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

(to be continued)

An isosceles trapezoid with
bases a and b and altitude h is given.

by Ðorde Dugošija

by Jan van de Craats

by Jan van de Craats

by Murray Klamkin

by Murray Klamkin

by Nikolai Vasil'ev and Igor F. Sharygin

by Murray Klamkin

by David Monk

by Jan van de Craats

by Jan van de Craats

by Igor F. Sharygin

by Laurentiu Panaitopol

by Frank Budden

by Igor F. Sharygin

by Gengzhe Chang and Dongxu Qi

by Sven Sigurðsson

by I.A. Kushnir

by Lucien Kieffer

by Dimitris Kontogiannis

by Esther Szekeres

by Eggert Briem

by Arkadii Skopenkov

by Johan Yebbou

by Johan Yebbou

To be continued ...

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 h

_{a}, h_{b}(the altitudes from A and B) and m_{a}, 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 .

(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 P

_{1}P_{2}P_{3}and a point P within the triangle. Lines P_{1}P , P_{2}P , P_{3}P intersect the opposite sides in points Q_{1}, Q_{2}, Q_{3}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 < 90

^{o}. 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

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 α

_{1}≥ α_{2}≥ ... ≥ α_{n}. Prove that α_{1}= α_{2}= ... = α_{n}.
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 D

_{o}the centroid of ∆ ABC. Lines parallel to D D_{o}are drawn through A, B and C: These lines intersect the planes BCD, CAD and ABD in points A_{1}, B_{1}and C_{1}, respectively. Prove that the volume of ABCD is one third the volume of A_{1}B_{1}C_{1}D_{o}. Is the result true if point D_{o}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 A

_{o}B_{o}C_{o}and A_{1}B_{1}C_{1 }be any two acute-angled triangles. Consider all triangles ABC that are similar to ∆ A_{1}B_{1}C_{1}(so that vertices A_{1},B_{1},C_{1}correspond to vertices A, B, C respectively) and circumscribed about triangle A_{o}B_{o}C_{o}(where A_{o }lies on BC, B_{o}on CA, and C_{o}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 r

_{1}, r_{2}and r be the radii of the inscribed circles of triangles AMC,BMC and ABC. Let q_{1}, q_{2 }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 (AD^{2}+ BD^{2}+ CD^{2}). For what tetrahedra does equality hold?
Consider a convex polyhedron P

_{1}with nine vertices A_{1}A_{2}, ... ,A_{9}. Let P_{i}be the polyhedron obtained from P_{1}by a translation that moves vertex A_{1}to A_{i}(i = 2, 3, ... , 9): Prove that at least two of the polyhedra P_{1},P_{2}, … , P_{9}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 = 45

^{0}, <BCP = <ACQ = 30^{0}, <ABR = <BAR = 15^{0}_{ }. Prove that <QRP = 90^{0}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 A

_{1}A_{2}A_{3}is given with sides a_{1}, a_{2}, a_{3}(a_{i}is the side opposite Ai). For all i = 1, 2, 3, M_{i}is the midpoint of side a_{i}, and T_{i}is the point where the incircle touches side a_{i}. Denote by S_{i }the reflection of Ti in the interior bisector of angle A_{i}. Prove that the lines M_{1}S_{1}, M_{2}S_{2}and M_{3}S_{3 }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 C

_{1}and C_{2}with centers O_{1 }and O_{2}, respectively. One of the common tangents to the circles touches C_{1 }at P_{1}and C_{2 }at P_{2}, while the other touches C_{1 }at Q_{1 }and C_{2 }at Q_{2}. Let M_{1}be the midpoint of P_{1}Q_{1 }and M_{2 }be the midpoint of P_{2}Q_{2}. Prove that <O_{1}AO_{2 }= <M_{1}AM_{2}.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 A

_{1}A_{2}A_{3}and a point P_{0}are given in the plane. We define A_{s}= A_{s-}_{3}for all s ≥ 4. We construct a set of points P_{1}, P_{2}, P_{3}, . . . , such that P_{k}_{+1}is the image of P_{k}under a rotation with center A_{k}_{+1}through angle 120^{o}clockwise (for k = 0, 1, 2, . . . ). Prove that if P_{1986}= P_{0}, then the triangle A_{1}A_{2}A_{3 }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 BC

_{}^{2}+ CA_{}^{2}+ AB_{}^{2}.
(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 B

_{1}and C_{1 }are defined similarly. Let A_{0}be the point of intersection of the line AA_{1}with the external bisectors of angles B and C. Points B_{0}and C_{0}are defined similarly. Prove that:
(i) The area of the triangle A

_{0}B_{0}C_{0}is twice the area of the hexagon AC_{1}BA_{1}CB_{1}.
(ii) The area of the triangle A

_{0}B_{0}C_{0}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 30

^{0}.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

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