the inscribed circle points, points $I_A$ and $I_C$ are the centers of the excircles, tangent to sides $BC$ and $AB$, respectively. Point $O$ is the center of the circumscribed circle of triangle $II_AI_C$. Prove that $OI \perp AC$

Given a circle and a chord $AB$, different from the diameter. Point $C$ moves along the large arc $AB$. The circle passing through passing through points $A, C$ and point $H$ of intersection of heights of of the triangle $ABC$, re-intersects the line $BC$ at point $P$. Prove that line $PH$ passes through a fixed point independent of the position of point $C$.

Restore the triangle with a compass and a ruler given the point of intersection of heights and the bases of the median and bisectors drawn to one side. (No research required.)

Is it true that the center of the inscribed circle of the triangle lies inside the triangle formed by the lines of connecting it's midpoints?

In the convex pentagon $ABCDE$: $\angle A = \angle C = 90^o$, $AB = AE, BC = CD, AC = 1$. Find the area of the pentagon.

$H$ is the intersection point of the heights $AA'$ and $BB'$ of the acute-angled triangle $ABC$. A straight line, perpendicular to $AB$, intersects these heights at points $D$ and $E$, and side $AB$ at point $P$. Prove that the orthocenter of the triangle $DEH$ lies on segment $CP$.

Inside the convex polyhedron, the point $P$ and several lines $\ell_1,\ell_2, ..., \ell_n$ passing through $P$ and not lying in the same plane. To each face of the polyhedron we associate one of the lines $l_1, l_2, ..., l_n$ that forms the largest angle with the plane of this face (if there are there are several direct ones, we will choose any of them). Prove that there is a face that intersects with its corresponding line.

Inside the circle with center $O$, points $A$ and $B$ are marked so that $OA = OB$. Draw a point $M$ on the circle from which the sum of the distances to points $A$ and $B$ is the smallest among all possible.

Tangents drawn to the circumscribed circle of an acute-angled triangle $ABC$ at points $A$ and $C$, intersect at point $Z$. Let $AA_1, CC_1$ be heights. Line $A_1C_1$ intersects $ZA, ZC$ at points $X$ and $Y$, respectively. Prove that the circumscribed circles of the triangles $ABC$ and $XYZ$ are tangent.

In triangle $ABC$ the bisector $AK$ is perpendicular on the median is $CL$. Prove that in the triangle $BKL$ also one of bisectors are perpendicular to one of the medians.

With a compass and a ruler, split a triangle into two smaller triangles with the same sum of squares of sides.

The bisectors $AA_1$ and $CC_1$ of the right triangle $ABC$ ($\angle B = 90^o$) intersect at point $I$. The line passing through the point $C_1$ and perpendicular on the line $AA_1$ intersects the line that passes through $A_1$ and is perpendicular on $CC_1$, at the point $K$. Prove that the midpoint of the segment $KI$ lies on segment $AC$.

Let $ABC$ be a triangle. On the extensions of sides $AB$ and $CB$ towards $B$, points $C_1$ and $A_1$ are taken, respectively, so that $AC = A_1C = AC_1$. Prove that circumscribed circles of triangles $ABA_1$ and $CBC_1$ intersect on the bisector of angle $B$.

In triangle $ABC, \angle C= 60^o, \angle A= 45^o$. Let $M$ be the midpoint of $BC, H$ be the orthocenter of triangle $ABC$. Prove that line $MH$ passes through the midpoint of arc $AB$ of the circumcircle of triangle $ABC$.

Let $ABC$ be a triangle. On its sides $AB$ and $BC$ are fixed points $C_1$ and $A_1$, respectively. Find a point$P$ on the circumscribed circle of triangle $ABC$ such that the distance between the centers of the circumscribed circles of the triangles $APC_1$ and $CPA_1$ is minimal.

Diagonals of an cyclic quadrilateral $ABCD$ intersect at point $O$. The circumscribed circles of triangles $AOB$ and $COD$ intersect at point $M$ on the $AD$ side. Prove that the point $O$ is the center of the inscribed circle of the triangle $BMC$.

Inside the angle $AOD$, the rays $OB$ and $OC$ are drawn  such that $\angle AOB = \angle COD.$ Two circles are inscribed inside the angles $\angle AOB$ and $\angle COD$ . Prove that the intersection point of the common internal tangents of these circles lies on the bisector of the angle $AOD$.

Is there a polyhedron whose area ratio of any two faces is at least $2$ ?

Similar triangles $ABM, CBP, CDL$ and $ADK$ are built on the sides of the quadrilateral $ABCD$ with perpendicular diagonals in the outer side (the neighboring ones are oriented differently). Prove that $PK = ML$.

In the acute-angled triangle $ABC$, let  $AP$ and $BQ$ be the heights,  $CM$ be the median . Point $R$ is the midpoint of $CM$. Line $PQ$ intersects line $AB$ at $T$. Prove that $OR \perp TC$, where $O$ is the center of the circumscribed circle of triangle $ABC$.

The trapezoid $ABCD$ is inscribed in the circle $w$ ($AD // BC$). The circles inscribed in the triangles $ABC$ and $ABD$ touch the base of the trapezoid $BC$ and  $AD$ at points $P$ and $Q$ respectively. Points $X$ and $Y$ are the midpoints of the arcs $BC$ and $AD$ of circle $w$ that do not contain points $A$ and $B$ respectively. Prove that lines $XP$ and $YQ$ intersect on the circle $w$.

In triangle $ABC, \angle A= 45^o, BH$ is the height, the point $K$ lies on the $AC$ side, and $BC = CK$. Prove that the center of the circumscribed circle of triangle $ABK$ coincides with the center of an excircle of triangle $BCH$.

Let $ABCD$ be a parallelogram. On side $AB$, point $M$ is taken so that $AD = DM$.
On side $AD$ point $N$ is taken so that $AB = BN$. Prove that $CM = CN$.

Is there a convex pentagon in which each diagonal is equal to a side?

In triangle $ABC$, the perpendicular bisectors of sides $AB$ and $BC$ intersect side $AC$ at points $P$ and $Q$, respectively, with point $P$ lying on the segment $AQ$. Prove that the circumscribed circles of the triangles $PBC$ and $QBA$ intersect on the bisector of the angle $PBQ$.

Segment $AD$ is the diameter of the circumscribed circle of an acute-angled triangle $ABC$. Through the intersection of the heights of this triangle, a straight line was drawn parallel to the side $BC$, which intersects sides $AB$ and $AC$ at points $E$ and $F$, respectively. Prove that the perimeter of the triangle $DEF$ is two times larger than the side $BC$.

Inside an isosceles right triangle $ABC$ with hypotenuse $AB$ a point $M$ is taken such that the angle $MAB$ is $15 ^o$ larger than the $MAC$ angle, and the $MCB$ angle is $15^o$ larger than the angle $MBC$. Find the $BMC$ angle.

In trapezoid $ABCD$: $BC <AD, AB = CD, K$ is midpoint of $AD, M$ is midpoint of $CD, CH$ is height. Prove that lines $AM, CK$ and $BH$ intersect at one point.

Is it possible to cut a regular triangular prism into two equal pyramids?

The bisectors $AA_1$ and $CC_1$ of triangle $ABC$ intersect at point $I$. The circumscribed circles of triangles $AIC_1$ and $CIA_1$ intersect the arcs $AC$ and $BC$ (not containing points $B$ and $A$ respectively) of the circumscribed circle of triangle $ABC$ at points $C_2$ and $A_2$, respectively. Prove  that lines $A_1A_2$ and $C_1C_2$ intersect on the circumscribed circle of triangle $ABC$.

The medians $AA_0, BB_0$, and $CC_0$ of the acute-angled triangle $ABC$ intersect at the point $M$, and heights $AA_1, BB_1$ and $CC_1$  at point $H$. Tangent to the circumscribed circle of triangle $A_1B_1C_1$ at $C_1$ intersects the line $A_0B_0$ at the point C'. Points $A'$  and $B'$ are defined similarly. Prove that $A', B'$  and $C'$ lie on one line perpendicular to the line $MH$.

Given a regular triangle $ABC$, whose area is $1$, and the point $P$ on its circumscribed circle. Lines $AP, BP, CP$ intersect, respectively, lines $BC, CA, AB$ at points $A', B', C'$. Find the area of the triangle $A'B'C'$.

A convex quadrangle $ABCD$ is given. Let $I$ and $J$ be the circles of circles inscribed in the triangles $ABC$ and $ADC$, respectively, and $I_a$ and $J_a$ are the centers of the excircles circles of triangles $ABC$ and $ADC$, respectively (inscribed in the angles $BAC$ and $DAC$, respectively). Prove that the intersection point $K$ of the lines $IJ_a$ and $JI_a$ lies on the bisector of the angle $BCD$.

In triangle $ABC$, the height $AH$ passes through middle of the median $BM$. Prove that in the triangle $BMC$ also one of the heights passes through the middle of one of the medians.

The square $ABCD$ and the equilateral triangle $MKL$ are located as shown in the figure. Find the angle $PQD$.

In triangle $ABC$, points $D, E$, and $F$ are marked on sides $AC, BC$, and $AB$ respectively, so that $AD = AB, EC = DC, BF = BE$. After that, they erased everything except points $E, F$ and $D$. Reconstruct the $ABC$ triangle (no study required).

In trapezoid $ABCD$, the bisectors of angles $A$ and $D$ intersect at point $E$ lying on the side of $BC$. These bisectors divide the trapezoid into three triangles into which the circles are inscribed. One of these circles touches the base $AB$ at the point $K$, and two others touch the bisector $DE$ at points $M$ and $N$. Prove that $BK = MN$.

On the $BE$ side of a regular $ABE$ triangle, a $BCDE$ rhombus is built outside it. The segments $AC$ and $BD$ intersect at point $F$. Prove that $AF <BD$.

In the acute-angled non-isosceles triangle $ABC$, the height $AH$ is drawn. Points $B_1$ and $C_1$ are marked on the sides $AC$ and $AB$, respectively, so that $HA$ is the angle bisector of $B_1HC_1$ and quadrangle $BC_1B_1C$ is cyclic. Prove that $B_1$ and $C_1$ are base of the heights of triangle $ABC$.

Two trapezoid angles and diagonals are respectively equal. Is it true that such are the trapezoid equal?

Line $\ell$ is perpendicular to one of the medians of the triangle. The perpendicular bisectors of the sides of this triangle intersect line $\ell$ at three points. Prove that one of them is the middle of the segment formed by the remaining two.

$O$ is the intersection point of the diagonals of the trapezoid $ABCD$. A line passing through $C$ and a point symmetric to $B$ with respect to $O$, intersects the base $AD$ at the point $K$. Prove that $S_{AOK} = S_{AOB} + S_{DOK}$.

In triangle $ABC$, point $M$ is the midpoint of $BC, P$ is the intersection point of the tangents at points $B$ and $C$ of the circumscribed circle, $N$ is the midpoint of the segment $MP$. The segment $AN$ intersects the circumscribed circle at point $Q$. Prove that $\angle PMQ = \angle MAQ$.

A triangle $ABC$ and spheres are given in space $S_1$ and $S_2$, each of which passes through points $A, B$ and $C$. For points $M$ spheres $S_1$ not lying in the plane of triangle $ABC$ are drawn lines $MA, MB$ and $MC$, intersecting the sphere $S_2$ for the second time at points $A_1,B_1$ and $C_1$, respectively. Prove that the planes passing through points $A_1, B_1$ and $C_1$, touch a fixed sphere or pass through a fixed point.

In an acute-angled isosceles triangle $ABC$, altitudes $CC_1$ and $BB_1$ intersect the line passing through the the vertex $A$ and parallel to the line $BC$, at points $P$ and $Q$. Let $A_0$ be the midpoint of side $BC$, and $AA_1$ the altitude. Lines $A_0C_1$ and $A_0B_1$ intersect line $PQ$ at points $K$ and $L$. Prove that the circles circumscribed around triangles $PQA_1, KLA_0, A_1B_1C_1$ and a circle with a diameter $AA_1$ intersect at one point.

Angles are equal in a hexagon, three main diagonals are equal and the other six diagonals are also equal. Is it true that it has equal sides?

In the rectangle there is a broken line, the neighboring links of which are perpendicular and equal to the smaller side of the rectangle (see the figure). Find the ratio of the sides of the rectangle.

A circle with center $O$ passes through the ends of the hypotenuse of a right-angled triangle and intersects its legs at points $M$ and $K$. Prove that the distance from point $O$ to line $MK$ is half the hypotenuse.

Let $M$ and $N$ be the midpoints of the hypotenuse $AB$ and the leg $BC$ of a right triangles $ABC$ respectively. The excircle of the triangle $ACM$ touches the side $AM$ at point $Q$, and line $AC$ at point $P$. Prove that points $P, Q$ and $N$ lie on one straight line.

Points $I_A, I_B, I_C$ are the centers of the excircles of $ABC$ related to sides $BC, AC$ and $AB$ respectively. Perpendicular from $I_A$ to $AC$ intersects the perpendicular from $I_B$ to $B_C$ at point $X_C$. The points $X_A$ and $X_B$. Prove that the lines $I_AX_A, I_BX_B$ and $I_CX_C$ intersect at the same point.

Given a square sheet of paper with a side of $2016$. Is it possible to bend its not more than ten times, build a segment of length $1$?

The line passing through the center $I$ of the inscribed circle of a triangle $ABC$, perpendicular to $AI$ and intersects sides $AB$ and $AC$ at points $C'$ and $B'$, respectively. In the triangles $BC'I$ and $CB'I$, the heights $C'C_1$ and $B'B_1$ were drawn, respectively. Prove that the middle of the segment $B_1C_1$ lies on a straight line passing through point $I$ and perpendicular to $BC$.

A regular heptagon $A_1A_2A_3A_4A_5A_6A_7$ is given. Straight $A_2A_3$ and $A_5A_6$ intersect at point $X$, and straight lines $A_3A_5$ and $A_1A_6$ - at point Y. Prove that lines $A_1A_2$ and $XY$ are parallel.

Two squares are arranged as shown in the picture. Prove that the areas of shaded quadrilaterals are equal.
In a convex $n$-gonal prism all sides are equal. For what $n$ is this prism right?

From point $A$ to circle $\omega$ tangent $AD$ and arbitrary a secant intersecting a circle at points $B$ and $C$ (B lies between points $A$ and $C$). Prove that the circle passing through points $C$ and $D$ and touching the straight line $BD$, passes through a fixed point (other than $D$).

Given an acute triangle $ABC$. Let $A'$ be a point symmetric to $A$ with respect to $BC, O_A$ is the center of the circle passing through $A$ and the midpoints of the segments $A'B$ and $A'C. O_B$ and $O_C$ points  are defined similarly. Find the ratio of the radii of the circles circumscribed around the triangles $ABC$ and $O_AO_BO_C$.

On side $AB$ of triangle $ABC$ is marked point $K$ such that $AB = CK$. Points $N$ and $M$ are the midpoints of $AK$ and $BC$, respectively. The segments $NM$ and $CK$ intersect in point $P$. Prove that $KN = KP$.

An isosceles trapezoid $ABCD$ with bases $BC$ and $AD$ is given. Circles with centers $O_1$ and $O_2$ are inscribed in triangles $ABC$ and $ABD$. Prove that direct $O_1O_2$ is perpendicular on $BC$.

Points $M$ and $N$ are the midpoints of sides $AB$ and $CD$, respectively of quadrilateral $ABCD$. It is known that $BC // AD$ and $AN = CM$. Is it true that $ABCD$ is parallelogram?

We consider triangles $ABC$, in which the point $M$ lies on the side $AB$, $AM = a, BM = b, CM = c$ ($c <a, c <b$). Find the smallest radius of the circumcircle of such triangles.

Two squares are arranged as shown.
Prove that the area of ​​the black triangle equal to the sum of the gray areas.

Around triangle $ABC$ with acute angle C is circumscribed a circle. On the arc $AB$, which does not contain point $C$, point $D$ is chosen. Point $D'$ is symmetric on point $D$ with respect to line $AB$. Straight lines $AD'$ and $BD'$ intersect segments $BC$ and $AC$ at  points $E$ and $F$. Let point $C$ move along its arc $AB$. Prove that the center of the circumscribed circle of a triangle $CEF$ moves on a straight line.

One square is inscribed in a circle, and another square is described around the same circle so that its vertices lie on the continuations of the sides of the first (see figure). Find the angle between the sides of these squares
Given pyramid with base $n-gon$. How many maximum number of edges can be perpendicular to base?

On the plane, a non-isosceles triangle is given, a circle described around it and the center of its inscribed circle are marked. Using only a ruler without tick marks and drawing no more than seven lines, build the diameter of the circumcircle.

Prove that a circle constructed with the side $AB$ of a triangle $ABC$ as a diameter touches the inscribed circle of the triangle $ABC$ if and only if the side $AB$ is equal to the radius of the exircle on that side.

The inscribed circle of the non-isosceles  triangle $ABC$ touches sides $AB, BC$ and $AC$ at points $C_1, A_1$ and $B_1$, respectively. The circumscribed circle of the triangle $A_1BC_1$ intersects the lines $B_1A_1$ and $B_1C_1$ at the points $A_0$ and $C_0$, respectively. Prove that the orthocenter of triangle $A_0BC_0$, the center of the inscribed circle of triangle $ABC$ and the midpoint  of the $AC$ lie on one straight line.

Given acute angled traingle $ABC$ and altitudes $AA_1$, $BB_1$, $CC_1$.  Let $M$ midpoint of $BC$. $P$ point of intersection of circles $(AB_1C_1)$ and $(ABC)$ . $T$ is point of intersection of tangents to $(ABC)$ at $B$ and $C$. $S$ point of intersection of $AT$ and $(ABC)$. Prove that $P,A_1,S$ and midpoint of $MT$ collinear.

Two parallelograms are arranged so as it shown on the picture. Prove that the diagonal of the one parallelogram passes through the intersection point of the other diagonal pour.

Bisectors of angle $C$ and externalangle $A$ of trapezoid $ABCD$ with bases $BC$ and $AD$ intersect at point $M$, and the bisector of angle $B$ and external angle $D$ intersect at point $N$. Prove that the midpoint of the segment $MN$ is equidistant from the lines $AB$ and $CD$.

On the extensions of sides $CA$ and $AB$ of triangle $ABC$ beyond points $A$ and $B$, respectively, the segments $AE = BC$ and $BF = AC$ are drawn. A circle is tangent to segment $BF$ at point $N$,  side $BC$ and the extension of side $AC$ beyond point $C$. Point $M$ is the midpoint of segment $EF$. Prove that the line $MN$ is parallel to the bisector of angle $A$.

Given a triangle $ABC$ ($AB> AC$) and a circle circumscribed around it. Construct with a compass and a ruler the midpoint of the arc $BC$ (not containing vertex $A$), with no more than two lines (straight or circles).

The circle circumscribed about an acute triangle $ABC$ and the vertex $C$ are fixed. Orthocenter $H$ moves in a circle with center at point $C$. Find the locus of the midpoints of the segments connecting the bases of heights drawn from vertices $A$ and $B$.

Cut each of the equilateral triangles with sides $2$ and $3$ into three parts and construct an equilateral triangle from all received parts.

In a right triangle $ABC$ with a right angle $C$,  let  $AK$ and $BN$ be the angle bisectors. Let $D,E$ be the projections of $C$ on $AK, BN$ respectively. Prove that the length of the segment $DE$ is equal to the radius of the inscribed circle.

The diagonals of the trapezoid $ABCD$  are perpendicular ($AD//BC, AD>BC$) . Point $M$ is the midpoint of the side of $AB$, the point $N$ is symmetric of  the center of the circumscribed circle of the triangle $ABD$  wrt  $AD$. Prove that $\angle CMN = 90^o$.

(A. Mudgal, India)
A circle is fixed, point $A$ is on it and point $K$ outside the circle. The secant passing through $K$ intersects circle at points $P$ and $Q$. Prove that the orthocenters of the triangle $APQ$ lie on a fixed circle.

On the side $AB$  of the triangle $ABC$, point $M$ is selected. In triangle $ACM$ point $I_1$ is the center of the inscribed circle, $J_1$ is the center of excircle wrt side $CM$. In the triangle $BCM$ point $I_2$ is the center of the inscribed cirlce, $J_2$ is the center of excircle wrt side $CM$. Prove that the line passing through the midpoints of the segments $I_1I_2$ and $J_1J_2$ is perpendicular to $AB$.

Two ants sit on the surface of a tetrahedron. Prove that they can meet by breaking the sum of a distance not exceeding the diameter of a circle is circumscribed around the edge of a tetrahedron.

Let $ABC$ be an acute-angled triangle with circumcenter $O$. The circumcircle of $\triangle{BOC}$ meets the  lines $AB, AC$ at points $A_1, A_2$, respectively. Let $\omega_{A}$ be the circumcircle of triangle $AA_1A_2$. Define $\omega_B$ and $\omega_C$ analogously. Prove that the circles $\omega_A, \omega_B, \omega_C$ concur on $\odot(ABC)$.

In the triangle $ABC, I$ is the center of the inscribed circle, point $M$ lies on the side of $BC$, with $\angle BIM = 90^o$. Prove that the distance from point $M$ to line $AB$ is equal to the diameter of the circle inscribed in triangle $ABC$

On the side $AC$  of the triangle $ABC$ in the external side is constructed the parallelogram $ACDE$ . Let $O$ be the intersection point of its diagonals, $N$ and $K$ be  midpoints of BC and BA respectively. Prove that lines $DK, EN$ and $BO$ intersect at one point.

In the acute triangle $ABC, \angle ABC = 60^o , O$ is the center of the circumscribed circle and  $H$ is the orthocenter. The bisector $BL$ intersects the circumscribed circle at the point $W, X$ is the intersection point of segments $WH$ and $AC$ . Prove that points $O, L, X$ and $H$ lie on the same circle.

The perpendicular bisector of  the bisector $BL$ of the triangle $ABC$ intersects the bisectors of its external angles $A$ and $C$ at points $P$ and $Q$, respectively. Prove that the circle circumscribed around the triangle $PBQ$ is tangent to the circle circumscribed around the triangle $ABC$.

Given the segment$PQ$ and a circle . A chord $AB$ moves around the circle, equal to $PQ$. Let $T$ be the intersection point of the perpendicular bisectors of the segments $AP$ and $BQ$. Prove that all points of $T$ thus obtained lie on one line.

In the acute triangle $ABC$, the point $I_c$ is the center of excircle on the side $AB$, $A_1$ and $B_1$ are the tangency points of the other two excircles with sides $BC$ and $CA$, respectively, $C'$  is the point on the circumcircle diametrically opposite to point $C$. Prove that the lines $I_cC'$ and $A_1B_1$ are perpendicular.

Circle inscribed in square $ABCD$ , is tangent to sides $AB$ and $CD$ at points $M$ and $K$ respectively. Line $BK$ intersects this circle at the point $L, X$ is the midpoint of $KL$. Find the angle $\angle MXK$.

The angles of one quadrilateral are equal to the angles another quadrilateral. In addition, the corresponding angles between their diagonals are equal. Are these quadrilaterals necessarily congruent ?

Restore the acute triangle $ABC$ given the vertex $A$, the base of the altitude drawn from the vertex $B$ and the center of the circle circumscribed around triangle $BHC$ (point $H$ is the orthocenter of triangle $ABC$).

Given a right triangle $ABC$ ($\angle C=90^o$). The $C$-excircle touches the hypotenuse $AB$ at point $C_1, A_1$ is the touchpoint  of $B$-excircle with  line $BC, B_1$  is the touchpoint  of $A$-excircle with line $AC$. Find the angle $\angle A_1C_1B_1$.

On sides $AB$ and $BC$ of a non-isosceles triangle $ABC$ are selected  points $C_1$ and $A_1$ such that the quadrilateral $AC_1A_1C$ is cyclic. Lines $CC_1$ and $AA_1$ intersect  at point $P$. Line $BP$ intersects the circumscribed circle of triangle $ABC$ at the point $Q$. Prove that the lines $QC_1$ and $CM$, where $M$ is the midpoint of $A_1C_1$, intersect at the circumscribed circles of triangle $ABC$.

The sum of the cosines of the flat angles of the trihedral angle is $-1$. Find the sum of its dihedral angles.