geometry problems from Czech-Polish-Slovak Mathematical Match (CPS)

with aops links in the names

(Waldemar Pompe, Poland)

(Kamil Duszenko, Poland)
Let ABC be a
right angled triangle with hypotenuse AB and P be a point on the shorter arc AC
of the circumcircle of triangle ABC. The line, perpendicular to CP and passing
through C, intersects AP, BP at points K and L respectively. Prove that the
ratio of area of triangles BKL and ACP is independent of the position of point
P.

(Dominik Burek & Tomasz Cieśla, Poland)

(Patrik Bak, Slovakia)

with aops links in the names

*Started in 1995 as Czech and Slovak Match.*

*In 2001 Polish joined and was renamed to Czech-Polish-Slovak Match.*

1995 - 2019

Consider all
triangles ABC in the cartesian
plane whose vertices are at lattice points (i.e. with integer coordinates) and
which contain exactly one lattice point (to be denoted P) in its interior. Let the line AP meet BC at E. Determine the maximum possible
value of the ratio AP/ PE .

The diagonals of
a convex quadrilateral ABCD are
orthogonal and intersect at point E.
Prove that the reflections of E in
the sides of quadrilateral ABCD lie
on a circle.

The base of a
regular quadrilateral pyramid p is a square with side length 2a and its lateral edge has length $a \sqrt{17}$. Let M be a point inside the pyramid.
Consider the five pyramids which are similar to p , whose top vertex is at M and whose bases lie in the planes
of the faces of p . Show that the sum of the surface areas of these five
pyramids is greater or equal to one fifth the surface of p , and find for which
M equality holds.

The points E and D are taken on the sides AC
and BC respectively of a
triangle ABC. The lines AD and BE intersect at F.
Show that the areas of the triangles ABC
and ABF satisfy
$\frac{S_{ABC}}{S_{ABF}}=\frac{AC}{AE}+\frac{BC}{BD} -1$.

Points K and L are chosen on the sides AB
and AC of an equilateral
triangle ABC such that BK = AL. Segments BL and
CK intersect at P. Determine the ratio AK / KB for which the segments AP and CK are perpendicular.

Let P be an interior point of the
parallelogram ABCD. Prove that ÐAPB+ÐCPD = 180

^{o}if and only if ÐPDC = ÐPBC.
Let ABCDEF be a convex hexagon such that AB = BC, CD = DE, EF = FA. Prove
that $\frac{BC}{BE}+\frac{DE}{DA}+\frac{FA}{FC}\ge \frac{3}{2}$ . When does
equality occur?

In a triangle ABC, T is the centroid and ÐTAB = ÐACT. Find the maximum possible value of sin ÐCAT +sin ÐCBT.

The altitudes
through the vertices A,B,C of an acute-angled triangle ABC meet the opposite sides at D,E,F, respectively. The line through D parallel to EF meets the lines AC and
AB at Q and R,
respectively. The line EF meets
BC at P. Prove that the circumcircle of the triangle PQR passes through the midpoint of BC.

Let ABC be a triangle, k its incircle and k

_{a}, k_{b}, k_{c}three circles orthogonal to k passing through B and C, A and C, and A and B respectively. The circles k_{a}, k_{b}meet again in C΄, in the same way we obtain the points B΄ and A΄. Prove that the radius of the circumcircle of A΄B΄C΄ is half the radius of k.
(IMO SHL 1999)

Let ABCD be an isosceles trapezoid with
bases AB and CD. The incircle of the triangle BCD touches CD at E. Point F is chosen on the bisector of the
angle DAC such that the lines EF and CD are perpendicular. The circumcircle of the triangle ACF intersects the line CD again at G. Prove that the triangle AFG is isosceles.

(USA, MO 1998-99)

A triangle ABC has acute angles at A and B. Isosceles triangles ACD
and BCE with bases AC and BC are constructed externally to triangle ABC such that ÐADC =ÐABC and ÐBEC = ÐBAC. Let S be the circumcenter of △ABC. Prove that the length of the polygonal line DSE equals the perimeter of triangle ABC if and only if ÐACB is right.

(Jaromír Šimša, Czech)

Distinct points A and B are given on the plane. Consider all triangles ABC in this plane on whose sides BC,CA points D,E respectively can be taken so that

(i) $\frac{BD}{BC} = \frac{CE}{ CA} = \frac{1}{3}$ .

(ii) points A,B,D,E lie on a circle in this order.

Find the locus of the intersection points of lines AD and BE.

(Jaroslav Švrček, Czech)

A triangle ABC has sides BC= a, CA= b, AB = c with a < b < c and area S. Determine the largest number u and the least number v
such that, for every point P inside
△ABC, the inequality u ≤ PD+PE +PF ≤ v holds,
where D,E,F are the
intersection points of AP,BP,CP with the opposite sides.

In an
acute-angled triangle ABC with
circumcenter O, points P and Q are taken on sides AC
and BC respectively such
that $\frac{ AP}{PQ }= \frac{BC}{AB}$ and $\frac{BQ}{PQ }= \frac{AC}{AB}$ . Prove that the points O,P,Q,C lie on a
circle.

In an
acute-angled triangle ABC the
angle at B is greater than 45

^{o}. Points D,E,F are the feet of the altitudes from A,B,C respectively, and K is the point on segment AF such that ÐDKF = ÐKEF.
(a) Show that
such a point K always exists.

(b) Prove that KD

^{2 }= FD^{2 }+AF ·BF.
Point P lies on
the median from vertex C of a triangle ABC. Line AP meets BC at X, and line BP
meets AC at Y. Prove that if quadrilateral ABXY is cyclic, then triangle ABC is
isosceles.

A point P in the interior of a cyclic
quadrilateral ABCD satisfies ÐBPC =ÐBAP+ÐPDC. Denote by E, F and G the feet
of the perpendiculars from P to
the lines AB, AD and DC, respectively. Show that the triangles FEG and PBC are similar.

(Jaroslav Švrček, Czech)

Points K,L,M on the sides
AB,BC,CA respectively
of a triangle ABC satisfy $\frac{AK}{KB} = \frac{BL}{LC} =
\frac{CM}{MA}$ . Show that the triangles ABC and KLM have
a common orthocenter if and only if △ABC is
equilateral.

(P. Černek)

A convex
quadrilateral ABCD is inscribed
in a circle with center O and
circumscribed to a circle with center I.
Its diagonals meet at P. Prove
that points O, I and P lie on a line.

Given a convex
quadrilateral ABCD, find the
locus of the points P inside
the quadrilateral such that S

_{PAB}· S_{PCD}= S_{PBC}· S_{PDA}.
Five points A,B,C,D,E lie in this order on a circle of radius r and satisfy AC=BD = CE = r. Prove that the orthocenters of triangles ACD,BCD,BCE form a
rectangular triangle.

(Tomáš Jurík, Slovakia)

Find out if
there is a convex pentagon A

_{1}A_{2}A_{3}A_{4}A_{5}such that, for each i =1,…,5, the lines A_{i}A_{i}_{+3 }and A_{i}_{+1}A_{i}_{+2 }intersect at a point B_{i}and the points B_{1},B_{2},B_{3},B_{4},B_{5}are collinear. (Here A_{i}_{+5 }= A_{i}.)
(Waldemar Pompe, Poland)

A convex
quadrilateral ABCD inscribed in
a circle

*k*has the property that the rays DA and CB meet at a point E for which CD^{2}= AD·ED. The perpendicular to ED at A intersects*k*again at point F. Prove that the segments AD and CF are congruent if and only if the circumcenter of △ABE lies on ED.
(Jaroslav Švrček, Czech)

Let ABCD be a convex quadrilateral. A
circle passing through the points A and
D and a circle passing through
the points B and C are externally tangent at a point P inside the quadrilateral. Suppose
that ÐPAB+ÐPDC ≤ 90

^{o}and ÐPBA+ ÐPCD ≤ 90^{o}. Prove that AB+CD ≥ BC+AD.
ABCDEF is a
convex hexagon, such that |ÐFAB| = |ÐBCD| = |ÐDEF| and |AB| = |BC|, |CD| =
|DE|, |EF| = |FA|. Prove that the lines AD, BE and CF are concurrent.

(Waldemar Pompe, Poland)

ABCDE is a
regular pentagon. Determine the smallest value of the expression $\frac{|PA| +
|PB|}{|PC| + |PD| + |PE|}$ where P is an arbitrary point lying in the plane of
the pentagon ABCDE.

(Waldemar Pompe, Poland)

Let ω denote the
excircle tangent to side BC of triangle ABC. A line

*ℓ*parallel to BC meets sides AB and AC at points D and E, respectively. Let ω′ denote the incircle of triangle ADE. The tangent from D to ω (different from line AB) and the tangent from E to ω (different from line AC) meet at point P. The tangent from B to ω′ (different from line AB) and the tangent from C to ω′ (different from line AC) meet at point Q. Prove that, independent of the choice of*ℓ*, there is a fixed point that line PQ always passes through.
(Tomáš Jurík, Slovakia)

Given a circle,
let AB be a chord that is not a diameter, and let C be a point on the longer
arc AB. Let K and L denote the reflections of A and B, respectively, about
lines BC and AC, respectively. Prove that the distance between the midpoint of
AB and the midpoint of KL is independent of the choice of C.

(Tomáš Jurík, Slovakia)

Let ABCD be a
convex quadrilateral for which AB + CD = √2·AC and BC + DA = √2·BD. Prove that
ABCD is a parallelogram.

(Jaromír Šimša, Czech)

Points A, B, C,
D lie on a circle (in that order) where AB and CD are not parallel. The length
of arc AB (which contains the points D and C) is twice the length of arc CD
(which does not contain the points A and B). Let E be a point where AC = AE and
BD = BE. Prove that if the perpendicular line from point E to the line AB
passes through the center of the arc CD (which does not contain the points A
and B), then ÐACB = 108

^{o}.
(Tomáš Jurík, Slovakia)

In convex
quadrilateral ABCD, let M and N denote the midpoints of sides AD and BC,
respectively. On sides AB and CD are points K and L, respectively, such that ÐMKA = ÐNLC. Prove that if lines BD, KM,
and LN are concurrent, then ÐKMN = ÐBDC and ÐLNM = ÐABD.

(Poland)

Let ABCD be a
cyclic quadrilateral with circumcircle ω. Let I, J and K
be the incentres of the triangles ABC, ACD and ABD respectively. Let E be the
midpoint of the arc DB of circle ω containing the
point A. The line EK intersects again the circle ω
at point F (F ≠E). Prove that the points C, F, I, J lie on a circle.

(Tomáš Jurík, Slovakia)

Suppose
ABCD is a cyclic quadrilateral with BC = CD. Let ω be the circle with center C tangential
to the side BD. Let I be the centre of the incircle of triangle ABD. Prove that
the straight line passing through I, which is parallel to AB, touches the
circle ω.

(Kamil Duszenko, Poland)

Let
ABC be a triangle inscribed in a circle. Point P is the center of the arc BAC. The circle with the diameter CP
intersects the angle bisector of angle \angle BAC at points K, L (|AK| <|AL|). Point M is the reflection of
L with respect to line BC. Prove that the circumcircle of the triangle BKM
passes through the center of the segment BC .

(Dominik Burek & Tomasz Cieśla, Poland)

Given is a
convex ABCD, which is |ÐABC| = |Ð ADC|= 135

^{ο}. On the AB, AD are also selected points M, N such that |ÐMCD| = |Ð NCB| = 90^{ }^{ο}. The circumcircles of the triangles AMN and ABD intersect for the second time at point K \ne A. Prove that lines AK and KC are perpendicular.
(Iran)

Let ABC be a triangle, and let P be the midpoint of AC. A circle intersects AP, CP, BC, AB sequentially at their inner points K, L, M, N. Let S be the midpoint of KL. Let also $2 \cdot | AN |\cdot |AB |\cdot |CL | = 2 \cdot | CM | \cdot| BC | \cdot| AK| = | AC | \cdot| AK |\cdot |CL |.$ Prove that if $P\ne S$, then the intersection of KN and ML lies on the perpendicular bisector of the PS.

(Jan Mazák, Slovakia)

On
a circle of radius r, the distinct points A, B, C, D, and E lie in this order,
satisfying AB = CD = DE > r. Show that the triangle with vertices lying in
the centroids of the triangles ABD, BCD, and ADE is obtuse.

(Tomáš Jurík, Slovakia)

Let
ABC be an acute triangle, which is not equilateral. Denote by O and H its
circumcenter and orthocenter, respectively. The circle

*k*passes through B and touches the line AC at A. The circle*l*with center on the ray BH touhes the line AB at A. The circles*k*and*l*meet in X (X≠A). Show that ÐHXO=180^{ο}- ÐBAC.
(Josef Tkadlec, Czech)

Let P be a
non-degenerate polygon with n sides, where n > 4. Prove that there exist
three distinct vertices A, B, C of P with the following property: If

*l*_{1},*l*_{2},*l*_{3}are the lengths of the three polygonal chains into which A, B, C break the perimeter of P, then there is a triangle with side lengths*l*_{1},*l*_{2}, and*l*_{3}.
Remark: By a
non-degenerate polygon we mean a polygon in which every two sides are disjoint,
apart from consecutive ones, which share only the common endpoint.

(Poland)

Let ABC be an
acute-angled triangle with AB < AC. Tangent to its circumcircle at A intersects the line BC at D. Let G be the
centroid of ABC and let AG meet again at
H ≠A. Suppose the line DG intersects the lines AB and AC at E and F,
respectively. Prove that ÐEHG = ÐGHF.

(Patrik Bak, Slovakia)

Let ω
be the circumcircle of an acute-angled triangle ABC. Point D lies on the arc BC
of ω
not containing point A. Point E lies in the interior of the triangle ABC, does
not lie on the line AD, and satisfies ÐDBE
= ÐACB and ÐDCE = ÐABC. Let F be a point on the line
AD such that lines EF and BC are parallel, and let G be a point on ω
different from A such that AF = FG. Prove that points D,E, F,G lie on one
circle.

(Patrik Bak, Slovakia)

Let ABC be a
triangle. Line

*l*is parallel to BC and it respectively intersects side AB at point D, side AC at point E, and the circumcircle of the triangle ABC at points F and G, where points F,D,E,G lie in this order on*l*. The circumcircles of triangles FEB and DGC intersect at points P and Q. Prove that points A, P,Q are collinear.
Let $ABC$ be an acute scalene triangle. Let $D$ and $E$ be points on the sides $AB$ and $AC$, respectively, such that $BD=CE$. Denote by $O_1$ and $O_2$ the circumcentres of the triangles $ABE$ and $ACD$, respectively. Prove that the circumcircles of the triangles $ABC, ADE$, and $AO_1O_2$ have a common point different from $A$.

(Patrik Bak, Slovakia)

Let $\omega$ be a circle. Points $A,B,C,X,D,Y$ lie on $\omega$ in this order such that $BD$ is its diameter and $DX=DY=DP$ , where $P$ is the intersection of $AC$ and $BD$. Denote by $E,F$ the intersections of line $XP$ with lines $AB,BC$, respectively. Prove that points $B,E,F,Y$ lie on a single circle.

Let $ABC$ be an acute triangle with $AB<AC$ and $\angle BAC=60^{\circ}$. Denote its altitudes by $AD,BE,CF$ and its orthocenter by $H$. Let $K,L,M$ be the midpoints of sides $BC,CA,AB$, respectively. Prove that the midpoints of segments $AH, DK, EL, FM$ lie on a single circle.

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