IMO 1959-1992 65p

geometry problems from International Mathematical Olympiads (also known as IMO)

Aops links are on the names.
The ''also'' links are within the blog.
All links of these problem within the blog are at the end of the blog 



1959-1992
(to be continued)

Construct a right triangle with given hypotenuse c such that the median drawn to the hypotenuse is the geometric mean of the two legs of the triangle.

An arbitrary point M is selected in the interior of the segment AB. The squares AMCD and MBEF are constructed on the same side of AB, with the segments AM and MB as their respective bases. The circles circumscribed about these squares, with centers P and Q, intersect at M and also at another point N. Let N΄ denote the point of intersection of the straight lines AF and BC.
(a) Prove that the points N and N΄ coincide.
(b) Prove that the straight lines MN pass through a fixed point S independent of the choice of M.
(c) Find the locus of the midpoints of the segments PQ as M varies between A and B.
by Cezar Cosnita
Two planes, P and Q, intersect along the line p. The point A is given in the plane P, and the point C in the plane Q, neither of these points lies on the straight line p: Construct an isosceles trapezoid ABCD (with AB parallel to CD) in which a circle can be inscribed, and with vertices B and D lying in the planes P and Q respectively.

In a given right triangle ABC, the hypotenuse BC, of length a, is divided into n equal parts (n an odd integer). Let ω be the acute angle subtending,  from A, that segment which contains the midpoint of the hypotenuse. Let h be the length of the altitude to the hypotenuse of the triangle. Prove $$\tan \omega =\frac{4nh}{\left( {{n}^{2}}-1 \right)\alpha }$$                                                                                 
  by Gheorghe D. Simionescu

Construct triangle ABC, given h­­a , hb (the altitudes from A and B) and ma, the median from vertex A.

Consider the cube ABCDA'B'C'D' (with face ABCD directly above face A'B'C'D').
(a) Find the locus of the midpoints of segments XY , where X is any point of AC and Y is any point of B'D'.
(b) Find the locus of points Z which lie on the segments XY of part (a) such that ΖΥ = 2 ΧΖ .

An isosceles trapezoid with bases a and b and altitude h is given.
(a) On the axis of symmetry of this trapezoid, find all points P such that both  legs of   the trapezoid subtend right angles at P.
(b) Calculate the distance of P from either base.
(c) Determine under what conditions such points P actually exist. (Discuss various cases that might arise).

Consider triangle P1P2P3 and a point P within the triangle. Lines P1P , P2P , P3P intersect the opposite sides in points Q1, Q2, Q3 respectively. Prove that, of the numbers $$\frac{{{P}_{1}}P}{P{{Q}_{1}}},\frac{{{P}_{2}}P}{P{{Q}_{2}}},\frac{{{P}_{3}}P}{P{{Q}_{3}}}$$  at least one is    2 and at least one is  ≥ 2.

Construct triangle ABC if AC = b, AB = c and <AMB = ω, where M is the midpoint of segment BC and ω < 90­o. Prove that a solution exists if and only if $$b\varepsilon \varphi \frac{\omega }{2}\le c<b$$ . In what case does the equality hold?

Consider a plane ε  and three non-collinear points A, B, C on the same side of ε, suppose the plane determined by these three points is not parallel to ε. In plane a take three arbitrary points A΄, B΄, C΄. Let L, M, N be the midpoints of segments AA΄, BB΄, CC΄΄, let G be the centroid of triangle LMN. (We will not consider positions of the points A΄, B΄, C΄such that the points L, M, N do not form a triangle.) What is the locus of point G as A΄, B΄, C΄. range independently over the plane ε ?
by Gheorghe D. Simionescu
Consider the cube ABCDA΄B΄C΄D΄ (ABCD and A΄B΄C΄D΄ are the upper and lower bases, respectively, and edges AA΄, BB΄, CC΄, DD΄ are parallel). The point X moves at constant speed along the perimeter of the square ABCD in the direction ABCDA, and the point Y moves at the same rate along the perimeter of the square B΄C΄CB in the direction B΄C΄CB΄ Β. Points X and Y begin their motion at the same instant from the starting positions A and B΄, respectively. Determine and draw the locus of the midpoints of the segments XY.

On the circle K there are given three distinct points A,B,C. Construct (using only straightedge and compasses) a fourth point D on K such that a circle can be inscribed in the quadrilateral thus obtained.

Consider an isosceles triangle. Let r be the radius of its circumscribed circle and ρ the radius of its inscribed circle. Prove that the distance d between the centers of these two circles is
$d=\sqrt{r(r-2p)}$

The tetrahedron SABC has the following property: there exist five spheres, each tangent to the edges SA, SB, SC, BC, CA, AB or to their extensions.
(a) Prove that the tetrahedron SABC is regular.
(b) Prove conversely that for every regular tetrahedron five such spheres exist.

Point Α  and segment BC are given. Determine the locus of points in space which are vertices of right angles with one side passing through Α, and the other side intersecting the segment BC.

In an n-gon all of whose interior angles are equal, the lengths of consecutive sides satisfy the relation α1  α2  ≥ ... ≥ αn . Prove that α1 =  α2  = ... = αn .

A circle is inscribed in triangle ABC with sides a, b, c, Tangents to the circle parallel to the sides of the triangle are constructed. Each of these tangents cuts off a triangle from ∆ ABC. In each of these triangles, a circle is inscribed. Find the sum of the areas of all four inscribed circles (in terms of a, b, c).

In tetrahedron ABCD, vertex D is connected with Do the centroid of ∆ ABC. Lines parallel to D Do are drawn through A, B and C: These lines intersect the planes BCD, CAD and ABD in points A­1, B­1 and C­1, respectively. Prove that the volume of ABCD is one third the volume of A­111o. Is the result true if point D­o is selected anywhere within ∆ ABC

Given the tetrahedron ABCD whose edges AB and CD have lengths a and b respectively. The distance between the skew lines AB and CD is d, and the angle between them is ω. Tetrahedron ABCD is divided into two solids by plane ε, parallel to lines AB and CD. The ratio of the distances of ε from AB and CD is equal to k. Compute the ratio of the volumes of the two solids obtained.

Consider ∆ OAB with acute angle AOB: Through a point M ≠ O perpendiculars are drawn to OA and OB, the feet of which are P and Q respectively. The point of intersection of the altitudes of ∆ OPQ is H. What is the locus of H if M is permitted to range over (a) the side AB, (b) the interior of ∆ OAB?
by Gheorghe D. Simionescu
Prove: The sum of the distances of the vertices of a regular tetrahedron from the center of its circumscribed sphere is less than the sum of the distances of these vertices from any other point in space.

In the interior of sides BC, CA, AB of triangle ABC, any points K, L, M  respectively, are selected. Prove that the area of at least one of the triangles AML, BKM, CLK is less than or equal to one quarter of the area of triangle ABC.

Let ABCD be a parallelogram with side lengths AB = α, AD = 1, and with <BAD = ω. If ∆ABD is acute, prove that the four circles of radius 1 with centers A,B,C,D cover the parallelogram if and only if  $$a\le \cos \omega +\sqrt{3}\sin \omega$$
Prove that if one and only one edge of a tetrahedron is greater than 1, then its volume is 1 / 8.

Let Ao BoCo and A1B1C1 be any two acute-angled triangles. Consider all triangles ABC that are similar to ∆ A1B1C1 (so that vertices A1,B1,C1 correspond to vertices A, B, C  respectively) and circumscribed about triangle AoBoCo  (where Ao  lies on BC, Bo  on CA, and Co on AB). Of all such possible triangles, determine the one with maximum area, and construct it.
by Tullio Viola
Prove that in every tetrahedron there is a vertex such that the three edges meeting there have lengths which are the sides of a triangle.

For each value of k =1,2,3,4,5  find necessary and sufficient conditions on the number a > 0 so that there exists a tetrahedron with k edges of length a, and the remaining 6-k edges of length 1.

A semicircular arc γ is drawn on AB as diameter. C is a point on γ other than A  and B, and D is the foot of the perpendicular from C to AB. We consider three circles, γ1, γ2, γ3, all tangent to the line ΑΒ. Of these, γ1 is inscribed in ∆ ABC, while γ2 and γ3 are both tangent to CD  and to γ, one on each side of CD . Prove that γ1, γ2 and γ3 have a second tangent in common.

Let M be a point on the side AB of ∆ABC. Let r1, r2 and r be the radii of the inscribed circles of triangles AMC,BMC and ABC. Let q1, q2  and q be the radii of the excircles of the same triangles that lie in the angle ACB. Prove that   $$\frac{{{r}_{1}}}{{{q}_{1}}}\cdot \frac{{{r}_{2}}}{{{q}_{2}}}=\frac{r}{q}$$
In the tetrahedron ABCD,  angle BDC is a right angle. Suppose that the foot H of the perpendicular from D to the plane ABC is the intersection of the altitudes of ∆ABC. Prove that (AB + BC + CA)2 6 (AD2 + BD2 + CD2). For what tetrahedra does equality hold?

Consider a convex polyhedron P1 with nine vertices A1A2, ... ,A9. Let Pi be the polyhedron obtained from P1 by a translation that moves vertex A1 to Ai (i = 2, 3, ... , 9): Prove that at least two of the polyhedra P1,P2, … , P9 have an interior point in common.

All the faces of tetrahedron ABCD are acute-angled triangles. We consider all closed polygonal paths of the form XYZTX defined as follows: X is a point on edge AB distinct from A and B; similarly, Y,Z, T are interior points of edges BC,CD,DA respectively. Prove:
(a) If <DAB + <BCD ≠ <CDA + <ABC ,then among the polygonal paths, there is none of minimal length.
(b) If <DAB + <BCD = <CDA + <ABC, then there are infinitely many shortest polygonal paths, their common length being 2AC sin(a/2),  where a = <BAC + <CAD + <DAB.

Prove that if n ≥ 4, every quadrilateral that can be inscribed in a circle can be dissected into n quadrilaterals each of which is inscribable in a circle.
Given four distinct parallel planes, prove that there exists a regular tetrahedron with a vertex on each plane.

Determine whether or not there exists a finite set M of points in space not lying in the same plane such that, for any two points A and B of M, one can select two other points C and D of M so that lines AB and CD are parallel and not coincident.

A soldier needs to check on the presence of mines in a region having the shape of an equilateral triangle. The radius of action of his detector is equal to half the altitude of the triangle. The soldier leaves from one vertex of the triangle. What path shouid he follow in order to travel the least possible distance and still accomplish his mission?

by Ðorde Dugošija
On the sides of an arbitrary triangle ABC  triangles ABR, BCP, CAQ are constructed externally with <CBP = <CAQ = 450   <BCP = <ACQ = 300, <ABR = <BAR = 15­­0 . Prove that  <QRP = 900 and QR = RP.

by Jan van de Craats
In a plane convex quadrilateral of area 32,  the sum of the lengths of two opposite sides and one diagonal is 16. Determine all possible lengths of the other diagonal.

Equilateral triangles ABK,BCL,CDM,DAN are constructed inside the square ABCD: Prove that the midpoints of the four segments KL,LM,MN,NK and the midpoints of the eight segments AK, BK,BL,CL,CM,DM,DN,AN are the twelve vertices of a regular dodecagon.

by Jan van de Craats
P is a given point inside a given sphere. Three mutually perpendicular rays from P intersect the sphere at points U, V, and W, Q denotes the  vertex diagonally opposite to P in the parallelepiped determined by PU, PV, and PW. Find the locus of Q for all such triads of rays from P.

by Murray Klamkin
In triangle ABC, AB = AC. A circle is tangent internally to the circumcircle of triangle ABC and also to sides AB, AC at P, Q, respectively.  Prove that the midpoint of segment PQ is the center of the incircle of triangle ABC.

by Murray Klamkin
Two circles in a plane intersect. Let A be one of the points of intersection. Starting simultaneously from A two points move with constant speeds, each point travelling along its own circle in the same sense. The two points return to A simultaneously after one revolution. Prove that there is a fixed point P in the plane such that, at any time, the distances from P to the moving points are equal.

by Nikolai Vasil'ev and Igor F. Sharygin
Given a plane π, a point P in this plane and a point Q not in π, find all points R in π such that the ratio $$\frac{QP+PR}{QR}$$  is a maximum.

by Murray Klamkin
P is a point inside a given triangle ABC. D,E, F are the feet of the perpendiculars from P to the lines BC, CA, AB respectively. Find all P for which $$\frac{BC}{PD}+\frac{CA}{PE}+\frac{AB}{PF}$$ is least.

by David Monk
Three congruent circles have a common point O and lie inside a given triangle. Each circle touches a pair of sides of the triangle. Prove that the incenter and the circumcenter of the triangle and the point O are collinear.

A non-isosceles triangle A1A2A3 is given with sides a1, a2, a3 (ai is the side opposite Ai). For all  i = 1, 2, 3, Mi is the midpoint of side ai, and Ti is the point where the incircle touches side ai. Denote by Si the reflection of Ti in the interior bisector of angle Ai. Prove that the lines M1S1, M2S2 and M3S3  are concurrent.

by Jan van de Craats
The diagonals AC and CE of the regular hexagon ABCDEF are divided by the inner points M and N, respectively, so that $$\frac{AM}{AC}=\frac{CN}{CE}=r$$. Determine r if B,M, and N are collinear.

by Jan van de Craats
Let A be one of the two distinct points of intersection of two unequal coplanar circles C1 and C2 with centers O1  and O2, respectively. One of the common tangents to the circles touches C1  at P1 and C2  at P2, while the other touches C1  at Q1  and C2  at Q2. Let M1 be the midpoint of P1Q1 and M2  be the midpoint of P2Q2. Prove that <O1AO2  = <M1AM2.

by Igor F. Sharygin
Let ABC be an equilateral triangle and E the set of all points contained in the three segments AB,BC and CA (including A, B and C). Determine whether, for every partition of E into two disjoint subsets, at least one of the two subsets contains the vertices of a right-angled triangle. Justify your answer.

Let ABCD be a convex quadrilateral such that the line CD is a tangent to the circle on AB as diameter. Prove that the line AB is a tangent to the circle on CD as diameter if and only if the lines BC and AD are parallel.

by Laurentiu Panaitopol
A circle has center on the side AB of the cyclic quadrilateral ABCD. The other three sides are tangent to the circle. Prove that AD + BC = AB.

by Frank Budden
A circle with center O passes through the vertices A and C of triangle ABC and intersects the segments AB and BC again at distinct points K and N respectively. The circumscribed circles of the triangles ABC and EBN intersect at exactly two distinct points B and M. Prove that angle <OMB is a right angle.

by Igor F. Sharygin
A triangle A1A2A3 and a point P0 are given in the plane. We define As = As-3 for all s ≥ 4. We construct a set of points P1, P2, P3, . . . , such that Pk+1 is the image of Pk under a rotation with center Ak+1 through angle 120o  clockwise  (for k = 0, 1, 2, . . . ). Prove that if P1986 = P0, then the triangle A1A2A3  is equilateral.

 by Gengzhe Chang and Dongxu Qi
Let A, B be adjacent vertices of a regular n-gon (n ≥ 5) in the plane having center at O. A triangle XY Z, which is congruent to and initially coincides with OAB, moves in the plane in such a way that Y and Z each trace out the whole boundary of the polygon, X remaining inside the polygon. Find the locus of X.

by Sven Sigurðsson
In an acute-angled triangle ABC the interior bisector of the angle A intersects BC at L and intersects the circumcircle of ABC again at N. From point L perpendiculars are drawn to AB and AC, the feet of these perpendiculars being K and M respectively. Prove that the quadrilateral AKNM and the triangle ABC have equal areas.

by I.A. Kushnir
Consider two concentric circles of radii R and r (R > r) with the same center. Let P be a fixed point on the smaller circle and B a variable point on the larger circle. The line BP meets the larger circle again at C. The perpendicular l to BP at P meets the smaller circle again at A. (If l is tangent to the circle at P then A P.)
(i) Find the set of values of BC­­2 + CA­­2 + AB­­2.
(ii) Find the locus of the midpoint of BC.

by Lucien Kieffer
ABC is a triangle right-angled at A, and D is the foot of the altitude from A. The straight line joining the incenters of the triangles ABD, ACD intersects the sides AB, AC at the points K, L respectively. S and T denote the areas of the triangles ABC and AKL respectively. Show that    S ≥ 2T.

by Dimitris Kontogiannis
In an acute-angled triangle ABC the internal bisector of angle A meets the circumcircle of the triangle again at A1. Points B1 and C1  are defined similarly. Let A0 be the point of intersection of the line AA1 with the external bisectors of angles B and C. Points B0 and C0 are defined similarly. Prove that:
(i) The area of the triangle A0B0C0 is twice the area of the hexagon AC1BA1CB1.
(ii) The area of the triangle A0B0C0 is at least four times the area of the triangle ABC.

by Esther Szekeres
Let ABCD be a convex quadrilateral such that the sides AB, AD, BC satisfy AB = AD + BC. There exists a point P inside the quadrilateral at a distance h from the line CD such that AP = h + AD and BP = h + BC. Show that:  $$\frac{1}{\sqrt{h}}\ge \frac{1}{\sqrt{AD}}+\frac{1}{\sqrt{BC}}$$.

by Eggert Briem
Chords AB and CD of a circle intersect at a point E inside the circle. Let M be an interior point of the segment EB. The tangent line at E to the circle through D, E, and M intersects the lines BC and AC at F and G, respectively. If $$\frac{AM}{AB}=t$$ , find  $$\frac{EF}{EG}$$ in terms of t.

by C.R. Pranesachar
Given a triangle ABC, let I be the center of its inscribed circle. The internal bisectors of the angles A,B,C  meet the opposite sides in A’, B’ , C’ respectively. Prove that $$\frac{1}{4}<\frac{AI\cdot BI\cdot CI}{AA'\cdot BB'\cdot CC'}\le \frac{8}{27}$$

by Arkadii Skopenkov
Let ABC be a triangle and P an interior point of ABC . Show that at least one of the angles <PAB, <PBC, <PCA is less than or equal  to 300.

by Johan Yebbou
In the plane let C be a circle, L a line tangent to the circle C  and M a point on L. Find the locus of all points P with the following  property: there exists two points Q,R on L such that M  is the
midpoint of QR and C is the inscribed circle of triangle PQR.

by Johan Yebbou

To be continued ...


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