**all**the Euclidean Geometry problems, with aops links ftom

__Quantum: The Magazine of Math And Science__

(English edition)

this post of the geometry problems with aops links,

Here are posted all the (Euclidean) geometry problems,

from this magazine with aops links in the names:

(b) Use six lines to cut a triangle into parts such that it is possible to compose seven congruent triangles from them.

(A. Soifer)

M6 Three circles with the same radius $r$ all pass through point $H$ (fig. ). Prove that the circle passing through the points where pairs of circles intersect (that is, points$A, B$,and $C$) also has the same radius $r$.

M8 When the sides of a convex polygon are moved outward by the same distance $1$, they fall on the corresponding parallel sides of a larger similar polygon. Prove that circles can be inscribed in these polygons.

(N. Vasilyev)

M11 The lengths of two sides of a triangle are $10$ and $15$. Prove that the bisector of the angle between them is no greater than $12$.

(N. Vasilyev)

M13 A quadrangle is inscribed in a parallelogram whose area is twice that of the quadrangle, as shown in figure. Prove that at least one of the quadrangle's diagonals is paraliel to one of the parallelogram's sides.
(E. Sallinen)

M17 On straight lines $AB$ and $BC$ containing two sides of a parallelogram ABCD points H and K are chosen so that the triangles $KAB$ and $HCB$ are isosceles ($KA = AB, HC= CB$, see figure). Prove that the triangle $KDH$ is also isosceles.

(V.Gutenmacher)

M19 A number of chords are drawn in a circle of radius $1$ so that each diameter crosses no more than $k$ chords. Prove that the sum of the lengths of all the chords is less than $\pi k$.

(A. Kolotov)

M33 Two congruent circles intersect at points $A$ and $B$. Two more circles of the same radius are drawn: one through $A$, the other through $B$ (fig.). Prove that the four points of the paired intersection of all four circles (other than $A$ and $B$ are the vertices of a parallelogram.

M21 A regular hexagon with side $1$ is drawn on the plane. Construct a segment of length $\sqrt7$ using only a straightedge.

(A. Aliayev)

M29 A cube contains a convex polyhedron whose projection onto any of the cube's faces covers the entire face. Show that the volume of the polyhedron is not less than $1/3$ that of the cube.

(V. Prasolov)

M33 Two congruent circles intersect at points $A$ and $B$. Two more circles of the same radius are drawn: one through $A$, the other through $B$ (fig.). Prove that the four points of the paired intersection of all four circles (other than $A$ and $B$ are the vertices of a parallelogram.

(V.and I. Kapovich)

M36 Point $I$ divides the diagonal $AC$ of a square $ABCD$ in the ratio $3:1$ (fig.), $K$ is the midpoint of side $AB$. Prove that argle $KLD$ is a right angle.

(Y. Bogaturov)

M38 Each of four sides of a convex pentagon is parallel to one of its diagonals (having no end points in common with the side-see figure).Prove that the same holds for the fifth side, too.

(E. Turkevich)

M43 A hexagon $A_1A_2...A_6$ is inscribed in a regular hexagon, one vertex of the former hexagon lying on each side of the latter (fig. ). Three nonadjacent sides $A_1A_2, A_3A_4$, and $A_5A_6$ of the first hexagon are extended to form a triangle. Prove that if the vertices of this triangle lie on the lines containing the diagonals of the regular hexagon, then the same is true for the vertices of the triangle formed by the three other sides $A_2A_3, A_4A_5, A_6A_1$.

(S. Orevkov)

M53 Inside a circle there are two intersecting circles. One of them touches the big circle in point A, the other in point B. Prove that if segment AB meets the smaller circles at one of their common points (fig. ), then the sum of their radii equals the radius of the big circle. Is the converse true?

(A. Vesyolov)

M59 Given three points $O,I$, and $E$ in the plane construct a triangle such that its circumcenter is at $O$, its incenter is at $I$, and one of its excenters is at $E$.

(B. Martynov)

M63 The diameter $AB$ of a semicircle is arbitrarily divided into two parts, $AC$ and $CB$, on which two other semicircles are constructed {fig. ). Find the diameter of a circle inscribed in a curvilinear triangle formed by the three semicircles, given only the distance $I$ from this circle's center to line $AB$.

M67 Two parabolas on the plane have perpendicular lines of symmetry and four common points. Prove that these four points lie on one circle.

(L. Kuptsov)

M70 In trapezoid $ABCD$, diagonal $AC$ is equal to leg $BC$ and $H$ is the midpoint of base $AB$. A variable line through $H$ intersects line $AD$ at $P$ and line $BD$ at $Q$. Show that angles $ACP$ and $QCB$ are either equal or supplementary.

(I. Sharygin)

M73 In a right triangle one half of the hypotenuse (from a vertex to the midpoint of the hypotenuse) subtends a right angle at the triangle's incenter. Find the ratio of side lengths of the triangle.

(B. Pitskel)

M79 A line through the vertex $B$ of an isosceles triangle $ABC$ ($AB = BC$) cuts its base $AC$ at $D$ so that the radius of the incircle of triangle ABD equals that of the excircle of triangle $CBD$ externally touching side $DC$ (and the extensions ol $BC$ and $BD$ - see figure). Prove that this radius is $1/4$ the height $h$ of the triangle dropped from a base vertex.

(I. Sharygin)

M82 On the sides $AC$ and $AB$ of an equilateral triangle $ABC$, points D and E are given such that $AD : DC = BE : EA = 1 :2$. The lines $BD$ and $CE$ meet at point $P$. Prove that angle $APC$ is a right angle.
(A. Krasnodemskaya)

M84 Let's define a "skewb" as a hexahedron all of whose six faces are (arbitrary) quadrilaterals joined like the faces of a cube (fig.) .Prove that if three "big" diagonals of a skewb (that is, lines through the pairs of vertices that don't lie in one face) meet at one point, then the fourth big diagonal also passes through this point.

(V. Dubrovsky)

M87 Each of the three midlines of a convex hexagon (the lines that join the midpoints of its opposite sides) divides its area in half. Prove that they meet at one point.

(V. Proizvolov)

M92 Prove that it's impossible to construct two trapezoids (not parallelograms!) such that the legs of each are congruent to the bases of the other.

(V. Proizvolov)

M96 Does there exist a (nonconvex) pentagon that can be cut into two congruent pentagons?

(S. Hosid)

M103 Equal sides of two acute isosceles triangles are the same length, and the radii of their incircles are the same too. Are these triangles necessarily congruent?

(A. Yegorov)

(E. Tsinovi)

(D. Nyamsuren [Mongolia])

M158 Circles $S_1$ and $S_2$ touch each other externally at point $F$. Line $\ell$ touches $S_1$ and $S_2$ at points $A$ arrd $B$, respectively, and the line parallel to $\ell$ and tangent to $S_2$ at $C$ meets $S_1$ at points $D$ and $E$. Prove that

(a) points $A, F, C$ are on the same line,

(b) the common chord of the circumcircles of triangles $ABC$ and $BDE$ passes through $F$.

M166 Two circles intersect at points $A$ and $B$. The tangents to them drawn through $A$ meet them again at $M$ and $N$. The lines $BM$ and $BN$ meet the circles for the second time at points $P$ and $Q$, respectively (fig.). Prove that $MP= NQ$.

M112 A line drawn through a point $K$ in a square $ABCD$ intersects two opposite sides $AB$ and $CD$ at points $P$ and $Q$ (fig.). Two circles are drawn: through points $K, B, P$ and through points $K,D,Q$. Prove that their second point of intersection (the point other thanK) lies on the diagonal $BD$.

(V. Dubrovsky)

M119 The base $A_1A_2...A_n$ of an n-sided pyramid $PA_1A_2...A_n$ has congruent sides $A_1A_2 = A_2A_3 =...=A_nA_1$, the angles $PA_1A_2, PA_2A_3,..., PA_nA_1$ are also congruent (fig. ). Prove that the pyramid is regular - that is, its base is a regular n-gon and its altitude falls on the center of the base.

(V. Senderov, V. Dubrovsky)

M125 A quadrilateral has both circumscribed and inscribed circles. Prove that the intersection point of its diagonals and the centers of the circles lie on the same straight line.

(V. Protasov)

M128 Points $A ,B, C$, and $D$ are chosen in the plane with $AB = BC = CD = 1$. The four points are repeatedly subjected to the following transformation that leaves points $B$ and $C$ fixed and preserves the lengths of $AB, BC, CD$, and $DA$.First, point $A$ is reflected about $BD$, then $D$ is reflected about $AC$, where $A$ is in the new, reflected position, then the new point $A$ is again reflected about $BD$ (with the new $D$), then $D$ is reflected, and so on. Prove that after a number of reflections points $A$ and $C$ will return to their starting positions

(M. Kontsevich)

M130 The opposite sides of a convex cluadrilateral are extended to intersect at two points. A line is drawn through each of these points. These two lines divide the quadrilateral into four smaller quadrilaterals. If some pair of these quadrilaterals that don't share a common side, each has an inscribed circle, show that the original quadrilateral also has an inscribed circle.

(I. Sharygin)

M135 (a) Three equilateral triangles $ABC_1, BCA_1$ and $CAB_1$, are constructed externally on the sides of an arbitrary triangle $ABC$, the midpoints of the segments $A_1B_1, B_1C_1, C_1A_1$, are labeled $C_2, A_2, B_2$, respectively. Prove that the lines $AA_2, BB_2, CC_2$ meet at the same point or are paralel.

(b) Prove this statement with "equilateral triangles" replaced by any similar isosceles triangles $ABC_1, BCA_1, CAB_1$ with bases $AB, BC, CA$.

(N. Sedrakian, S. Tkachov)

M137 Prove that the sum of the distances from an arbitrary point in the plane to three vertices of an isosceles trapezoid is always greater than the distance from this point to the fourth vertex.

(S. Rukshin)

M143 (a) Three lines are drawn through a point in a triangle parallel to its sides. The segments intercepted on these lines by the triangle turn out to have the same length {see figure, in which the three equal segments are colored red). Given the triangle's side lengths $a, b$, and $c$, find the length of the segments.

(b) Four planes are drawn through a point in a tetrahedron parallel to its faces. The sections of the tetrahedron created by these planes turn out to have the same area. Given the areas $a,b,c$, and $d$ of the faces, find the area of the sections.

(A. Yagubiants, V. Dubrovsky)

M152 A point $P$ is marked in a square $A_1A_2A_3A_4$ and joined to its vertices. Prove that the perpendiculars dropped from $A_{i-1}$, on line $PA_i, i=1,2, 3, 4$ (of course, $A_0$ here should be read as $A_4$) all meet at the same point.

(A. Vilenkin)

M158 Circles $S_1$ and $S_2$ touch each other externally at point $F$. Line $\ell$ touches $S_1$ and $S_2$ at points $A$ arrd $B$, respectively, and the line parallel to $\ell$ and tangent to $S_2$ at $C$ meets $S_1$ at points $D$ and $E$. Prove that

(a) points $A, F, C$ are on the same line,

(b) the common chord of the circumcircles of triangles $ABC$ and $BDE$ passes through $F$.

(A. Kalinin)

M166 Two circles intersect at points $A$ and $B$. The tangents to them drawn through $A$ meet them again at $M$ and $N$. The lines $BM$ and $BN$ meet the circles for the second time at points $P$ and $Q$, respectively (fig.). Prove that $MP= NQ$.

(I. Nagel)

M173 Find all the points $X$ on the side $BC$ of a triangle $ABC$ such that the triangle $XPQ$, where $P$ and $Q$ are the

(a) circumcenters,

(b) centroids,

(c) orthocenters of the triangles $AXB$ and $AXC$, is similar to $ABC$.

M 179 You have a ruler with two marks on it. With it, you can draw lines as with an ordinary ruler and also mark off segments

equal in length to the distance between the two given marks. You are not allowed any other constructions. With this instrument, construct

(a) a right angle,

(b) a line perpendicular to a given line.

(a) circumcenters,

(b) centroids,

(c) orthocenters of the triangles $AXB$ and $AXC$, is similar to $ABC$.

(E. Turkevich)

M 179 You have a ruler with two marks on it. With it, you can draw lines as with an ordinary ruler and also mark off segments

equal in length to the distance between the two given marks. You are not allowed any other constructions. With this instrument, construct

(a) a right angle,

(b) a line perpendicular to a given line.

(V. Gutenmacher)

M182 Let $A$ be one of the intersection points of two circles in the plane. In each of the circles a diameter is drawn parallel to the tangent to the other circle at $A$. Prove that the endpoints of the diameters lie on a circle.

(S. Berlov)

M187 Prove that of the $n$ quadrilaterals cut from a convex $n$-gon by its diagonals (subtending triples of consecutive sides) no more than $n/2$ can have inscribed circles. Give an example of an octagon that has four such quadrilaterals.

(N. Sedrakyan)

M194 A quadrilateral $ABCD$ can be inscribed in a circle. Let straight lines $AB$ and $CD$ meet at $M$, and let $BC$ and $AD$ meet at $K$, so that $B$ lies on the segment $AM$ and $D$ on the segment $AK$. Let $P$ be the foot of the perpendicular from $M$ to line $AK$, and let $I$ be the foot of the perpendicular from $K$ to line $AM$. Prove that $LP$ bisects $BD$.

M200 A trapezoid $ABCD$ (in which $AD$ and $BC$ are bases) is inscribed in a circle. Its diagonals intersect at point $M$. Let a straight line perpendicular to the bases of $ABCD$ meet $BC$ at $K$ and meet the circle at $I$ (where $I$ is the point of intersection for which $M$ lies on line segment $KL$). Let $MK = a$ and $LM = b$. Express in terms of $a$ and $b$ the radius of the circle tangent to segments $AM$ and $BM$, and also tangent internally to the circle circumscribed about $ABCD$.

(I. Sharygin)

M204 Find the location of the midpoints of all the chords drawn in a given circle so that their endpoints lie on different sides of a given straight line intersecting this circle.

(I. Sharygin)

M205 A circle on the plane is given. It's center is not shown. Find the center using a compass (without a straightedge) in such a way that the total number of arcs or circles you draw does not exceed six.

(V. Panfyorov)

M207 Four points $K, P, H$, and $M$ are taken on a side of a triangle. These points are the midpoint, the endpoint of the bisector of the opposite angle, the point of tangency with the inscribed circle, and the base of the corresponding altitude, respectively (fig.). Show that if $KP = a$ and $KM=b$ then $KH =\sqrt{ab}.$

(I. Sharygin)

M210 In triangle $ABC$, bisectors $AA_1, BB_1$, and $CC_1$, of the interior angles are drawn (fig. ) .Prove that if $\angle ABC=120^o$ then $\angle A_1B_1C_1 = 90^o$.

(A. Yegorov)

M212 In triangle $ABC$ sides $CB$ and $CA$ are equal to $a$ and $b$, respectively. The bisector of the angle $ACB$ intersects side $AB$ at point $K$, and the circle circumscribed about the triangle intersects it at point $M$. The second point at which the circle circumscribed about the triangle $AMK$ meets line $CA$ is $P$. Find the lengh of $AP$.

M214 Prove that one can fold a given paper triangle so that it covers without overlap the surface of a unit regular tetrahedron (that is, a triangular pyramid whose edges are all equal to $1$), if

(a) the triangle is isosceles, with legs of length $2$ and vertex angle equal to $120^o$,

(b) two sides of the triangle are equal to $2$ and $2\sqrt3$ and the angle between them is $150^o$.

M218 Let $M$ be the point where the diagonals of a parallelogram $ABCD$ meet. Consider three circles passing through $M$: the first and the second circles are tangent to $AB$ at points $A$ and $B$, respectively, and the third circle passes through $C$ and $D$. Denote the points, other than $M$, where the third circle intersects the first and the second ones by $P$ and $Q$, respectively. Prove that $PQ$ is tangent to the first and second circles.

M222 In triangle $ABC, \angle BAC$ equals $\alpha$. The circle inscribed in the triangle touches its sides at points $K, L$, and $M$, and $M$ lies on $BC$. Show that the ratio of the length of altitude $MM_1$ of triangle $KLM$ to the length of altitude $AA_1$ of triangle $ABC$ equals $sin (\alpha /2)$.

M230 In triangle $ABC, \angle B \ne 90^o$, and $AB:BC = k$.Let $M$ be the midpoint of $AC$. Lines symmetric to $BM$ with respect to $AB$ and $BC$ meet line $AC$ at point $D$ and $E$ respectively. Find the ratio $BD:BE$.

M232 Point $M$ is taken inside a parallelogram $ABCD$. We know $\angle MBC = 20^o, \angle MCB = 50^o,\angle MDA=70^o$, and $\angle MAD=40^o$. Find the angles of the parallelogram .

M235 Find the greatest possible value of the area of an orthogonal projection of a cylinder with radius $r$ and altitude $h$ on a plane.

M237 Consider the rhombus $ABCD$. Find the locus of points $M$ such that $\angle AMB + \angle CMD=180^o$.

M236 Consider two circles that intersect at points $A$ and $B$. Let a line through $B$ meet the circles at points $K$ and $M$ (see fig.). Let $E$ and $F$ be the midpoints of arcs $AK$ and $AM$, respectively (the arcs that don't contain $B$), and let $L$ be the midpoint of segment $KM$. Prove that $\angle ELF$ is a right angle.

M240 A math student is lost in a vast forest whose border is a line. (Imagine that the forest covers a half-plane.) The student knows that she is no more than $2$ miles from the border. Propose a route for her such that she would come out of the forest having walked no more than $13$ miles. (Of course the student doesn't know where the border lies, and no matter how close to it she passes she cannot see it. We say that the student comes out of the forest when she reaches the border.)

M242 Let $M$ be the point of intersection of the diagonals of the inscribed quadrilateral $ABCD$, where $\angle AMB$ is acute. An isosceles triangle $BCK$ is constructed on the base $BC$ such that $\angle KBC + \angle AMB =90^o$. Prove that $KM$ is perpendicular to $AD$.

M244 The planet Brick is a rectangular parallelepiped with edges of $1,2$, and $4$ km. The Prince of Brick built a brick house at the center of one of the largest faces. What is the distance from the house to the farthest point on the planet? (The distance between two points is defined as the length of the shortest connecting path along the planet's surface.)

M248 A triangular pyramid $ABCD$ has the equilateral triangle $ABC$ as its base, and $AD= BC$. If the three plane angles at vertex $D$ are equal, what values can these angles take on?

M250 The perimeter of \triangle $ABC$ is $k$ times greater than side $BC$ , and $AB < AC$. A diameter of the inscribed circle is drawn perpendicular to $BC$.In what proportion does the median to $BC$ divide this diameter?

M252 Two nonintersecting circles with radii $R$ and$ r$ are each tangent to both sides of the same angle. Construct an isosceles triangle such that its base lies on one side of the angle, the vertex is on the other side, and each leg touches one of the circles. Express the length of the altitude to the base of this triangle in terms of $R$ and $r$.

M253 An equilateral triangle made of a piece of cardboard lies on a plane. Three nails are driven at points $K, L$, and $M$ at its sides in such away that the triangle cannot move (fig.). It is given that points $K$ and $L$ divide their corresponding sides in the proportion of $2:1$ and $3:2$ as in figure. In what proportion does point $M$ divide its side of the triangle?

M254 A plane intersects a unit cube and divides it into two polyhedrons. It is known that the distance between any two points of one polyhedron does not exceed $3/2$. What value can the area ol this section take?

M259 A triangular pyramid is given such that all plane angles at one of the vertices are right. It is known that apoint exists such that its distance from the given vertex is $3$, and the distances from th: other vertices are $\sqrt5 , \sqrt6$, and $\sqrt7$, respectively. Find the radius of the sphere circumscribed around this pyramid.

M260 A circle inscribed in triangle $ABC$ touches $BC$ at point $T$, and $M$ is the midpoint of the altitude drawn to $BC$. Point $P$ is the second point of intersection of line $TM$ with the inscribed circle. Prove that the circle that passes through points $B, C$, and $P$ touches the circle inscribed in triangle $ABC$.

M267 In a triangle $ABC$, angle $BAC$ is $50^o$. A point $P$ is selected inside the triangle in such a way that angles $APB, BPC$, and $CPA$ are $120^o$. Segment $AP = a$. Find the area of triangle $BPC$.

M270 Points $D$ and $F$ are chosen on the bisector of angle $A$ of a triangle $ABC$ in such a way that $\angle DBC = \angle FBA$. Prove that

(a) $\angle DCB = \angle FCA$,

(b) the circle that passes through $D$ and $F$ and is tangent to $BC$ is also tangent to the circle circumscribed around triangle $ABC$.

M273 An isosceles triangle $ABC$ ($AB = BC$) is given in a plane. Find the locus of points $M$ in the plane such that $ABCM$ is a convex quadrilateral and $\angle MAC + \angle CMB=90^o$.

M274 The distances from all vertices of a cube and from the centers of its faces to a certain plane ($14$ quantities in all) take two different values, and the lower value is $1$. What can the length of the cube's edge be?

M275 In a triangle $ABC$, angle $B$ is obtuse and its measure is $\alpha$. The bisectors of angles $A$ ard $C$ intersect the opposite sides at points $P$ and $M$, respectively. Points $K$ and $I$ are taken on side $AC$ such that $\angle ABK= \angle CBL=2\alpha- 180^o$. Find the angle between lines $KP$ and $LM$.

M276 Five edges of a triangular pyramid are of length $1$. Find the sixth edge if it is given that the radius of the sphere circumscribed about this pyramid is $1$.

M280 Let $M$ be the midpoint of side $BC$ of a triangle $ABC$ and let $Q$ be the point of intersection of its bisectors. It is given that $MQ=QA$. Find the minimum possible value of angle $MQA$,

M284 A chord $AB$ is drawn in circle $O$, of radius $r$. Points $P$ and $Q$ are taken on its extension beyond points $A$ and $B$, respectively, such that $AP=BQ$. As $P$ and $Q$ vary along line $AB$, they determine two pairs of tangents to circle $O$. These four tangents, in turn, determine four new points of intersection with each other. Find the locus of all such points of intersection.

M285 Side $BC$ of triangle $ABC$ has length $a$, and the opposite angle has degree-measure $\alpha$. The line passing through the midpoint $D$ of $BC$ and the center of the circle inscribed in the triangle intersects $AB$ and $AC$ at points $M$ and $P$, respectively. Find the area of the (nonconvex) quadrilateral $BMPC$.

M286 Quadrilateral $ABCD$ is inscribed in a circle. Let $M$ be the point of intersection of its diagonais, and $L$ be the midpoint of arc $AD$ (which does not contain the other vertices of the quadrilateral). Prove that the distances from $L$ to the centers of the circles inscribed in triangles $ABM$ and $CDM$ are equal.

M290 Two circles and an isosceles triangie are arranged as shown in figure. Find the altitude of the triangle drawn to its base if the sum of the circles' diameters is $2$.

M292 In triangle $ABC$ angle $A$ is equal to $\alpha$.. The circle that passes through $A$ and $B$ and is tangent to line $BC$ intersects the median drawn to side $BC$ (or its extension) at a point $M$, which is different from $A$. Express the measure of angle $BMC$ in terms of $\alpha$.

M294 In triangle $ABC, \angle BAC=\alpha$ and $\angle ABC=2\alpha$. The circle with center $C$ and radius $CA$ intersects the line containing the bisector of the exterior angle at vertex $B$ at points $M$ and $N$. Express the measures of the angles of triangle $AMN$ in terms of $\alpha$.

M298 Let the perpendicular to side $AD$ of the parallelogram $ABCD$ passing through vertex $B$ intersect line $CD$ at point $M$, and the perpendicular to side $CD$ passing through vertex $B$ intersect line $AD$ at point $N$. Prove that the perpendicular dropped from $B$ onto diagonal $AC$ passes through the midpoint of segment $MN$.

M299 A circle lies entirely inside a given angle. Construct another circle, tangent to the first and to the sides of the given angle. How many such circles are there?

M300 Let a line $m$ be perpendicular to a plane $L$. Three spheres that are tangent in pairs are also tangent to line $m$ and plane $L$. The radius of the largest sphere is $r$. Find the minimum possible radius of the smallest sphere.

M304 In triangle $ABC$ the altitude $CM$ is drawn. The line that is symmetric to the altitude drawn from vertex $A$ about line $CM$ intersects line $BC$ at point $K$. Find angle $OMK$, where $O$ is the center of the circle circumscribed about triangle $ABC$ (the points $O$, $M$, and $K$ are all distinct).

M307 Two intersecting circles are given in the plane.$ A$ is one of the points where the circles intersect. In each circle a diameter is drawn that is parallel to the tangent to the other circle at point $A$, and these diameters do not intersect. Prove that the four endpoints of these diameters lie on the same circle.

M309 Three potnts $A, B$, and $C$ are given in the plane. Draw a line through point $C$ such that the product of the distances from points $A$ and $B$ to this line is greatest. Does such a line always exist?

M312 For a given chord $MN$ of a circle, consider all triangles $ABC$ such that $AB$ is a diameter of the given circle not intersecting $MN$ and the sides $AC$ and $BC$ pass through the endpoints of $MN$. Prove that the altitudes of all such triangles drawn from vertex $C$ to side $AB$ meet in a point.

M315 A point $D$ is taken on the base $AC$ of isosceles triangle $ABC$ such that the circle inscribed in triangle $ABD$ has the same radius as the circle that touches the extensions of segments $BC$ and $BD$ and segment $CD$ (the escribed circle of triangle $BCD$). Prove that this radius equals $1/4$ of the triangle's altitude drawn to a leg.

M320 Prove that in (a) a regular $12$-gon and (b) a regular $54$-gon there exist four diagonals that meet at a point and do not pass through the center of the polygon.

(V. Protasov)

(a) the triangle is isosceles, with legs of length $2$ and vertex angle equal to $120^o$,

(b) two sides of the triangle are equal to $2$ and $2\sqrt3$ and the angle between them is $150^o$.

(I. Sharygin)

M218 Let $M$ be the point where the diagonals of a parallelogram $ABCD$ meet. Consider three circles passing through $M$: the first and the second circles are tangent to $AB$ at points $A$ and $B$, respectively, and the third circle passes through $C$ and $D$. Denote the points, other than $M$, where the third circle intersects the first and the second ones by $P$ and $Q$, respectively. Prove that $PQ$ is tangent to the first and second circles.

M222 In triangle $ABC, \angle BAC$ equals $\alpha$. The circle inscribed in the triangle touches its sides at points $K, L$, and $M$, and $M$ lies on $BC$. Show that the ratio of the length of altitude $MM_1$ of triangle $KLM$ to the length of altitude $AA_1$ of triangle $ABC$ equals $sin (\alpha /2)$.

M230 In triangle $ABC, \angle B \ne 90^o$, and $AB:BC = k$.Let $M$ be the midpoint of $AC$. Lines symmetric to $BM$ with respect to $AB$ and $BC$ meet line $AC$ at point $D$ and $E$ respectively. Find the ratio $BD:BE$.

M232 Point $M$ is taken inside a parallelogram $ABCD$. We know $\angle MBC = 20^o, \angle MCB = 50^o,\angle MDA=70^o$, and $\angle MAD=40^o$. Find the angles of the parallelogram .

(M.Volchkevich)

M235 Find the greatest possible value of the area of an orthogonal projection of a cylinder with radius $r$ and altitude $h$ on a plane.

(M. Volchkevch)

M237 Consider the rhombus $ABCD$. Find the locus of points $M$ such that $\angle AMB + \angle CMD=180^o$.

M236 Consider two circles that intersect at points $A$ and $B$. Let a line through $B$ meet the circles at points $K$ and $M$ (see fig.). Let $E$ and $F$ be the midpoints of arcs $AK$ and $AM$, respectively (the arcs that don't contain $B$), and let $L$ be the midpoint of segment $KM$. Prove that $\angle ELF$ is a right angle.

M240 A math student is lost in a vast forest whose border is a line. (Imagine that the forest covers a half-plane.) The student knows that she is no more than $2$ miles from the border. Propose a route for her such that she would come out of the forest having walked no more than $13$ miles. (Of course the student doesn't know where the border lies, and no matter how close to it she passes she cannot see it. We say that the student comes out of the forest when she reaches the border.)

M242 Let $M$ be the point of intersection of the diagonals of the inscribed quadrilateral $ABCD$, where $\angle AMB$ is acute. An isosceles triangle $BCK$ is constructed on the base $BC$ such that $\angle KBC + \angle AMB =90^o$. Prove that $KM$ is perpendicular to $AD$.

M244 The planet Brick is a rectangular parallelepiped with edges of $1,2$, and $4$ km. The Prince of Brick built a brick house at the center of one of the largest faces. What is the distance from the house to the farthest point on the planet? (The distance between two points is defined as the length of the shortest connecting path along the planet's surface.)

M248 A triangular pyramid $ABCD$ has the equilateral triangle $ABC$ as its base, and $AD= BC$. If the three plane angles at vertex $D$ are equal, what values can these angles take on?

M250 The perimeter of \triangle $ABC$ is $k$ times greater than side $BC$ , and $AB < AC$. A diameter of the inscribed circle is drawn perpendicular to $BC$.In what proportion does the median to $BC$ divide this diameter?

M252 Two nonintersecting circles with radii $R$ and$ r$ are each tangent to both sides of the same angle. Construct an isosceles triangle such that its base lies on one side of the angle, the vertex is on the other side, and each leg touches one of the circles. Express the length of the altitude to the base of this triangle in terms of $R$ and $r$.

(I. F. Sharygin)

(A. Shen)

M254 A plane intersects a unit cube and divides it into two polyhedrons. It is known that the distance between any two points of one polyhedron does not exceed $3/2$. What value can the area ol this section take?

(N. P. Dolbilin)

M260 A circle inscribed in triangle $ABC$ touches $BC$ at point $T$, and $M$ is the midpoint of the altitude drawn to $BC$. Point $P$ is the second point of intersection of line $TM$ with the inscribed circle. Prove that the circle that passes through points $B, C$, and $P$ touches the circle inscribed in triangle $ABC$.

M264 Given two points in a plane and a straightedge whose length is less than the distance between them (but no compass!), construct the line passing through the two points.

You may want to use a special case of Desargues' theorem:

Suppose we have two triangles, $ABC$ and $A_1B_1C_1$ positioned in such a way that $AA_1, BB_1$, and $CC_1$ intersect in a point. Let lines $AB$ and $A_1B_1$ meet at point $K$, lines $BC$ and $B_1C_1$, meet at $P$, and $CA$ and $C_1A_1$ at $M$. Then, points $K, P$, and $M$ are collinear (fig. ).

M265 Let $AA_1, BB_1$, and $CC_1$ be the bisectors of the interior angles of triangle $ABC$ (where $A_1, B_1$, and $C_1$ are on the sides of the triangle). If $\angle AA_1C =\angle AC_1B_1$ find $\angle BCA$.

M270 Points $D$ and $F$ are chosen on the bisector of angle $A$ of a triangle $ABC$ in such a way that $\angle DBC = \angle FBA$. Prove that

(a) $\angle DCB = \angle FCA$,

(b) the circle that passes through $D$ and $F$ and is tangent to $BC$ is also tangent to the circle circumscribed around triangle $ABC$.

M273 An isosceles triangle $ABC$ ($AB = BC$) is given in a plane. Find the locus of points $M$ in the plane such that $ABCM$ is a convex quadrilateral and $\angle MAC + \angle CMB=90^o$.

M274 The distances from all vertices of a cube and from the centers of its faces to a certain plane ($14$ quantities in all) take two different values, and the lower value is $1$. What can the length of the cube's edge be?

M275 In a triangle $ABC$, angle $B$ is obtuse and its measure is $\alpha$. The bisectors of angles $A$ ard $C$ intersect the opposite sides at points $P$ and $M$, respectively. Points $K$ and $I$ are taken on side $AC$ such that $\angle ABK= \angle CBL=2\alpha- 180^o$. Find the angle between lines $KP$ and $LM$.

M276 Five edges of a triangular pyramid are of length $1$. Find the sixth edge if it is given that the radius of the sphere circumscribed about this pyramid is $1$.

M280 Let $M$ be the midpoint of side $BC$ of a triangle $ABC$ and let $Q$ be the point of intersection of its bisectors. It is given that $MQ=QA$. Find the minimum possible value of angle $MQA$,

M284 A chord $AB$ is drawn in circle $O$, of radius $r$. Points $P$ and $Q$ are taken on its extension beyond points $A$ and $B$, respectively, such that $AP=BQ$. As $P$ and $Q$ vary along line $AB$, they determine two pairs of tangents to circle $O$. These four tangents, in turn, determine four new points of intersection with each other. Find the locus of all such points of intersection.

M285 Side $BC$ of triangle $ABC$ has length $a$, and the opposite angle has degree-measure $\alpha$. The line passing through the midpoint $D$ of $BC$ and the center of the circle inscribed in the triangle intersects $AB$ and $AC$ at points $M$ and $P$, respectively. Find the area of the (nonconvex) quadrilateral $BMPC$.

M286 Quadrilateral $ABCD$ is inscribed in a circle. Let $M$ be the point of intersection of its diagonais, and $L$ be the midpoint of arc $AD$ (which does not contain the other vertices of the quadrilateral). Prove that the distances from $L$ to the centers of the circles inscribed in triangles $ABM$ and $CDM$ are equal.

M290 Two circles and an isosceles triangie are arranged as shown in figure. Find the altitude of the triangle drawn to its base if the sum of the circles' diameters is $2$.

M292 In triangle $ABC$ angle $A$ is equal to $\alpha$.. The circle that passes through $A$ and $B$ and is tangent to line $BC$ intersects the median drawn to side $BC$ (or its extension) at a point $M$, which is different from $A$. Express the measure of angle $BMC$ in terms of $\alpha$.

M294 In triangle $ABC, \angle BAC=\alpha$ and $\angle ABC=2\alpha$. The circle with center $C$ and radius $CA$ intersects the line containing the bisector of the exterior angle at vertex $B$ at points $M$ and $N$. Express the measures of the angles of triangle $AMN$ in terms of $\alpha$.

M298 Let the perpendicular to side $AD$ of the parallelogram $ABCD$ passing through vertex $B$ intersect line $CD$ at point $M$, and the perpendicular to side $CD$ passing through vertex $B$ intersect line $AD$ at point $N$. Prove that the perpendicular dropped from $B$ onto diagonal $AC$ passes through the midpoint of segment $MN$.

M299 A circle lies entirely inside a given angle. Construct another circle, tangent to the first and to the sides of the given angle. How many such circles are there?

M300 Let a line $m$ be perpendicular to a plane $L$. Three spheres that are tangent in pairs are also tangent to line $m$ and plane $L$. The radius of the largest sphere is $r$. Find the minimum possible radius of the smallest sphere.

M304 In triangle $ABC$ the altitude $CM$ is drawn. The line that is symmetric to the altitude drawn from vertex $A$ about line $CM$ intersects line $BC$ at point $K$. Find angle $OMK$, where $O$ is the center of the circle circumscribed about triangle $ABC$ (the points $O$, $M$, and $K$ are all distinct).

M307 Two intersecting circles are given in the plane.$ A$ is one of the points where the circles intersect. In each circle a diameter is drawn that is parallel to the tangent to the other circle at point $A$, and these diameters do not intersect. Prove that the four endpoints of these diameters lie on the same circle.

(S. Berlov)

M309 Three potnts $A, B$, and $C$ are given in the plane. Draw a line through point $C$ such that the product of the distances from points $A$ and $B$ to this line is greatest. Does such a line always exist?

(N. Vasilyev)

M312 For a given chord $MN$ of a circle, consider all triangles $ABC$ such that $AB$ is a diameter of the given circle not intersecting $MN$ and the sides $AC$ and $BC$ pass through the endpoints of $MN$. Prove that the altitudes of all such triangles drawn from vertex $C$ to side $AB$ meet in a point.

(E. Kulanin)

(I. Sharygin and N. Vasilyev)

M320 Prove that in (a) a regular $12$-gon and (b) a regular $54$-gon there exist four diagonals that meet at a point and do not pass through the center of the polygon.

(S. Tokarev)

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