Czech-Polish-Slovak Match 1995 - 2022 (CPS) 54p

geometry problems from Czech-Polish-Slovak Mathematical Match (CPS)
with aops links in the names

Czech - Polish - Slovak Match (CPS) geometry problems 1995- 2017 EN in pdf with aops links

collected inside aops here

Started in 1995 as Czech and Slovak Match.

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

1995 - 2022

CS Match 1995.3

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 .

CS Match 1995.5

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.

CS Match 1996.3

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.

CS Match 1996.6

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 $\frac{S_{ABC}}{S_{ABF}}=\frac{AC}{AE}+\frac{BC}{BD} -1$.

CS Match 1997.1

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.

CS Match 1998.1

Let P be an interior point of the parallelogram ABCD. Prove that ÐAPB+ÐCPD = 180^o  if and only if ÐPDC = ÐPBC.

CS Match 1998.3

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?

CS Match 1998.5

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

CS Match 1999.2

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.

CS Match 2000.2

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)

CS Match 2000.5

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) 

CPS Match 2001.2

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)

CPS Match 2001.4

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)

CPS Match 2002.2

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.

CPS Match 2002.5

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.

CPS Match 2003.2

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 KD² = FD² +AF ·BF.

CPS Match 2003.4

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.

CPS Match 2004.3

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)

CPS Match 2004.5

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)

CPS Match 2005.2

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.

CPS Match 2005.5

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 .

CPS Match 2006.1

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)

CPS Match 2006.5

Find out if there is a convex pentagon A₁A₂A₃A₄A₅ such that, for each i =1,…,5, the lines A_iA_i+3 and A_i+1A_i+2 intersect at a point B_i and the points B₁,B₂,B₃,B₄,B₅ are collinear. (Here A_i+5 = A_i.)

(Waldemar Pompe, Poland)

CPS Match 2007.3

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)

CPS Match 2007.6

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.

(Waldemar Pompe, Poland)

CPS Match 2008.2

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)

CPS Match 2008.5

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)

CPS Match 2009.3

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)

CPS Match 2009.4

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)

CPS Match 2010.6

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)

CPS Match 2011.3

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)

CPS Match 2011.5

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)

CPS Match 2012.3

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)

CPS Match 2012.4

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)

CPS Match 2013.1

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)

CPS Match 2013.6

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)

CPS Match 2014.3

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)

CPS Match 2014.4

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)

CPS Match 2015.1

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)

CPS Match 2015.5

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)

CPS Match 2016.1

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.

(Poland)

CPS Match 2016.6

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)

CPS Match 2017.2

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)

CPS Match 2017.4

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)

CPS Match 2018.2

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)

CPS Match 2019.1

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.

(Patrik Bak, Slovakia)

CPS Match 2019.6

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.

(Dominik Burek, Poland)

CPS Match 2020.1

Let $ABCD$ be a parallelogram whose diagonals meet at $P$. Denote by $M$ the midpoint of $AB$.

Let $Q$ be a point such that $QA$ is tangent to the circumcircle of $MAD$ and $QB$ is tangent to the

circumcircle of $MBC$. Prove that points $Q,M,P$ are collinear.

(Patrik Bak, Slovakia)

CPS Match 2020.6

Let $ABC$ be an acute triangle. Let $P$ be a point such that $PB$ and $PC$ are tangent to circumcircle

of $ABC$. Let $X$ and $Y$ be variable points on $AB$ and $AC$, respectively, such that

$\angle XPY = 2\angle BAC$ and $P$ lies in the interior of triangle $AXY$. Let $Z$ be the reflection

of $A$ across $XY$. Prove that the circumcircle of $XYZ$ passes through a fixed point.

(Dominik Burek, Poland)

CPS Match 2021.2

In an acute triangle $ABC$, the incircle $\omega$ touches $BC$ at $D$. Let $I_a$ be the excenter of $ABC$ opposite to $A$, and let $M$ be the midpoint of $DI_a$. Prove that the circumcircle of triangle $BMC$ is tangent to $\omega$.

Patrik Bak (Slovakia)

CPS Match 2021.6

Let $ABC$ be an acute triangle and suppose points $A, A_b, B_a, B, B_c, C_b, C, C_a,$ and $A_c$ lie on its perimeter in this order. Let $A_1 \neq A$ be the second intersection point of the circumcircles of triangles $AA_bC_a$ and $AA_cB_a$. Analogously, $B_1 \neq B$ is the second intersection point of the circumcircles of triangles $BB_cA_b$ and $BB_aC_b$, and $C_1 \neq C$ is the second intersection point of the circumcircles of triangles $CC_aB_c$ and $CC_bA_c$. Suppose that the points $A_1, B_1,$ and $C_1$ are all distinct, lie inside the triangle $ABC$, and do not lie on a single line. Prove that lines $AA_1, BB_1, CC_1,$ and the circumcircle of triangle $A_1B_1C_1$ all pass through a common point.

Josef Tkadlec (Czech Republic), Patrik Bak (Slovakia)

CPS Match 2022.3

Circles $\Omega_1$ and $\Omega_2$ with different radii intersect at two points, denote one of them by $P$. A variable line $l$ passing through $P$ intersects the arc of $\Omega_1$ which is outside of $\Omega_2$ at $X_1$, and the arc of $\Omega_2$ which is outside of $\Omega_1$ at $X_2$. Let $R$ be the point on segment $X_1X_2$ such that $X_1P = RX_2$. The tangent to $\Omega_1$ through $X_1$ meets the tangent to $\Omega_2$ through $X_2$ at $T$. Prove that line $RT$/is tangent to a fixed circle, independent of the choice of $l$.

CPS Match 2022.5

Let $ABC$ be a triangle with $AB < AC$ and circumcenter $O$. The angle bisector of $\angle BAC$ meets the side $BC$ at $D$. The line through $D$ perpendicular to $BC$ meets the segment $AO$ at $X$. Furthermore, let $Y$ be the midpoint of segment $AD$. Prove that points $B, C, X, Y$ are concyclic.