geometry problems from International Schoolchildren Tournament "Mathematical Multiathlon" (Russia) with aops links in the names
Математическое многоборье
collected inside aops: Seniors , Juniors
2008 - 2017 (lasted only these years)
Reggata Juniors
Inside the rectangle ABCD, whose sides AB = CD = 15 and BC = AD = 10, a point P is given such that AP = 9, BP = 12. Find CP.
Given a convex pentagon ABCDE such that AB = AE = DC = BC + DE = 1 and \angle ABC = \angle DEA = 90^o. What is the area of this pentagon?
An isosceles triangle ABC with a base AC is inscribed in circle \omega. It turned out that the radius of the inscribed circle ABC is equal to the radius of the circle tangent to the smaller arc BC of the circle \omega and the side of the BC in its midpoint (see fig.). Find the ratio of the sides of the triangle ABC.
In a convex polygon A_1A_2...A_{2006} opposite sides are parallel (A_1A_2\parallel A_{1004}A_{1005}, ...). Prove that the diagonals A_1A_{1004},A_2A_{1005},...,A_{1003}A_{2006} intersect at one point if and only if every two opposite sides are equal.
In a triangle ABC with an angle BAC equal to 24^o, the points X and Y are taken on the sides AB and AC, respectively. In this case, a circle centered at Y passing through A also passes through X, and a circle centered at X passing through B also passes through C and Y. Find \angle ABC.
AL, BM, CN are medians of triangle ABC, intersecting at point K. It is known that the quadrilateral CLKM is cyclic and AB = 2. Find the length of the median CN.
Can a right isosceles triangle be split into 6 different right isosceles triangles?
Let E be the intersection point of the diagonals of the convex quadrilateral ABCD. It is known that the perimeters of triangles ABE, BCE, CDE, DAE are the same, and the radii of the inscribed circles of triangles ABE, CE, CDE are equal to 3, 4, 6, respectively. Find the radius of the inscribed circle of triangle DAE.
An equilateral triangle ABC is given. Point D is such that \angle BDC = 90^o and points B, A lie in different half-planes wrt line BC. Point M is the midpoint of side AB. Find the angle BDM.
In triangle ABC, point M is the midpoint of side AB and BD is the angle bisector. Prove that \angle MDB = 90^o if and only if AB=3BC.
In the triangle ABC from the vertex A, the altitude AH was drawn. It turned out that CH:HB=CA^2: AB^2 \ne 1. What values can the angle A take?
In a right-angled triangle ABC with a right angle A, bisectors BB_1 and CC_1 are drawn. From points B_1 and C_1, perpendiculars B_1B_2 and C_1C_2 are drawn on the hypotenuse BC. What is the angle B_2AC_2 ?
Point P lies on side BC of square ABCD. A square APRS was built with side the segment AP . Prove that the angle RCD is 45^o. (The vertices of both squares are labeled clockwise.)
In an isosceles triangle ABC (AB = AC), the angle BAC is 40^o. Points S and T lie on sides AB and BC respectively, such that \angle BAT = \angle BCS = 10^o. Segments AT and CS meet at P. Prove that BT = 2PT.
The angle bisector BD is drawn in an isosceles triangle ABC with base BC. It turned out that BD + DA = BC. Find the angles of triangle ABC.
The h_a and h_b are drawn on the adjacent sides a and b of parallelogram ABCD, respectively. It is known that a + h_a = b + h_b. Consider segments AB, AC, AD, BC, BD, CD. What is the largest number of different ones among them?
On the sides AB and BC of triangle ABC, there are points X and Y, respectively, such that \angle BYX = \angle AYC and \frac{BY}{Y C} = \frac{2BX}{XA} . Prove that triangle ABC is right-angled.
On the lateral side AB of a right trapezoid ABCD (AB\perp BC), a semicircle is constructed (having it as diameter) that touches the side CD at point K. The diagonals of the trapezoid meet at point O. Find the length of the segment OK if the lengths of the bases of the trapezoid ABCD are equal to 2 and 3.
In trapezoid ABCD with bases AB and CD, it turned out that AD = DC = CB <AB. Points E and F lie on the sides CD and BC are respectively such that \angle ADE = \angle AEF. Prove that 4CF \le BC.
The quadrilateral ABCD is inscribed in a circle. It is known that AB = AC and BC = CD. The diagonals of the quadrilateral ABCD meet at the point O, and the point X is the midpoint of the arc CD not containing the point A. Prove that XO \perp AB.
Points M, N, P are the midpoints of sides AB, CD and DA of the inscribed quadrilateral ABCD. It is known that \angle MPD=150^o, \angle BCD=140^o. Find the angle \angle PND.
(D. Maksimov)
There are 2013 line segments of unit length on the plane, each intersecting with everyone. Prove that all of them can be covered with a circle of radius 1.5.
(A. Shapovalov)
A circle center O is inscribed in the quadrilateral ABCD. AB is parallel to and longer than CD and has midpoint M. The line OM meets CD at F. CD touches the circle at E. Show that DE = CF iff AB = 2CD.
In parallelogram ABCD, BD=BC. A point M on AC is such that 3AM=AC. Prove that AM=BM.
(13th Ural Tournament, Major League)
A circle with center O is inscribed in the triangle ABC. The point L lies on the extension of the side AB beyond A. The tangent from L intersects the side AC in the point K. Find \angle KOL if \angle BAC = 50^o.
(revision of the problem of the Soros Olympiad 1995)
From the point A_0 black and red rays are drawn with an angle 7^o between them. Then a polyline A_0A_1...A_{20} is drawn (possibly self-intersecting, but with all vertices different), in which all segments have length 1, all vertices with even numbers lie on the black ray and the ones with odd numbers - on the red ray. What is the index of the vertex that is farthest from A_0?
(old Mathematical Kangaroo)
In the convex quadrilateral ABCD, the bisectors of the angles A and C are parallel and intersect the diagonal BD in two distinct points P and Q, so that BP = DQ. Prove that the quadrilateral ABCD is a parallelogram.
(A.Chapovalov)
A paper rectangle ABCD (AB = 3, BC = 9) is folded in such way that vertices A and C coincide. What is the area of the obtained pentagon?
(additional problems from Mathematical Kangaroo 2014)
Different points A,B and C are marked on the straight line in such a way that AB=AC=1. On the segments AB and AC, a square ABDE and an equilateral triangle ACF are constructed in one half-plane, respectively. Find the angle between lines BF and CE.
In a quadrilateral ABCD, point M is the midpoint of side AB. Prove that if the angle DMC is right , then AD+BC \ge CD .
An angle bisector AD is drawn in an acute-angled triangle ABC. The perpendicular drawn from point B on line AD intersects the circumscribed circle of the triangle ABD at a point E other than B. Prove that points A,E and the center of the circumcircle O of triangle ABC are collinear.
On side AB of triangle ABC, points K and L are selected in such a way that \angle ACK = \angle KCL = \angle LCB. The point M on the side BC is such that \angle BKM =\angle MKC . It turned out that ML is the bisector of the angle KMB. Find \angle CLM.
Given a parallelogram ABCD. Points E ,F are marked on lines AB ,BC respectively such that AF=AB and CE=CB, and the points E and F do not coincide with B. Prove that DE=DF.
The vertices of a convex polygon are located at the nodes of an lattice points, moreover, none of its sides passes along the lattice lines. Prove that the sum of the lengths of the vertical segments of the lattice lines enclosed within the polygon, equal to the sum of the horizontal lines
In a regular hexagon ABCDEF, the point M is the midpoint of the diagonal AC, N is the midpoint of side DE. Prove that triangle FMN is equilateral.
P and Q are the midpoints of sides BC and AD of rectangle ABCD , respectively. Diagonals of rectangle PQDC meet at point R. It turned out that AP is a bisector of the angle BAR. Find the length of the side BC if AB=1.
Is it possible to mark points A, B, C, D, E on a straight line so that the distances between them in centimeters are equal: AB = 6, BC = 7, CD = 10, DE = 9, AE = 12 ?
On the sides BC and AB of the triangle ABC, there are points L and K, respectively, such that AL is the bisector of the angle BAC, ZACK = ZABC, ZCLK = ZBKC. Prove that AC = KB
The diagonals of the convex quadrilateral ABCD are perpendicular and intersect at the point O, and BC = AO. Point F is such that CF \perp CD and CF = BO. Prove that triangle ADF is isosceles.
In an acute-angled triangle ABC on the side AC, a point P is chosen such that 2AP = BC. Points X and Y are symmetric to point P wrt vertices A and C. It turned out that BX = BY. What is the angle C of the original triangle?
Team Juniors
Find the locus of the points of intersection of the medians of the triangles, all the vertices of each of which lie on different sides of the given square.
Two unequal circles \omega_1 and \omega_2 of radii r_1 and r_2, respectively, intersect at points A and B. On the plane, a point O is taken, for which \angle OAB is 90^o, and a circle \omega with center O is drawn, tangent to the internally to the circles \omega_1 and \omega_2. Find the radius of the circle \omega.
Point D is selected inside the triangle ABC. The circumscribed circles of triangles CAD and CBD intersect the segments CB and CA, at points E and F , respectively. It turned out that BE = AF. Prove that CD is the bisector of the angle \angle ACB.
You are given a parallelogram ABCD. On rays DB and AC, there were such points K and L, respectively, that KL \parallel BC, \angle BCD = 2 \angle KLD. Prove that AK \perp DL.
In an acute-angled triangle ABC, a square is inscribed with side m so that its two vertices lie on the side AB, and one vertex on the sides BC and AC . Denote h_c the altitude of the triangle ABC drawn from the vertex C , and c the side AB. Prove that \frac{1}{h_c}=\frac{1}{m}+\frac{1}{c}.
In the non-isosceles triangle ABC, let M be the point of intersection of the medians and I be the point of intersection of the angle bisectors . It turned out that line MI is perpendicular to the side BC . Prove that AB + AC = 3BC.
The angle bisector AL is drawn in triangle ABC. It is known that AB = 2007, BL = AC. Find the sides of triangle ABC if known to be integers.
Points P and Q on sideAB of convex quadrilateral ABCD are such that AP = QB. Point X is a non-D intersection point of the circumscribed circles of triangles APD and DQB, and point Y is a non-C intersection point of the circumscribed circles of triangles ACP and QCB. Prove that points C, D, X and Y lie on the same circle.
A paper triangle with sides a, b, c bent in a straight line so that the vertex opposite side of length c, hit this side. It is known that in the resulting quadrilateral two angles are equal, to the fold line. Find the lengths of the segments into which the vertex that gets there divides side c.
Given an isosceles triangle ABC (AB = BC). On the side AB, point K is selected, and on the side BC point L so that AK + CL = \frac12AB. Find the locus of the midpoints of the line segments KL.
In a regular heptagon ABCDEFG, the sides are 1. The diagonals AD and CG meet at a point H. Prove that FH = \sqrt2.
Inside the triangle ABC is chosen point P such that \angle ABP=\angle CPM, where M is the midpoint of the segment AC. The line MP intersects the circumcircle of the triangle APB at points P and Q. Prove that QA=PC.
Several chords are drawn in a circle so that every pair of them intersects inside the circle. Prove that all the drawn chords can be intersected by the same diameter.
(A.Chapovalov)
The point I_b is the center of an excircle of the triangle ABC, that is tangent to the side AC. Another excircle is tangent to the side AB in the point C_1. Prove that the points B, C, C_1 and the midpoint of the segment BI_b lie on the same circle.
(inspired by olympiad of the Faculty of Mathematics and Mechanics of SPBU)
Let AB be a diameter of circle \omega. \ell is the tangent line to \omega at B. Given two points C, D on \ell such that B is between C and D. E, F are the intersections of \omega and AC, AD, respectively, and G, H are the intersections of \omega and CF, DE, respectively. Prove that AH = AG.
Let ABC be an isosceles triangle (AB = AC). On the extensions of the sides BC, AB and AC, points P, X, Y are selected in such a way that PX \parallel AC and PY \parallel AB and point P lies on the ray CB . Point T is the midpoint of the arc BC of the circumscribed circle of the triangle ABC (T \ne A). Prove that PT \perp XY.
Construct a right-angled triangle for a given hypotenuse c, if it is known that the median drawn to c is the geometric mean of its legs.
A quadrilateral ABCD is inscribed in a circle. Lines AB and CD meet at point E, lines AD and BC met at point F. The bisector of angle AEC intersects side BC at point M and side AD at point N, and the bisector of angle BFD intersects side AB at point P and side CD at point Q. Prove that the quadrilateral MNPQ is a rhombus.
In a right-angled triangle ABC, the altitude CH is drawn to the hypotenuse. The bisector BD of angle B intersects the altitude at point E. Let K be the intersection point of line segments AE and HD. Prove that the quadrilateral CDKE and the triangle AHK have equal areas.
(Peru Geometrico)
Geometry Round Juniors
Inside the triangle ABC, a point M is taken such that \angle CMB = 100^o . Perpendicular bisectors of BM and CM intersect the sides AB and AC, respectively, at points P and Q. Points P, Q and M lie on one straight line. Find the value of \angle CAB .
Given a quadrilateral ABCD inscribed in a semicircle \omega with diameter AB. Lines AC and BD intersect at point E, lines AD and BC intersect at point F. Line EF intersects semicircle \omega at point G, and line AB at point H. Prove that E is the midpoint of segment GH if and only if G is midpoint of FH.
An acute-angled triangle ABC was drawn on the plane, in which the angle is A\ne 60^o. The point of intersection of the altitudes H and the center of the circumscribed circle O was marked in it, then straight lines m = BH, n = CH were drawn. After that, all but the straight lines were erased from the drawing, those m and n, points O (that is, there were two straight lines and one point). Reconstruct triangle ABC using a compass and ruler.
Let ABCD be a convex quadrilateral with \angle DAC = 30^o, \angle BDC = 50^o, \angle CBD = 15^o, and \angle BAC = 75^o. The diagonals of the quadrilateral intersect at point P. Find the value of \angle APD .
ABC is an acute-angled triangle, AD is its angle bisector, and BM is its altitude . Prove that \angle DMC> 45^o.
ABCD is trapezoid with bases BC and AD. Equilateral triangles ADK and BCL are built on the bases ouside ABCD. Prove that AC, BD and KL meet at one point.
Cut a square into 5 pieces, from which you can make 3 pairs of different squares.
In \vartriangle ABC, AB = AC, \angle BAC = 100^o. Inside \vartriangle ABC, a point M is taken such that \angle MCB = 20^o, \angle MBC = 30^o. Find \angle BAM.
The median BD is drawn in the triangle ABC, and the point of intersection of the medians G is marked on it. A straight line parallel to BC and passing through point G intersects AB at point E. It turned out that \angle AEC = \angle DGC. Prove that \angle ACB = 90^o.
Points X and Y are selected on the sides AB and CD of rectangle ABCD, respectively. The segments AY and DX intersect at the point P, and the segments CX and BY intersect at the point Q. Prove that PQ\ge \frac12 AB.
In an acute-angled triangle ABC, the altitude AD is drawn. Points M and N are symmetrical to point D wrt lines AC and AB, respectively. Ray AO (where O is the center of the circumscribed circle of triangle ABC) intersects BC at point E. Prove that \angle CME = \angle BNE.
The bisector of angle A of triangle ABC intersects the circumscribed circle of triangle ABC at point M, and the side BC at point A_1. A circle \omega was drawn through points B and C with center at point M. Chord XY of circle \omega passes through point A_1. Prove that the centers of the inscribed circles of triangles ABC and AXY coincide.
In triangle ABC, the median AA_1 is drawn and point M, the point of intersection of the medians, is marked on it . Point K lies on side AB is such that MK \parallel AC. It turned out that AM = CK. Find the angle ACB .
2011 points are marked on the plane. A pair of marked points A and B is said to be isolated if all other points are strictly outside the circle constructed on AB as the diameter. What is the smallest number of isolated pairs possible?
In triangle ABC, points M and L on side BC are the feet of the median and angle bisector, respectively, drawn from vertex A. Points P and Q are the feet of perpendiculars drawn from point L on sides AB and AC, respectively. Point X lies on the median AM such that XL\perp BC. Prove that points P, X and Q are collinear.
In the convex quadrilateral ABCD, it turned out that AB + CD =\sqrt2 AC and BC + DA = \sqrt2 BD. Prove that ABCD is a parallelogram.
In a parallelogram ABCD, the angle bisector at A meets side BC in its midpoint M. Assume that \angle BDC = 90^o. Find the angles of the parallelogram ABCD.
In a trapezoid ABCD with the parallel sides AD and BC, the diagonals are orthogonal. The line parallel to AD and passing through the intersection of the diagonals meets the lateral sides AB and CD at points K and L respectively. Point M on side AB is such that AM = BK. Prove that LM = AB.
The incircle of an isosceles triangle ABC with AB = BC is tangent to BC and AB at E and F respectively. A half-line trough A inside the angle EAB intersects the incircle at points P and Q. The lines EP and EQ meet the line AC at P' and Q'. Prove that P'A = Q'C.
Prove or disprove that any triangle of area 3 can be covered by an axially symmetric convex polygon of area 5.
Points D on the side AC and E on the side BC of triangle ABC are such that \angle ABD=\angle CBD=\angle CAE and \angle ACB=\angle BAE. Let F be the intersection point of segments BD and AE. Prove that AF=DE.
(Ф.Ивлев, Ф.Бахарев)
A hexagon ABCDEF is inscribed into a circle. X is the intersection point of the segments AD and BE, Y is the intersection of AD and CF, and Z is theintersection of BE and CF. Given AX=DY and CY=FZ, prove that BX=EZ.
(Д.Максимов, Ф.Петров)
A quadrilateral ABCD is inscribed into a circle, given AB>CD and BC>AD. Points K and M are chosen on the rays AB and CD respectively in such a way that AK=CM=\frac 12 (AB+CD). Points L and N are chosen on the rays BC and DA respectively in such a way that BL=DN=\frac 12 (BC+AD). Prove that the KLMN is a rectangle of the same area as ABCD.
(Eisso J.Atzema, proposed by В.Дубровский)
Point O is the circumcenter and point H is the orthocenter in an acute non-isosceles triangle ABC. Circle \omega_A is symmetric to the circumcircle of AOH with respect to AO. Circles \omega_B and \omega_C are defined similarly. Prove that circles \omega_A, \omega_B and \omega_C have a common point, which lies on the circumcircle of ABC.
(Ф.Бахарев, inspired by Iran TST 2013)
In a right triangle ABC with the right angle B, the angle bisector CL is drawn. The point L is equidistant from the points B and the midpoint of the hypotenuse AC. Find the angle BAC.
(F.Nilov)
In a convex quadrilateral ABCD the equality \angle BCA +\angle CAD = 180^o holds. Prove that AB + CD \ge AD + BC.
(A. Smirnov inspired by Serbian regional olympiad 2014)
You are given a circle and its chord AB. At the ends of the chord to the circle are drawn tangent and equal segments AK and BL, lying on different sides wrt line AB. Prove that line AB divides the segment KL in half.
Consider a pentagonal star formed by diagonals of an arbitrary convex pentagon.
Let's circle 10 segments of its outer contour one by one solid and dotted lines (see figure).
Prove that the product of the lengths of solid segments is equal to the product of the lengths of the dotted segments .
On the bisectors of angles A, B, C, D of the convex quadrilateral ABCD taken points A ', B', C ', D', respectively, so that line A'B' is parallel to AB, line B'C' is parallel to BC and line C'D' is parallel to CD. Prove that
a) the line D'A' is parallel to DA;
b) B'D '|| AC, if additionally it is known that A'C '|| BD .
Angle B of triangle ABC is twice the angle C. Circle of radius AB with the center A intersects the perpendicular bisector of the segment BC at point D, lying inside the angle BAC. Prove that \angle DAC = \frac13 \angle A.
Is it possible to construct a triangle (with a compass and a unmarked ruler) given two given angles \alpha and \beta and
a) known perimeter P
b) any any altitude of the triangle?
If it is possible, give the construction algorithm (sequence of actions) and indicate the number of different triangles in each case; if not ,justify your answer.
Find the sides of a right triangle if it's perimeter P and it's area S are known
In the sea off the coast of Kamchatka, three suspicious fishing vessels are regularly recorded, traditionally located at the tops of one isosceles triangle with the angle 120^o and lengths of the sides 20 in km. Find the locus of points (GMT) in the space where it is possible to place technical means of control (with a range of no more than 30 km) for illegal catch of Kamchatka crab, so that all points of this GMT are equidistant from vessels - potential violators of environmental legislation.
Two straight-line railways intersect at point N at an angle of 60^o. Inside this angle there is an airfield (point A) at distances of 10 km and 20 km from these roads. Find the locations of points B and C (places of loading and unloading cargo) on these roads so that the cost of cyclic transportation along the highway from A to B, then to C with a return to A is minimal. Considering that the transportation of 1 ton of cargo along the highway ABCA costs 100 rubles for 1 km, determine
a) is it possible for a cargo of 10 t to keep within the amount of 50 thousand rubles, excluding the costs of loading and unloading cargo?
b) the same question for the amount of 60 thousand rubles taking into account the cost of a full reloading of goods (2,500 rubles per 10 t) at 2 points out of 4? in 3 points out of 4? at all 4 points?
The bisector BD of angle B is drawn in an isosceles triangle ABC (AB = AC). The perpendicular to BD at point D intersects line BC at point E. Find BE if CD = d.
(A. A. Egorov)
The billiard table has a parallelogram shape. Two balls, placed in the middle of one of the sides, hit so that they bounced off different adjacent sides, after which both hit the same point on the opposite side. One ball traveled twice the distance before bouncing than after. Find the ratio of the lengths of the path segments before and after the bounce for the other ball.
(I.N.Sergeev)
Equilateral triangle ABC and right-angled triangle ABD are constructed on segment AB on opposite sides of it, in which \angle ABD = 90^o, \angle BAD = 30^o. The circumcircle of the first triangle intersects the median DM of the second at point K. Find the ratio AK: KB.
(A variation of the Nguyen Dung Thanh problem from Cut-the-knot)
Points K, L, M and N are taken on the sides AB, BC, CD, and AD of the convex quadrilateral ABCD, respectively, so that KLMN is a rectangle and AK <KB, BL> LC and CM <MD. Can the area of this rectangle be more than half the area of the quadrilateral ABCD?
(I.N.Sergeev)
Reggata Seniors
Inside the rectangular parallelepiped ABCDA'B'CD' there are two balls, so that the first one touches all three faces containing A, and the second three faces containing C'. In addition, the balls touch each other. The radii of the balls are 10 and 11, respectively. The lengths of the edges AB and AD are 25 and 26. Find the length of the edge AA'.
ABC is an equilateral triangle. On one side of the plane ABC, the perpendicular to it segments were laid AA'= AB and BB' = AB / 2. Find the angle between the planes ABC and A_1B_1C.
A quadrilateral ABCD is inscribed in a circle \omega and such that AB = AD and BC = CD (deltoid). It turned out that the radius of the inscribed circle ABC is equal to the radius of the circle tangent to the smaller arc BC of the circle \omega and the side of the BC in its midpoint (see fig. ). Find the ratio of the side AB to the radius of the inscribed circle of the triangle ABC.
Let AB be the diameter of the circle \omega. Line \ell touches \omega at point B. Points C and D lie on line \ell, and point B lies between points C and D. Lines AC and AD intersect circle \omega at points E and F, respectively. Lines CF and DE intersect the circle \omega at points G and H. Prove that AH = AG.
AB is the diameter of the unit circle centered at point O. C and D are points on the circle such that AC and BD intersect inside the circle at point Q and \angle AQB = 2\angle COD. Find the distance from O from line CD.
The circumscribed circle of triangle ABC is fixed. Find the locus of the points of intersection of the medians of the triangles ABC.
The radius of the circumscribed circle of the triangle is 2, and the lengths of all altitudes are integers. Find the sides of the triangle.
B_1 is the midpoint of the side AC of the triangle ABC , C_1 is the midpoint of the side AB of the triangle ABC . The circumscribed circles of triangles ABB_1 and ACC_1 intersect at point P. Line AP intersects the circumscribed circle of the triangle AB_1C_1 at point Q. Find \frac{AP}{AQ}.
The altitude AH and the ange bisector AL were drawn in the triangle ABC. It turned out that BH: HL: LC = 1: 1: 4 and the side AB is equal to 1. Find the lengths of the other sides of the triangle ABC.
Points A, B, C, D, X are given on the circle, and \angle AXB = \angle BXC = \angle CXD. The distances AX = a, BX = b, CX = c are known. Find DX.
A segment AB is given. For any point C, we mark a point B_1 on the segment AC such that AB_1: B_1C = 1: 2, and a point A_1 on the segment BC such that BA_1: A_1C = 2: 1. Find the locus of the intersection points AA_1 and BB_1, provided that the points A, A_1, B, B_1 lie on the same circle.
Given a rectangular trapezoid ABCD (BC\parallel AD and AB \perp AD), the diagonals of which are perpendicular and intersect at point M. On the sides AB and CD, points K and L are selected, respectively, that MK is perpendicular to CD, and ML is perpendicular to AB. It is known that AB = 1. Find the length of the segment KL.
In a regular pyramid ABCDS (S vertex), the length of AS is 1, and the angle ASB is equal to 30^o. Find the length of the shortest path from A to A that intersects all lateral edges other than AS.
In the quadrilateral ABCD with angles \angle A = 60^o, \angle B = 90^o, \angle C = 120^o, M isthe point of intersection of the diagonals . It turned out that BM = 1 and MD = 2. Find the area of ABCD.
Points A, B, C are given on the straight line, and point B lies between A and C, AB = 3 and BC = 5. Let BMN be an equilateral triangle. Find the smallest value of AM + CN.
In the inscribed hexagon ABCDEF, it turned out that AB = BC, CD = DE and EF = FA. Prove that S_{ABCDEF} = 2S_{BDF}.
Points K, L and M are taken on the sides AB, BC and AC of a triangle ABC, respectively. Suppose that the circumradii of the triangles AKM, BKL, CLM and KLM are equal. Prove that the triangles ABC and KLM are similar.
Through a point X inside a square ABCD, segments PQ and EF parallel to the sides AD and AB respectively are constructed, with the endpoints on the sides of the square (P on AB, F on AD). If S_{ECQX} = 2S_{PXFA}, determine \angle EAQ.
In a tetrahedron SABC, the circumradii of the faces SAB, SBC and SAC are equal to 108. The radius of the inscribed sphere of the tetrahedron equals 35, and the distance between its center and S equals 125. Find the radius of the circumsphere of the tetrahedron, assuming that its center lies inside the tetrahedron.
Points K, L, M and N lie on the sides AB, BC, CD and DA of a square ABCD, respectively. If \angle KLA = \angle LAM = \angle AMN = 45^o, prove that KL^2 + AM^2 = LA^2 +MN^2.
Points M and N are chosen on a hypotenuse AC of right isosceles triangle ABC. Given that \angle MBN=45^o, prove that it is possible to construct a right triangle from the segments AM, MN and NC.
(folklore)
Given a point Q inside a convex polyhedron M. A line \ell passes through Q and intersects the surface of M at points A and B. Prove that for infinitely many of lines \ell the equality AQ= BQ holds.
(Putnam 1977 B4 )
A quadrilateral ABCD is inscribed in a circle. K, L, M and N are chosen on the segments AB, BC, CD and DA respectively in such a way that AK=KB=6, BL=3, LC=12, CM=4, MD=9, DN =18, NA=2. Prove that quadrilateral KLMN can be inscribed into a circle.
(Ф.Бахарев)
All the vertices of a regular polygon lie on the surface of a cube, but its plane does not contain any of the cube’s faces. What is the maximum possible number of vertices of such a polygon?
(A.Chapovalov)
The incircle of the triangle ABC touches the side AC at D. The \angle BDC is equal to 60^o. Prove that the inscribed circles of triangle ABD and CBD touch BD at the same point and find the ratio of the radii of these circles.
(M.Volchkevich)
On the sides BC, CA and AB of an acute triangle ABC the points A_1, B_1 and C_1 are chosen respectively. The circumscribed circles of the triangles AB_1C_1, BC_1A_1 and CA_1B_1 intersect at P. The points O_1, O_2 and O_3 are the centres of these circles. Prove that 4S_(O_1O_2O_3) \ge S_(ABC).
(A. Smirnov inspired by Macedonia 2014)
P is any point inside a triangle ABC. The perimeter of the triangle AB+BC+CA=2s. Prove that s < AP+BP+CP < 2s.
Let ABC be an acute-angled triangle in which \angle ABC is the largest angle. Let O be its circumcentre. The perpendicular bisectors of BC and AB meet AC at X and Y respectively. The internal angle bisectors of \angle AXB and \angle BYC meet AB and BC at D and E respectively. Prove that BO is perpendicular to AC if DE is parallel to AC .
Let \vartriangle ABC be an acute scalene triangle, and let N be the center of the circle which pass through the feet of altitudes. Let D be the intersection of tangents to the circumcircle of \vartriangle ABC at B and C. Prove that A, D and N are collinear iff \angle BAC = 45^o.
ABC is an acute angle triangle such that AB>AC and \angle BAC=60^o. Let’s denote by O the center of the circumscribed circle of the triangle and H the intersection of altitudes of this triangle. Line OH intersects AB in point P and AC in point Q. Find the value of the ratio PO:HQ.
Given triangle ABC with right angle C, find all points X on the hypotenuse AB such that CX^2 =AX \cdot BX.
Given triangle ABC, point O is the center of its excircle that is tangent to the side BC and extensions of the sides AB and AC. Prove that points A,B,C and the center of the circumcircle of triangle ABO are concyclic.
Given isosceles triangle ABC (AB=BC), D is the midpoint of its base AC, E is the projection of D on the side BC, and F is the second intersection of circumcircle of the triangle ADB with the segment AE. Prove that the line BF bisects the segment DE.
Triangle ABC with angle bisector BL is given such that BL=AB. Point M on the extension of BL beyond point L is such that \angle BAM+\angle BAL=180^o. Prove that BM=BC.
AD is a diameter of the circumcircle of the quadrilateral ABCD. Point E is symmetric to the point A with respect to the midpoint of BC. Prove that DE \perp BC.
Incenter I of the acute triangle ABC lies on the bisector of an acute angle between the altitudes AA_1 and CC_1. Bisector of the angle B intersects the opposite side of the triangle at the point L. Prove that points A_1, I, L, C lie on one circle.
Given point P inside the triangle ABC such that points symmetrical to P with respect to the midpoint of BC and to the bisector of angle A lie on one line with the point A. Prove that projections of the point P on the sides AB and AC are equidistant from the midpoint of BC.
Point L on the bisector of the angle A of the triangle ABC is such that \angle LBC = \angle LCA = \angle LAB. Prove that the lengths of the sides of the triangle form a geometric progression.
Geometry Round Seniors
On the plane, two disjoint circles of equal radius are given and point O is the midpoint of a segment with ends at the centers of these circles. A straight line \ell, parallel to the line of the centers of these circles, intersects them at points A, B, C and D. A straight line m passing through O intersects them at points E, F, G and H. Find the radius of these circles if it is known that AB=BC=CD=14 and EF=FG=GH=6.
Two circles were drawn on the plane \omega_1 and \omega_2 and points A and B on them so that the segment AB has the greatest possible length. AR and AS are tangents from point A to circle \omega_2. BP and BQ are tangents from point B to circle \omega_1. The circle \alpha_1 touches the circle \omega_1 internally and the rays AR and AS. The circle \alpha_2 touches the circle \omega_2 internally and the rays BP and BQ. Prove that the radii of the circles \alpha_1 and \alpha_2 are equal.
Given a triangle ABC and points D and E on the sides AB and AC such that DE \parallel BC. Let P be an arbitrary point inside \vartriangle ADE. Lines PB and PC meet DE at points F and G, respectively. Let the circumcircles of triangle PDG and triangle PFE meet for the second time at point Q. Prove that points A, P, Q are collinear.
Are there several points and several straight lines on the plane such that each point lies exactly on 2008 lines and exactly 2008 points lie on each line?
ABC is an acute-angled triangle, AD is its angle bisector, and BM is its altitude . Prove that \angle DMC> 45^o.
A quadrilateral ABCD is inscribed in a circle with a diameter of BD. Point A_1 is symmetrical to point A wrt straight line BD, point B_1 is symmetric to point B wrt straight line AC. Let P be the point of intersection of lines CA_1 and BD, Q be the point of intersection of lines DB_1 and AC. Prove that AC \perp PQ.
An arbitrary tetrahedron is given. The segments connecting the vertices with the intersection points of the medians of the opposite faces intersect at the point M. Prove that at least two of the projections of the point M on the faces of the tetrahedron fall inside the corresponding faces.
Let \omega be the circumscribed circle of a non-isosceles triangle ABC. Circle \gamma touches lines AB and AC at points P and Q, respectively, and circle \omega internally at point T. Lines AT and PQ intersect at point S. Prove that the circumcircle of triangle IST, where I is the center of the incircle of triangle ABC, touches \omega.
Let H be the orthocenter of an isosceles triangle ABC (AC = AB). The point L lies on the base BC such that LH\parallel AC, and the point M lies on the side AC such that ML \parallel AB. Ray LH intersects AB at point K. Prove that \angle LMK = 90^o.
Circles \omega_1 and \omega_2 intersect at points P and Q. Segments AC and BD are chords of circles \omega_1 and \omega_2, respectively, and segment AB and ray CD intersect at point P. Ray BD and segment AC intersect at point X. Point Y lies on circle \omega_1 that PY\parallel BD. The point Z lies on the circle \omega_2 such that PZ\parallel AC. Prove that points Q, X, Y and Z are collinear.
The inscribed circle in triangle ABC touches the side BC at point A_1. Segment AA_1 intersects the inscribed circle for the second time at point P. Segments CP and BP intersect the inscribed circle at points M and N, respectively. Point Q is the midpoint of the segment MN. Prove that \angle MQA_1= \angle MQP.
Point P is marked inside the triangle ABC. Let h_a, h_b and h_c be the respective altitudes of the triangle ABC. Prove that\frac{PA}{h_b+h_c}+\frac{PB}{h_a+h_c}+\frac{PC}{h_a+h_b}\ge 1
In rectangle ABCD, point P is the midpoint of side AB, and point Q is the foot of the perpendicular drawn from point C on PD. Prove that BQ = BC.
Given a tetrahedron ABCD. The sphere passing through the vertices A, B and C intersects intersects the side edges DA, DB, and DC at points A_1, B_1, and C_1. Refelct these points wrt the midpoint of the corresponding edges and obtainpoints A_2, B_2 and C_2. Prove that points A, B and C are equidistant from the center of the circumscribed sphere of the tetrahedron DA_2B_2C_2 .
Points A, B, C, D lie on the circle in the indicated order, with AB and CD not parallel. The length of the arc AB containing points C and D is twice the length of the arc CD that does not contain points A and B. Point E is specified by the conditions AC = AE and BD = BE. It turned out that the perpendicular from point E to line AB passes through the midpoint of the arc CD not containing points A and B. Find \angle ACB.
2011 points are marked on the plane. A pair of marked points A and B is said to be isolated if all other points are strictly outside the circle constructed on AB as on the diameter. What is the largest number of isolated pairs that can be?
Let ABC be a triangle with a right angle at C and let M be the midpoint of the side AB. Point Q on side CB is such that \frac{BQ}{QC }= 2. Prove that \angle QAB = \angle QMC.
The incircle of an isosceles triangle ABC with AB = BC is tangent to BC and AB at E and F respectively. A half-line trough A inside the angle EAB intersects the incircle at points P and Q. The lines EP and EQ meet the line AC at P' and Q'. Prove that P'A = Q'C.
Let P be an arbitrary point inside a tetrahedron ABCD. Denote by R the circumradius of the tetrahedron and by x the distance from P to the circumcenter. Prove the inequalities(R + x) (R - x)^3 \le PA\cdot PB\cdot PC\cdot PD \le (R + x)^3 (R - x)
Let ABC be a scalene triangle. Denote by I_A the center of the excircle at side BC and by A_1 its tangency point with the corresponding side. Points I_B, I_C, B_1, C_1 are defined analogously. Show that the circumcircles of triangles AI_AA_1, BI_BB_1 and CI_CC_1 have two common points.
AA_1 and BB_1 are the altitudes in an acute triangle ABC, and O is its circumcenter. Prove that the areas of the triangles AOB_1and BOA_1 are equal.
(folklore)
Two circles \omega and \gamma have the same center, and \gamma lies inside \omega. Let O be an arbitrary point on \omega. OA and OB are the tangent lines through O to \gamma. Circle with center O and radius OA meets \omega at points C and D. Prove that the line CD contains a midline of the triangle OAB.
(F. Ivlev)
For which a>1 there exists a convex polyhedron such that the ratio of areas of any two of its faces is greater than a?
(A. Shapovalov)
Let H, I, O be the orthocenter, incenter and circumcenter of an acute triangle ABC respectively. Prove that \angle AIH= 90^o if and only if OI \parallel BC.
(F. Ivlev)
A closed six-segmented polyline in space is given. Each of its segments is parallell to one of the orthogonal coordinate axes. Prove that its vertices lie on one sphere or in one plane.
(Iran 2014)
A set M of points in the 3-dimensional space is called interesting, if for any plane there exist at least 100 points in M outside this plane. For which minimal d any interesting set contains an interesting subset with at most d points?
(IMC 2013.9)
Two congruent circles b and c touching each other are inscribed in the angles B and C of a square ABCD. A tangent to circle b is drawn from vertex A and a tangent to c is drawn from D (see fig. ). Prove that the circle inscribed in the triangle bounded by these tangents and the side AD is congruent to the given circles.
(a) Let A_1A_2A_3A_4A_5 be a pentagram, i.e., a closed self-intersecting five-edge polyline (see fig. ). Denote by B_1, B_2, ..., B_5 the points of intersection of its nonadjacent edges: B_n is the intersection point of A_{i}A_{i+1} and A_kA_{k+1}, where the indices of all points are different and we assume A_6=A_1. Prove that A_1B_2\cdot A_2B_3\cdot A_3B_4\cdot A_4B_5\cdot A_5B_1 = B_1A_2\cdot B_2A_3\cdot B_3A_4\cdot B_4A_5\cdot B_5A_1.
(b) Prove that this equation holds for any polyline A_1A_2A_3A_4A_5 that has no parallel edges if we define B_n as the intersection points of the same edges as in (a) or their extensions.
Let n + 1 points, A_1, A_2, ..., A_n and B, be given in the plane such that no three of them are collinear. It is known that for any two points A_i and A_j one can find a third point A_k that completes a triangle A_iA_jA_k containing the point B in its interior. Prove that n is odd.
On the sides BC and AC of a triangle ABC points B' and A', respectively, are taken. The circumcircle of the triangle ABC meets the line through C parallel to A'B' in a point D (other than C). The circumcircle of the triangle A'B'C meets the line through C parallel to AB in a point E (other than C). Prove that the lines AB, A'B', and DE are concurrent.
A circle inscribed in the right angle of an isosceles right triangle divides the hypotenuse into three equal parts. The leg of the triangle is of length 1. Find the radius of the circle.
The length of each bisectors of a triangle is at least one unit. Prove that its area is at least \sqrt3 / 3.
A chain of three equilateral triangles ABC, CDE, and EFG, whose vertices are written counterclockwise, is arranged so that D is the midpoint of AG. Prove that triangle BFD is also equilateral.
For any four vertices of a convex polyhedron, the tetrahedron with these vertices lies inside the polyhedron. Does there exist a polyhedron such that none of such tetrahedrons lies inside it?
Find the angles of a triangle ABC, knowing that its altitude CD and angle bisector BE meet at a point M such that CM = 2MD and BM = ME.
(A. A. Egorov)
Is it possible to construct a 27-gon that has an inscribed circle and side lengths 1, 2, ..., 27?
You can arrange its sides in any desired order.
(I. A. Sheipak)
Eight wooden balls are placed in a cubic box measuring 1\times 1\times 1. Can the sum of their radii be greater than 2?
(V. N. Dubrovsky, K. A. Knop)
Let KMLN be a square and let CML be a right triangle constructed externally on the side ML as hypotenuse. The sides CM and CL are extended to meet the line KN at A and B, respectively. Denote by P the intersection point of segments AL and KM, and by Q, the intersection point of BM and NL. Prove that CPQ is an isosceles triangle.
(Stan Fulger)
Team Seniors
Two unequal circles \omega_1 and \omega_2 of radii r_1 and r_2, respectively, intersect at points A and B. On the plane, a point O is taken, for which \angle OAB is 90^o, and a circle \omega with center O is drawn, tangent to the internally to the circles \omega_1 and \omega_2. Find the radius of the circle \omega.
A point on a side of a equilateral triangle is projected onto the other two sides. Prove that the line connecting the starting point with the center of the triangle bisects the segment between the projections.
Triangle ABC with an obtuse angle C is inscribed in a circle with center O. It turned out that AC + BC = 2OC. The segment OC intersects the side AB at point D. Prove that the inscribed circles of triangles ADC and BDC are equal.
Points P and Q are taken inside the triangle ABC such that \angle ABP = \angle QBC and \angle ACP = \angle QCB. Point D lies on the segment BC. Prove that \angle APB + \angle DPC = 180^o if and only if \angle AQC + \angle DQB = 180^o.
A circle \Omega is given. Circles \omega_1, \omega_2, \omega_3, \omega_4, \omega_5 and \omega_6 touch \Omega internally. In addition, the circles \omega_1, \omega_3 and \omega_5 touch each other externally, the circle \omega_2 touches the circles \omega_1 and \omega_3 externally, the circle \omega_4 touches the circles \omega_3 and \omega_5 externally, the circle \omega_6 touches the circles \omega_5 and \omega_1 externally (see fig.). Prove that there is a circle tangent to the common internal tangent pairs of circles \omega_i and \omega_{i+1}, i=1,2,3,4,5,6 (we assume \omega_7= \omega_1).
Line segment AL is the angle bisector of triangle ABC, I_1 and I_2 are centers of the circles, inscribed in triangles ABL and ACL, respectively. Line I_1I_2 crosses the sides AB and AC at points C_1 and B_1, respectively. Prove that lines BB_1, CC_1 and AL intersect at one point.
Given a triangle ABC and concentric circles \omega_b,\omega_c centered at A. An arbitrary ray starting from A intersects these circles at points B' and C', respectively. The perpendicular bisector of the segments BB' and CC' meet at the point X. Prove that the points X, constructed in this way for all rays emerging from A, lie on one straight line.
In a regular heptagon ABCDEFG, the sides are 1. The diagonals AD and CG meet at a point H. Prove that FH = \sqrt2.
Let \gamma_1 and \gamma_2 be two circles tangent at point T. Straight lines a and b, intersect the circle \gamma_1 for the second time at points A and B, respectively, and the circle \gamma_2 at points A_1 and B_1, respectively. Let X be an arbitrary point of the plane that does not lie on the given lines a, b and circles \gamma_1 , \gamma_2 . Circles circumscribed around triangles ATX and BTX intersect the circle \gamma_2 at points A_2 and B_2, respectively. Prove that lines TX, A_1B_2 and A_2B_1 meet in one point.
Two medians divide a triangle into a quadrilateral and three isosceles triangles. Prove that initial triangle is also isosceles.
(A. Shapovalov)
Prove that in acute-angled triangle the sum of its medians not greater than the sum of radii of its excircles.
(F. Ivlev)
The point I_b is the center of an excircle of the triangle ABC, that is tangent to the side AC. Another excircle is tangent to the side AB in the point C_1. Prove that the points B, C, C_1 and the midpoint of the segment BI_b lie on the same circle.
(inspired by olympiad of the Faculty of Mathematics and Mechanics of SPBU)
The spheres S_1, S_2 and S_3 are externally tangent to each other and all are tangent to some plane at the points A, B and C. The sphere S is externally tangent to the spheres S_1, S_2 and S_3 and is tangent to the same plane at the point D. Prove that the projections of the point D onto the sides of the triangle ABC are vertices of an equilateral triangle.
(folklore)
Through the center of an equilateral triangle ABC an arbitrary line \ell is drawn that intersects the sides AB and BC in points D and E. A point F is constructed, so that AE = FE and CD = FD. Prove that the distance from F to the line \ell does not depend on the choice of \ell .
(M.Volchkevich)
Let AB be a diameter of circle \omega. \ell is the tangent line to \omega at B. Given two points C, D on \ell such that B is between C and D. E, F are the intersections of \omega and AC, AD, respectively, and G, H are the intersections of \omega and CF, DE, respectively. Prove that AH = AG.
Let ABC be a triangle such that it’s circumcircle radius is equal to the radius of A-excircle. Suppose that the A-excircle touches BC, AC, AB at M, N, L. Prove that O (center of circumcircle) is the orthocenter of MNL
In acute triangle ABC angle B is greater than C. Let M be the midpoint of BC. D and E are the feet of the altitudes from C and B respectively. K and L are midpoints of ME and MD respectively. If the line KL intersects the line through A parallel to BC at point T, prove that TA = TM.
In a triangle ABC with angle A= 15^o the sides denoted by a,b,c and satisfy the equation b =\sqrt {a(a+c)} . Find the area of ABC if the radius of inscribed circle r=1.
(Write the answer as a sum of numbers.)
A tin sheet with area 2a^2 is used to make a closed box with the shape of a cuboid with maximum volume V_m=\max V.
1) Find the sides values of such box. The sheet can be deformated only with straight lines and cutted. But it is impossible to connect (glue or welding) the different parts of the sheet.
2) The same question about \max V and the sides values of opened box (without lid) that can be made from a tin sheet with sizes b\cdot c . Find \max V if the perimeter is known.
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