geometry problems Austrian Polish Mathematical Competition (APMC)
with aops links
This competition was replaced in 2007 Middle European Mathematical Olympiad (also known as MEMO) where more countries participated.
with aops links
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 = 90ο.
The circumcenter and incenter of a given tetrahedron coincide. Prove
that all its faces are congruent.
Let A,B,C,D be four
points in space, and M and N be the midpoints of AC and BD, respectively. Show that AB2+BC2+CD2+DA2 = AC2+BD2+4MN2 .
Prove that for every point P inside
a regular tetrahedron ABCD the
sum of the angles APB,APC,APD,BPC,BPD,CPD exceeds 540ο.
Let B1,B2,B3 be points on sides A2A3,A3A1,A1A2 respectively of a
triangle A1A2A3 (not coinciding with any vertices). Prove that the
perpendicular bisectors of the three segments AiBi never concur.
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 q1,q2 be two rays from C. Ray qi cuts the circle in Di and Ei
with Di between
C and Ei, i =
1,2. Rays AD1,AD2,AE1,AE2 meet m in the respective points M1,M2,N1,N2. Prove that M1M2 = N1N2.
In a triangle ABC, r is the inradius, rA the radius of the
circle touching segments AB,AC and the incircle of △ABC, and rB and rC are defined
analogously. Prove that rA+rB+rC ≥ r, equality holding if and only if △ABC is equilateral.
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.
Prove that if the feet of the altitudes of a tetrahedron are the
incenters of the corresponding faces, then the tetrahedron is regular.
A regular heptagon A1A2 … A7 is inscribed in
circle C. Point P is taken on
the shorter arc A7A1. Prove that PA1+PA3+PA5+PA7 = PA2+PA4+PA6.
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 SA,SB,SC,SD denote the centroids of
the tetrahedra PBCD,PCDA,PDAB,PABC,
respectively. Show that the volume of the tetrahedron SASBSCSD equals 1/
64 the volume of ABCD.
A non-right triangle A1A2A3 is given. Circles Γ1 and Γ2 are tangent at A3,
Γ2 and Γ3 are tangent at A1,
and Γ3 and Γ1 are tangent at A2.
Points O1,O2,O3 are the centers of Γ1, Γ2, Γ3, respectively. Supposing that the triangles A1A2A3
and O1O2O3 are similar, determine their angles.
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 .
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.
Three rays h1,h2,h3 emanating from a point O are given, not all in the same plane. Show that if for any
three points A1,A2,A3 on h1,h2,h3 respectively, distinct from O, the triangle A1A2A3
is acute-angled, then the rays h1,h2,h3 are pairwise orthogonal.
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, Pa,Pb,Pc denote the orthogonal projections of P to BC,CA,AB respectively. Consider
f (P) =(APc+BPa+CPb )/ (PPa+PPb+PPc ). Show that f (P) is constant if and only if ABC is an equilateral triangle.
The distinct
points X1, X2, X3, X4, X5,
X6 all lie on the same side of the line AB. The six triangles
ABXi are all similar. Show that the Xi lie on a
circle.
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.
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.
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.
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.
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
r1 the inradius of
triangle PAC, and by r2 the exradius of
triangle PBC opposite P. Find the sum r1+r2 as a function in a.
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.
In an equilateral triangle ABC,
A1,B1,C1 are the midpoints of the sides BC,CA,AB, respectively.
Three parallel lines p,q and r pass through A1,B1 and C1 and intersect the lines B1C1,C1A1 and A1B1 at points A2,B2,C2,
respectively. Prove that the lines AA2,BB2,CC2 have a common point D which lies on the circumcircle of
the triangle ABC.
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 = 90ο.
Prove that the angle between the diagonals BE and CF is equal to 45ο.
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 l1 and l2 intersect at point P. Circles S1 and S2
are tangent to l1 at P,
and circles T1 and T2 are tangent to l2
at P. Circle S1 meets T1 at points A,P and T2
at B,P, while circle S2
meets T2 at C,P and T1
at D,P. Show that the points A,B,C and D are
concyclic if and only if the lines l1 and l2
are perpendicular.
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.
Given a parallelepiped P,
let VP be its
volume, SP the area
of its surface and LP 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 Pt is given by the
formula V(Pt) =VP + SP t + π/4 LP t2 +
4π/3 t3.
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.
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.
Three lines k, l, m are drawn
through a point P inside a
triangle ABC such that k meets AB at A1
and AC at A2 ≠A1 and PA1 = PA2, l meets BC at B1
and BA at B2 ≠ B1 and PB1 = PB2, m meets CA at C1
and CB at C2≠C1 and PC1=PC2. Prove that the lines k,l,m are uniquely determined by these conditions. Find point P for which the triangles AA1A2, BB1B2, CC1C2 have the same area and show that this point is
unique.
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.
In a unit cube, CG is
the edge perpendicular to the face ABCD.
Let O1 be the incircle
of square ABCD and O2 be the circumcircle of
triangle BDG. Determine min{XY|XäO1,YäO2}
Triangle A0B0C0 is given in the plane. Consider all triangles ABC such that:
(i) The lines AB,BC,CA pass through C0,A0,B0, respectvely,
(ii) The triangles ABC and
A0B0C0 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 M1,M2, ... , M10 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 PMi
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 P1P2 … P2n with an
even number of vertices there exists a diagonal PiPj 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.
ABC is a
triangle. Take a = BC etc as usual. Take points T1, T2 on
the side AB so that AT1 = T1T2 = T2B.
Similarly, take points T3, T4 on the side BC so that
BT3 = T3T4 = T4C, and
points T5, T6 on the side CA so that CT5 =
T5T6 = T6A. Show that if a' = BT5,
b' = CT1, c' = AT3, then there is a triangle A'B'C' with
sides a', b', c' (a' = B'C' etc). In the same way we take points Ti'
on the sides of A'B'C' and put a" = B'T6', b" = C'T2',
c" = A'T4'. Show that there is a triangle A"B"C"
with sides a", b", c" and that it is similar to ABC. Find a"/a.
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 ≤ (a2 +
b2 + c2)/2 + 2F√3, then the triangle is equilateral.
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 Hk ∈ LM, Hl
∈ KM, Hm ∈ KL, so that (ADcD) = (KHmH),
(BDcD) = (LHmH), (BDaD) = (LHkH), (CDaD)
= (MHkH), (CDbD) = (MHlH) , (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.
Great! Thank you!
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