Czech-Polish-Slovak Match 1995 - 2019 (CPS) 48p

geometry problems from Czech-Polish-Slovak Mathematical Match (CPS)
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 = 180o  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 ka, kb, kc three circles orthogonal to k passing through B and C, A and C, and A and B respectively. The circles ka, kb 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 45o. 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 KD2 = FD2 +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 SPAB · SPCD = SPBC · SPDA .

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 A1A2A3A4A5 such that, for each i =1,…,5, the lines AiAi+3 and Ai+1Ai+2 intersect at a point Bi and the points B1,B2,B3,B4,B5 are collinear. (Here Ai+5 = Ai.)

(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 CD2 = 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 ≤ 90o and ÐPBA+ ÐPCD ≤ 90o. Prove that AB+CD ≥ BC+AD.

(Waldemar Pompe, Poland)
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 parallel to BC meets sides AB and AC at points D and E, respectively. Let ωdenote 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 , 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 = 108o.

(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.

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.

(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.

(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|= 135ο. 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.

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=180ο- 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  l1, l2, l3 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 l1, l2, and  l3.
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.

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.

(Patrik Bak, Slovakia)
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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