geometry problems Austrian Polish Mathematical Competition (APMC)

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**without**aops links
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__underlined__19 out of 56 problems have aops links)#

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This competition was replaced in 2007 Middle European Mathematical Olympiad (also known as MEMO) where more countries participated.

A parallelogram is inscribed into a regular hexagon so that the centers
of symmetry of both figures coincide. Prove that the area of the parallelogram
is at most 2/3 the area of the hexagon.

On sides AB and BC of a square ABCD the respective points E and F have been chosen so that BE = BF. Let BN be the altitude in triangle BCE. Prove that ÐDNF = 90

^{ο}.
The circumcenter and incenter of a given tetrahedron coincide. Prove
that all its faces are congruent.

Austrian-Polish 1979
Team 2

Let A,B,C,D be four
points in space, and M and N be the midpoints of AC and BD, respectively. Show that AB

^{2}+BC^{2}+CD^{2}+DA^{2}= AC^{2}+BD^{2}+4MN^{2}.
Austrian-Polish 1980 Individual 3

Prove that for every point P inside
a regular tetrahedron ABCD the
sum of the angles APB,APC,APD,BPC,BPD,CPD exceeds 540

^{ο}.
Austrian-Polish 1980 Individual 5

Let B

_{1},B_{2},B_{3}be points on sides A_{2}A_{3},A_{3}A_{1},A_{1}A_{2}respectively of a triangle A_{1}A_{2}A_{3}(not coinciding with any vertices). Prove that the perpendicular bisectors of the three segments A_{i}B_{i}never concur.
Austrian-Polish 1980 Team 3

Through the endpoints A and
B of a diameter AB of a given circle, the tangents

*l*and m have been drawn. Let C≠A be a point on*l*and let q_{1},q_{2}be two rays from C. Ray q_{i}cuts the circle in D_{i}and E_{i}with D_{i}between C and E_{i}, i = 1,2. Rays AD_{1},AD_{2},AE_{1},AE_{2}meet m in the respective points M_{1},M_{2},N_{1},N_{2}. Prove that M_{1}M_{2}= N_{1}N_{2}.
In a triangle ABC, r is the inradius, r

_{A}the radius of the circle touching segments AB,AC and the incircle of △ABC, and r_{B}and r_{C}are defined analogously. Prove that r_{A}+r_{B}+r_{C}≥ r, equality holding if and only if △ABC is equilateral.
Austrian-Polish 1982 Team 2

Let P be a point inside
a regular tetrahedron ABCD with
edge length 1. Show That d(P,AB)+d(P,AC)+d(P,AD)+d(P,BC)+d(P,BD)+d(P,CD) ≥ 3/2 √2 , with equality only when P is
the centroid of ABCD. Here d(P,XY) denotes
the distance from point P to
line XY.

Austrian-Polish 1983 Individual 6

Six straight lines are given in space. Among any three of them, two are
perpendicular. Show that the given lines can be labeled l

_{1}, … ,*l*_{6}in such a way that*l*_{1},*l*_{2},*l*_{3}are pairwise perpendicular, and so are*l*_{4},*l*_{5},*l*_{6}.
Austrian-Polish 1983 Team 1

Let P

_{1},P_{2},P_{3},P_{4}be four distinct points in the plane. Suppose*I*_{1},*I*_{2}, … ,*I*_{6}are closed segments in that plane with the following property: Every straight line passing through at least one of the points P_{i}meets the union I_{1}È I_{2}È … È I_{6}in exactly two points. Prove or disprove that the segments*I*j necessarily form a hexagon.
Austrian-Polish 1984 Individual 1

Prove that if the feet of the altitudes of a tetrahedron are the
incenters of the corresponding faces, then the tetrahedron is regular.

Austrian-Polish 1984 Individual 4

A regular heptagon A

_{1}A_{2}… A_{7}is inscribed in circle C. Point P is taken on the shorter arc A_{7}A_{1}. Prove that PA_{1}+PA_{3}+PA_{5}+PA_{7}= PA_{2}+PA_{4}+PA_{6}.
Prove that in a convex quadrilateral of area 1 the sum of the lengths of
all sides and diagonals is not less than 4+√8.

For a point P inside a
tetrahedron ABCD, points S

_{A},S_{B},S_{C},S_{D}denote the centroids of the tetrahedra PBCD,PCDA,PDAB,PABC, respectively. Show that the volume of the tetrahedron S_{A}S_{B}S_{C}S_{D}equals 1/ 64 the volume of ABCD.
Austrian-Polish 1986
Individual 1

A non-rectangular triangle A

_{1}A_{2}A_{3}is given. Circles Γ_{1}and Γ_{2}are tangent at A_{3}, Γ_{2}and Γ_{3}are tangent at A_{1}, and Γ_{3}and Γ_{1}are tangent at A_{2}. Points O_{1},O_{2},O_{3}are the centers of Γ_{1}, Γ_{2}, Γ_{3}, respectively. Supposing that the triangles A_{1}A_{2}A_{3}and O_{1}O_{2}O_{3}are similar, determine their angles.
Austrian-Polish 1986
Individual 6

Let M be the set of all tetrahedra whose inscribed and circumscribed spheres
are concentric. If the radii of these spheres are denoted by r and R respectively, find the possible values of R/r over all tetrahedra from M .

Austrian-Polish 1987
Individual 1

Three pairwise orthogonal chords of a sphere S are drawn through a given
point P inside S. Prove that
the sum of the squares of their lengths does not depend on their directions.

Austrian-Polish 1988
Individual 3

Let ABCD be a convex
quadrilateral with no two parallel sides. Consider the two angles made by two
pairs of opposite sides. Their angle bisectors intersect the sides of ABCD in P,Q,R,S, where PQRS is
a convex quadrilateral. Prove that the quadrilateral ABCD is cyclic if and only if PQRS is a rhombus.

Austrian-Polish 1988
Individual 6

Three rays h

_{1},h_{2},h_{3}emanating from a point O are given, not all in the same plane. Show that if for any three points A_{1},A_{2},A_{3}on h_{1},h_{2},h_{3}respectively, distinct from O, the triangle A_{1}A_{2}A_{3}is acute-angled, then the rays h_{1},h_{2},h_{3}are pairwise orthogonal.
Austrian-Polish 1989
Individual 5

Let A be a vertex of a
cube

*ω*circumscribed about a sphere*κ*of radius 1. We consider lines g through A containing at least one point of*κ*. Let P be the intersection point of g and*κ*closer to A, and Q be the second intersection point of g and*ω*. Determine the maximum value of AP·AQ and characterize the lines g yielding the maximum.
An acute triangle ABC is
given. For each point P of the
interior or boundary of △ABC, P

_{a},P_{b},P_{c}denote the orthogonal projections of P to BC,CA,AB respectively. Consider
f (P) =(AP

_{c}+BP_{a}+CP_{b})/ (PP_{a}+PP_{b}+PP_{c }). Show that f (P) is constant if and only if ABC is an equilateral triangle.
Austrian-Polish 1990
Individual 1

The distinct
points X

_{1}, X_{2}, X_{3}, X_{4}, X_{5}, X_{6}all lie on the same side of the line AB. The six triangles ABX_{i}are all similar. Show that the X_{i}lie on a circle.
Austrian-Polish 1991
Individual 3

Given two distinct points A

_{1},A_{2}in the plane, determine all possible positions of a point A_{3}with the following property: There exists an array of (not necessarily distinct) points P_{1},P_{2}, … ,P_{n}for some n ≥ 3 such that the segments P_{1}P_{2},P_{2}P_{3}, … ,P_{n}P_{1}have equal lengths and their midpoints are A_{1},A_{2},A_{3},A_{1},A_{2},A_{3},… in this order.
Austrian-Polish 1991
Individual 6

Suppose that there is a point P
inside a convex quadrilateral ABCD
such that the triangles PAB,PBC,PCD,PDA have
equal areas. Prove that one of the diagonals bisects the area of ABCD.

Austrian-Polish 1992
Individual 5

Given a circle

*k*with center M and radius r, let AB be a fixed diameter of*k*and let K be a fixed point on the segment AM. Denote by*t*the tangent of*k*at A. For any chord CD through K other than AB, denote by P and Q the intersection points of BC and BD with*t*, respectively. Prove that AP·AQ does not depend on CD.
Austrian-Polish 1992 Team 1

Consider triangles ABC in
space.

(a) What condition must the angles α,β,γ of △ABC fulfill in order that there is a point P in space such that ÐAPB, ÐBPC, ÐCPA are right
angles?

(b) Let d be the longest
of the edges PA,PB,PC and let h be
the longest altitude of △ABC. Show that 1/3√6 h ≤ d
≤ h.

Austrian-Polish 1993
Individual 2

Consider all tetrahedra ABCD in
which the sum of the areas of the faces ABD,
ACD, BCD does not exceed 1. Among such tetrahedra, find those with
the maximum volume.

Austrian-Polish 1993 Team 3

Point P is taken on the
extension of side AB of an
equilateral triangle ABC so that
A is between B and P. Denote by

*a*the side length of triangle ABC, by r_{1}the inradius of triangle PAC, and by r_{2}the exradius of triangle PBC opposite P. Find the sum r_{1}+r_{2}as a function in*a*.
Austrian-Polish 1994 Team 3

On the plane are given four distinct points A,B,C,D on a line g in
this order, at the mutual distances AB
= a, BC = b, CD = c.

(a) Construct (if possible) a point P outside line g such
that ÐAPB =ÐBPC =ÐCPD.

(b) Prove that such a point P exists if and only if (a+b)(b+c) < 4ac.

Austrian-Polish 1995
Individual 2

Let X = {A

_{1},A_{2},A_{3},A_{4}} be a set of four distinct points in the plane. Show that there exists a subset Y of X with the property that there is no (closed) disk K such that K∩X =Y.
In an equilateral triangle ABC,
A

_{1},B_{1},C_{1}are the midpoints of the sides BC,CA,AB, respectively. Three parallel lines p,q and r pass through A_{1},B_{1}and C_{1}and intersect the lines B_{1}C_{1},C_{1}A_{1}and A_{1}B_{1}at points A_{2},B_{2},C_{2}, respectively. Prove that the lines AA_{2},BB_{2},CC_{2}have a common point D which lies on the circumcircle of the triangle ABC.
Austrian-Polish 1996
Individual 2

A convex hexagon ABCDEF satisfies
the following conditions:

(i) opposite sides are parallel, i.e. AB // DE, BC // EF, CD //FA,

(ii) the distances between opposite sides are equal,

(iii) ÐFAB = ÐCDE = 90

^{ο}.
Prove that the angle between the diagonals BE and CF is equal to 45

^{ο}.
Austrian-Polish 1996
Individual 5

A sphere S divides every edge of a convex polyhedron P into three equal
parts. Show that there exists a sphere tangent to all the edges of P.

Lines

*l*_{1 }and*l*_{2}intersect at point P. Circles S_{1}and S_{2}are tangent to*l*_{1}at P, and circles T_{1}and T_{2}are tangent to*l*_{2}at P. Circle S_{1}meets T_{1}at points A,P and T_{2}at B,P, while circle S_{2}meets T_{2}at C,P and T_{1}at D,P. Show that the points A,B,C and D are concyclic if and only if the lines*l*_{1}and*l*_{2}are perpendicular.
Austrian-Polish 1997
Individual 4

In a trapezoid ABCD with
AB // CD, the diagonals AC
and BD intersect at
point E. Let F and G be the orthocenters of the triangles EBC and EAD.
Prove that the midpoint of GF lies
on the perpendicular from E to AB.

Austrian-Polish 1997 Team 3

Given a parallelepiped P,
let V

_{P}be its volume, S_{P}the area of its surface and L_{P}the sum of the lengths of its edges. For a real number t ≥ 0, let Pt be the solid consisting of all points X whose distance from some point of P is at most t. Prove that the volume of the solid P_{t }is given by the formula V(P_{t}) =V_{P }+ S_{P }t + π/4 L_{P }t^{2 }+ 4π/3 t^{3}.
Austrian-Polish 1998
Individual 6

Different points A,B,C,D,E,F lie on circle

*k*in this order. The tangents to*k*in the points A and D and the lines BF and CE have a common point P. Prove that the lines AD,BC and EF also have a common point or are parallel.
Austrian-Polish 1998 Team 3

Given a triangle ABC,
points K,L,M are the midpoints of the sides BC,CA,AB, and points X,Y,Z are the midpoints
of the arcs BC,CA,AB of the circumcircle not containing A,B,C respectively. If R denotes the circumradius and r the inradius of the triangle, show
that r+KX+LY+MZ=2R.

Austrian-Polish 1999
Individual 4

Three lines

*k**, l, m*are drawn through a point P inside a triangle ABC such that*k*meets AB at A_{1}and AC at A_{2}≠A_{1}and PA_{1}= PA_{2},*l*meets BC at B_{1}and BA at B_{2}≠ B_{1}and PB_{1}= PB_{2},*m*meets CA at C_{1}and CB at C_{2}≠C_{1}and PC_{1}=PC_{2}. Prove that the lines*k**,l,m*are uniquely determined by these conditions. Find point P for which the triangles AA_{1}A_{2}, BB_{1}B_{2}, CC_{1}C_{2}have the same area and show that this point is unique.
Austrian-Polish 1999 Team 2

Let P,Q,R be points on the same side of a line

*g*in the plane. Let M and N be the feet of the perpendiculars from P and Q to g respectively. Point S lies between the lines PM and QN and satisfies and satisfies PM = PS and QN = QS. The perpendicular bisectors of SM and SN meet in a point R. If the line RS intersects the circumcircle of triangle PQR again at T, prove that S is the midpoint of RT.
Austrian-Polish 2000
Individual 2

In a unit cube, CG is
the edge perpendicular to the face ABCD.
Let O

_{1}be the incircle of square ABCD and O_{2}be the circumcircle of triangle BDG. Determine min{XY|XäO_{1},YäO_{2}}
Austrian-Polish 2000 Team 1

Triangle A

_{0}B_{0}C_{0}is given in the plane. Consider all triangles ABC such that:
(i) The lines AB,BC,CA pass through C

_{0},A_{0},B_{0}, respectvely,
(ii) The triangles ABC and
A

_{0}B_{0}C_{0}are similar.
Find the possible positions of the circumcenter of
triangle ABC.

Prove that if a,b,c,d are lengths
of the successive sides of a quadrilateral (not necessarily convex) and S its area, then S ≤ 1/2(ac+bd). When
does equality hold?

A prism with the regular octagonal base and all edges of the length 1 is
given. Let M

_{1},M_{2}, ... , M_{10 }be the centers of the faces of the prism. For a point P inside the prism denote by Pi the second intersection point of line PM_{i}with the surface of the prism. Assume that the interior of each face contains exactly one of the points Pi. Prove that $\sum\limits_{i=1}^{10}{\frac{{{M}_{i}}P}{{{M}_{i}}{{P}_{i}}}}=5$ .
Prove that in any convex polygon P

_{1}P_{2}… P_{2n}with an even number of vertices there exists a diagonal P_{i}P_{j }which is not parallel to any of its sides.
Let S be the centroid of
a tetrahedron ABCD. A line
through S intersects the surface
of the tetrahedron at points K and
L. Prove that 1 /3 ≤ KS / LS ≤ 3.

The diagonals of a convex quadrilateral ABCD meet at E.
Let U and H be the circumcenter and orthocenter
of triangle ABE, respectively.
Similarly, let V and K be the circumcenter and orthocenter
of triangle CDE, respectively.
Prove that E lies on line UK if and only if it lies on line VH.

Austrian-Polish 2003
Individual 3

ABC is a
triangle. Take a = BC etc as usual. Take points T

_{1}, T_{2}on the side AB so that AT_{1}= T_{1}T_{2}= T_{2}B. Similarly, take points T_{3}, T_{4}on the side BC so that BT_{3}= T_{3}T_{4}= T_{4}C, and points T_{5}, T_{6}on the side CA so that CT_{5}= T_{5}T_{6}= T_{6}A. Show that if a' = BT_{5}, b' = CT_{1}, c' = AT_{3}, then there is a triangle A'B'C' with sides a', b', c' (a' = B'C' etc). In the same way we take points T_{i}' on the sides of A'B'C' and put a" = B'T_{6}', b" = C'T_{2}', c" = A'T_{4}'. Show that there is a triangle A"B"C" with sides a", b", c" and that it is similar to ABC. Find a"/a.
Austrian-Polish 2003
Individual 5

A triangle
with sides a, b, c has area F. The distances of its centroid from the vertices
are x, y, z. Show that if (x + y + z)

^{2}≤ (a^{2}+ b^{2}+ c^{2})/2 + 2F√3, then the triangle is equilateral.
Austrian-Polish 2003
Individual 6

ABCD is a
tetrahedron such that we can find a sphere k(A,B,C) through A, B, C which meets
the plane BCD in the circle diameter BC, meets the plane ACD in the circle
diameter AC, and meets the plane ABD in the circle diameter AB. Show that there
exist spheres k(A,B,D), k(B,C,D) and k(C,A,D) with analogous properties.

In a triangle ABC let D be the intersection of the angle bisector of ,
angle at C, with the side AB. And let F be the area of the triangle ABC. Prove
the following inequality:

$\text{2F}\left(
\frac{\text{1}}{\text{AD}}\text{-}\frac{\text{1}}{\text{BD}} \right)\le
\text{AB}$ .

Given is a convex quadrilateral ABCD with AB = CD. Draw the triangles ABE
and CDF outside ABCD so that ÐABE = ÐDCF and ÐBAE =ÐFDC. Prove that the midpoints of AD, BC and EF are collinear.

ABCD is a tetrahedron. Let K be the center of the incircle of CBD. Let M
be the center of the incircle of ABD. Let L be the gravycenter of DAC. Let N be
the gravycenter of BAC.

Suppose AK, BL, CM, DN have one common point. Is ABCD necessarily
regular?

Let D be an interior point of the triangle ABC. CD and AB intersect at
Dc, BD and AC intersect at Db, AD and BC intersect at Da. Prove that there
exists a triangle KLM with orthocenter H and the feet of altitudes H

_{k }∈ LM, H_{l}∈ KM, H_{m}∈ KL, so that (AD_{c}D) = (KH_{m}H), (BD_{c}D) = (LH_{m}H), (BD_{a}D) = (LH_{k}H), (CD_{a}D) = (MH_{k}H), (CD_{b}D) = (MH_{l}H) , (ADbD) = (KHlH) where (PQR) denotes the area of the triangle PQR.
Let ABCDS be a (not necessarily straight) pyramid with a rectangular
base ABCD and acute triangular faces ABS,BCS,CDS,DAS. We consider all cuboids
which are inscribed inside the pyramid with its base being in the plane ABCD
and its upper vertexes are in the triangular faces (one in each). Find the
locus of the midpoints of these cuboids.

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