Czech & Slovak MO, III A 1991 - 2021 61p

geometry problems from Czech And Slovak Mathematical Olympiads, Round III, Category A
with aops links in the names

1991 - 2021

1991 Czech & Slovak MO, III A p4
Prove that in all triangles $ABC$ with $\angle A = 2\angle B$ the distance from $C$ to $A$ and to the perpendicular bisector of $AB$ are in the same ratio.

1992 Czech & Slovak MO, III A p2
Let $S$ be the total area of a tetrahedron whose edges have lengths $a,b,c,d, e, f$ . Prove that $S \le \frac{\sqrt3}{6} (a^2 +b^2 +...+ f^2)$\

1992 Czech & Slovak MO, III A p6
Let $ABC$ be an acute triangle. The altitude from $B$ meets the circle with diameter $AC$ at points $P,Q$, and the altitude from $C$ meets the circle with diameter $AB$ at $M,N$. Prove that the points $M,N,P,Q$ lie on a circle.

1993 Czech & Slovak MO, III A p3
Let $AKL$ be a triangle such that $\angle ALK > 90^o +\angle LAK$. Construct an equilateral trapezoid $ABCD$ (i.e. with three sides equal) with $AB \perp CD$ such that $K$ lies on the side $BC, L$ on the diagonal $AC$ and the lines $AK$ and $BL$ intersect at the circumcenter of the trapezoid.

1993 Czech & Slovak MO, III A p6
Show that there exists a tetrahedron which can be partitioned into eight congruent tetrahedra, each of which is similar to the original one.

1994 Czech & Slovak MO, III A p2
A cuboid of volume $V$ contains a convex polyhedron $M$. The orthogonal projection of $M$ onto each face of the cuboid covers the entire face. What is the smallest possible volume of polyhedron $M$?

1994 Czech & Slovak MO, III A p5
In an acute-angled triangle $ABC$, the altitudes $AA_1,BB_1,CC_1$ intersect at point $V$. If the triangles $AC_1V, BA_1V, CB_1V$ have the same area, does it follow that the triangle $ABC$ is equilateral?

1995 Czech & Slovak MO, III A p1
Suppose that tetrahedron $ABCD$ satisfies $\angle BAC+\angle CAD+\angle DAB = \angle ABC+\angle CBD+\angle DBA = 180^o$. Prove that $CD \ge AB$.

1995 Czech & Slovak MO, III A p5
Let $A,B$ be points on a circle $k$ with center $S$ such that $\angle ASB = 90^o$ . Circles $k_1$ and $k_2$ are tangent to each other at $Z$ and touch $k$ at $A$ and $B$ respectively. Circle $k_3$ inside $\angle ASB$ is internally tangent to $k$ at $C$ and externally tangent to $k_1$ and $k_2$ at $X$ and $Y$, respectively. Prove that $\angle XCY = 45^o$

1996 Czech & Slovak MO, III A p2
Let $AP,BQ$ and $CR$ be altitudes of an acute-angled triangle $ABC$. Show that for any point $X$ inside the triangle $PQR$ there exists a tetrahedron $ABCD$ such that $X$ is the point on the face $ABC$ at the greatest distance from $D$ (measured along the surface of the tetrahedron).

1996 Czech & Slovak MO, III A p4
Points $A$ and $B$ on the rays $CX$ and $CY$ respectively of an acute angle $XCY$ are given so that $CX < CA = CB < CY$. Construct a line meeting the ray $CX$ and the segments $AB,BC$ at $K,L,M$, respectively, such that $KA \cdot YB = XA \cdot MB = LA\cdot LB \ne 0$.

1996 Czech & Slovak MO, III A p6
Let $K,L,M$ be points on sides $AB,BC,CA$, respectively, of a triangle $ABC$ such that $AK/AB = BL/BC = CM/CA = 1/3$. Show that if the circumcircles of the triangles $AKM, BLK, CML$ are equal, then so are the incircles of these triangles.

1997 Czech & Slovak MO, III A p1
Let $ABC$ be a triangle with sides $a,b,c$ and corresponding angles $\alpha,\beta\gamma$ . Prove that if $\alpha = 3\beta$ then $(a^2 -b^2)(a-b) = bc^2$ . Is the converse true?

1997 Czech & Slovak MO, III A p3
A tetrahedron $ABCD$ is divided into five polyhedra so that each face of the tetrahedron is a face of (exactly) one polyhedron, and that the intersection of any two of the polyhedra is either a common vertex, a common edge, or a common face. What is the smallest possible sum of the numbers of faces of the five polyhedra?

1997 Czech & Slovak MO, III A p6
In a parallelogram $ABCD$, triangle $ABD$ is acute-angled and $\angle BAD = \pi /4$. Consider all possible choices of points $K,L,M,N$ on sides $AB,BC, CD,DA$ respectively, such that $KLMN$ is a cyclic quadrilateral whose circumradius equals those of triangles $ANK$ and $CLM$. Find the locus of the intersection of the diagonals of $KLMN$

1998 Czech & Slovak MO, III A p3
A sphere is inscribed in a tetrahedron $ABCD$. The tangent planes to the sphere parallel to the faces of the tetrahedron cut off four smaller tetrahedra. Prove that sum of all the $24$ edges of the smaller tetrahedra equals twice the sum of edges of the tetrahedron $ABCD$.

1998 Czech & Slovak MO, III A p5
A circle $k$ and a point $A$ outside it are given in the plane. Prove that all trapezoids, whose non-parallel sides meet at $A$, have the same intersection of diagonals.

1999 Czech & Slovak MO, III A p2
In a tetrahedron $ABCD, E$ and $F$ are the midpoints of the medians from $A$ and $D$. Find the ratio of the volumes of tetrahedra $BCEF$ and $ABCD$.

1999 Czech & Slovak MO, III A p3
Show that there exists a triangle $ABC$ such that $a \ne b$ and $a+t_a = b+t_b$, where $t_a,t_b$ are the medians corresponding to $a,b$, respectively. Also prove that there exists a number $k$ such that every such triangle satisfies $a+t_a = b+t_b = k(a+b)$. Finally, find all possible ratios $a : b$ in such triangles.

1999 Czech & Slovak MO, III A p5
Given an acute angle $APX$ in the plane, construct a square $ABCD$ such that $P$ lies on the side $BC$ and ray $PX$ meets $CD$ in a point $Q$ such that $AP$ bisects the angle $BAQ$.

2000 Czech & Slovak MO, III A p2
Let be given an isosceles triangle $ABC$ with the base $AB$. A point $P$ is chosen on the altitude $CD$ so that the incircles of $ABP$ and $PECF$ are congruent, where E and F are the intersections of $AP$ and $BP$ with the opposite sides of the triangle, respectively. Prove that the incircles of triangles $ADP$ and $BCP$ are also congruent.

2000 Czech & Slovak MO, III A p5
Monika made a paper model of a tetrahedron whose base is a right-angled triangle. When she cut the model along the legs of the base and the median of a lateral face corresponding to one of the legs, she obtained a square of side a. Compute the volume of the tetrahedron.

2001 Czech & Slovak MO, III A p2
Given a triangle $PQX$ in the plane, with $PQ = 3, PX = 2.6$ and $QX = 3.8$. Construct a right-angled triangle $ABC$ such that the incircle of $\vartriangle ABC$ touches $AB$ at $P$ and $BC$ at $Q$, and point $X$ lies on the line $AC$.

2001 Czech & Slovak MO, III A p5
A sheet of paper has the shape of an isosceles trapezoid $C_1AB_2C_2$ with the shorter base $B_2C_2$. The foot of the perpendicular from the midpoint $D$ of $C_1C_2$ to $AC_1$ is denoted by $B_1$. Suppose that upon folding the paper along $DB_1, AD$ and $AC_1$ points $C_1,C_2$ become a single point $C$ and points $B_1,B_2$ become a point $B$. The area of the tetrahedron $ABCD$ is $64$ cm$^2$ . Find the sides of the initial trapezoid.

2002 Czech & Slovak MO, III A p2
Consider an arbitrary equilateral triangle $KLM$, whose vertices $K, L$ and $M$ lie on the sides $AB, BC$ and $CD$, respectively, of a given square $ABCD$. Find the locus of the midpoints of the sides $KL$ of all such triangles $KLM$.

2002 Czech & Slovak MO, III A p5
A triangle $KLM$ is given in the plane together with a point $A$ lying on the half-line opposite to $KL$. Construct a rectangle $ABCD$ whose vertices $B, C$ and $D$ lie on the lines $KM, KL$ and $LM$, respectively. (We allow the rectangle to be a square.)

2003 Czech & Slovak MO, III A p2
On sides $BC,CA,AB$ of a triangle $ABC$ points $D,E,F$ respectively are chosen so that $AD,BE,CF$ have a common point, say $G$. Suppose that one can inscribe circles in the quadrilaterals $AFGE,BDGF,CEGD$ so that each two of them have a common point. Prove that triangle $ABC$ is equilateral.

2003 Czech & Slovak MO, III A p4
Let be given an obtuse angle $AKS$ in the plane. Construct a triangle $ABC$ such that $S$ is the midpoint of $BC$ and $K$ is the intersection point of $BC$ with the bisector of $\angle BAC$.

2004 Czech & Slovak MO, III A p5
Let $L$ be an arbitrary point on the minor arc $CD$ of the circumcircle of square $ABCD$. Let $K,M,N$ be the intersection points of $AL,CD$; $CL,AD$; and $MK,BC$ respectively. Prove that $B,M,L,N$ are concyclic.

2005 Czech & Slovak MO, III A p3
In a trapezoid $ABCD$ with $AB // CD, E$ is the midpoint of $BC$. Prove that if the quadrilaterals $ABED$ and $AECD$ are tangent, then the sides $a = AB, b = BC, c =CD, d = DA$ of the trapezoid satisfy the equalities $a+c = \frac{b}{3} +d$ and $\frac1a +\frac1c = \frac3b$ .

2005 Czech & Slovak MO, III A p4
An acute-angled triangle $AKL$ is given on a plane. Consider all rectangles $ABCD$ circumscribed to triangle $AKL$ such that point $K$ lies on side $BC$ and point $L$ lieson side $CD$. Find the locus of the intersection $S$ of the diagonals $AC$ and $BD$.

2006 Czech & Slovak MO, III A p3
In a scalene triangle $ABC$,the bisectors of angle $A,B$ intersect their corresponding sides at $K,L$ respectively.$I,O,H$ denote respectively the incenter,circumcenter and orthocenter of triangle $ABC$. Prove that $A,B,K,L,O$ are concyclic iff $KL$ is the common tangent line of the circumcircles of the three triangles $ALI,BHI$ and $BKI$.

2006 Czech & Slovak MO, III A p4
Given a segment $AB$ in the plane. Let $C$ be another point in the same plane,$H,I,G$ denote the orthocenter,incenter and centroid of triangle $ABC$. Find the locus of $M$ for which $A,B,H,I$ are concyclic.

2007 Czech & Slovak MO, III A p2
In a cyclic quadrilateral $ABCD$, let $L$ and $M$ be the incenters of $ABC$ and $BCD$ respectively. Let $R$ be a point on the plane such that $LR\bot AC$ and $MR\bot BD$.Prove that triangle $LMR$ is isosceles.

2007 Czech & Slovak MO, III A p5
In an acute-angled triangle $ABC$ ($AC\ne BC$), let $D$ and $E$ be points on $BC$ and $AC$, respectively, such that the points $A,B,D,E$ are concyclic and $AD$ intersects $BE$ at $P$. Knowing that $CP\bot AB$, prove that $P$ is the orthocenter of triangle $ABC$.

2008 Czech & Slovak MO, III A p2
Two disjoint circles $W_1(S_1,r_1)$ and $W_2(S_2,r_2)$ are given in the plane. Point $A$ is on circle $W_1$ and $AB,AC$ touch the circle $W_2$ at $B,C$ respectively. Find the loci of the incenter and orthocenter of triangle $ABC$.

2009 Czech & Slovak MO, III A p2
Rectangle $ABCD$ is inscribed in circle $O$. Let the projections of a point $P$ on minor arc $CD$ onto $AB,AC,BD$ be $K,L,M$, respectively. Prove that $\angle LKM=45$if and only if $ABCD$ is a square.

2009 Czech & Slovak MO, III A p6
Given two fixed points $O$ and $G$ in the plane. Find the locus of the vertices of triangles whose circumcenters and centroids are $O$ and $G$ respectively.

2010 Czech & Slovak MO, III A p2
A circular target with a radius of $12$ cm was hit by $19$ shots. Prove that the distance between two hits is less than $7$ cm.

2010 Czech & Slovak MO, III A p4
A circle $k$ is given with a non-diameter chord $AC$. On the tangent at point $A$ select point $X \ne A$ and mark $D$ the intersection of the circle $k$ with the interior of the line $XC$ (if any). Let $B$ a point in circle $k$ such that quadrilateral $ABCD$ is a trapezoid . Determine the set of intersections of lines $BC$ and $AD$ belonging to all such trapezoids.

2011 Czech & Slovak MO, III A p1
In a certain triangle $ABC$, there are points $K$ and $M$ on sides $AB$ and $AC$, respectively, such that if $L$ is the intersection of $MB$ and $KC$, then both $AKLM$ and $KBCM$ are cyclic quadrilaterals with the same size circumcircles. Find the measures of the interior angles of triangle $ABC$.

2011 Czech & Slovak MO, III A p5
In acute triangle ABC, which is not equilateral, let $P$ denote the foot of the altitude from $C$ to side $AB$; let $H$ denote the orthocenter; let $O$ denote the circumcenter; let $D$ denote the intersection of line $CO$ with $AB$; and let $E$ denote the midpoint of $CD$. Prove that line $EP$ passes through the midpoint of $OH$.

2012 Czech & Slovak MO, III A p2
Find out the maximum possible area of the triangle $ABC$ whose medians have lengths satisfying inequalities $m_a \le 2, m_b \le 3, m_c \le  4$.

2012 Czech & Slovak MO, III A p4
Inside the parallelogram $ABCD$ is a point $X$. Make a line that passes through point $X$ and divides the parallelogram into two parts whose areas differ from each other the most.

2013 Czech & Slovak MO, III A p3
In the parallelolgram A$BCD$ with the center $S$, let $O$ be the center of the circle of the inscribed triangle $ABD$ and let $T$ be the touch point with the diagonal $BD$. Prove that the lines $OS$ and $CT$ are parallel.

2013 Czech & Slovak MO, III A p5
Given the parallelogram $ABCD$ such that the feet $K, L$ of the perpendiculars from point $D$ on the sides $AB, BC$ respectively are internal points. Prove that $KL \parallel AC$ when $|\angle BCA| + |\angle ABD| = |\angle BDA| + |\angle ACD|$.

2014 Czech & Slovak MO, III A p2
A segment $AB$ is given in (Euclidean) plane. Consider all triangles $XYZ$ such, that $X$ is an inner point of $AB$, triangles $XBY$ and $XZA$ are similar (in this order of vertices) and points $A, B, Y, Z$ lie on a circle in this order. Find a set of midpoints of all such segments $YZ$.

2014 Czech & Slovak MO, III A p5
Given is the acute triangle $ABC$. Let us denote $k$ a circle with diameter $AB$. Another circle, tangent to $AB$ at point $A$ and passing through point $C$ intersects the circle $k$ at point $P, P \ne A$. Another  circle which touches AB at point $B$ and passes point $C$, intersects the circle $k$ at point $Q, Q \ne B$. Prove that the intersection of the line $AQ$ and $BP$ lies on one of the sides of angle $ACB$.

2015 Czech & Slovak MO, III A p3
In triangle $\triangle ABC$ with median from $B$ not perpendicular to $AB$ nor $BC$, we call $X$ and $Y$ points on $AB$ and $BC$, which lie on the axis of the median from $B$. Find all such triangles, for which $A,C,X,Y$ lie on one circumrefference.

2015 Czech & Slovak MO, III A p5
In given triangle $\triangle ABC$, difference between sizes of each pair of sides is at least $d>0$. Let $G$ and $I$ be the centroid and incenter of $\triangle ABC$ and $r$ be its inradius. Show that $[AIG]+[BIG]+[CIG]\ge\frac{2}{3}dr,$where $[XYZ]$ is (nonnegative) area of triangle $\triangle XYZ$.

2016 Czech & Slovak MO, III A p2
Let us denote successively $r$ and $r_a$ the radii of the inscribed circle and the exscribed circle wrt to side BC of triangle $ABC$. Prove that if it is true that $r+r_a=|BC|$ , then the triangle $ABC$ is a right one.

2016 Czech & Slovak MO, III A p5
In the triangle $ABC$, $| BC | = 1$ and there is exactly one point $D$ on the side $BC$ such that $|DA|^2 =  |DB| \cdot |DC|$. Determine all possible values of the perimeter of the triangle $ABC$.

2017 Czech & Slovak MO, III A p5

Given is the acute triangle $ABC$ with the intersection of altitudes $H$.  The angle bisector of angle $BHC$ intersects side $BC$ at point $D$. Mark $E$ and $F$ the symmetrics of the point $D$ wrt lines $AB$ and $AC$. Prove that the circle circumscribed around the triangle $AEF$ passes through the midpoint of the arc $BAC$

2018 Czech & Slovak MO, III A p3
In triangle $ABC$ let be $D$ an intersection of $BC$ and the $A$-angle bisector. Denote $E,F$ the circumcenters of $ABD$ and $ACD$ respectively. Assuming that the circumcenter of $AEF$ lies on the line $BC$ what is the possible size of the angle $BAC$ ?

2018 Czech & Slovak MO, III A p5
Let $ABCD$ an isosceles trapezoid with the longer base $AB$. Denote $I$ the incenter of $\Delta ABC$ and $J$ the excenter relative to the vertex $C$ of $\Delta ACD$. Show that the lines $IJ$ and $AB$ are parallel.

2019 Czech & Slovak MO, III A p2
Let be $ABCD$ a rectangle with $|AB|=a\ge b=|BC|$. Find points $P,Q$ on the line $BD$ such that $|AP|=|PQ|=|QC|$. Discuss the solvability with respect to the lengths $a,b$.

2019 Czech & Slovak MO, III A p4
Let be $ABC$ an acute-angled triangle. Consider point $P$ lying on the opposite ray to the ray $BC$ such that $|AB|=|BP|$. Similarly, consider point $Q$ on the opposite ray to the ray $CB$ such that $|AC|=|CQ|$. Denote $J$ the excenter of $ABC$ with respect to $A$ and $D,E$ tangent points of this excircle with the lines $AB$ and $AC$, respectively. Suppose that the opposite rays to $DP$ and $EQ$ intersect in $F\neq J$. Prove that $AF\perp FJ$.

2020 Czech & Slovak MO, III A p2

The triangle $ABC$ is given. Inside its sides $AB$ and $AC$, the points $X$ and $Y$ are respectively selected Let $Z$ be the intersection of the lines $BY$ and $CX$. Prove the inequality $$[BZX] + [CZY]> 2 [XY Z]$$ , where $[DEF]$ denotes the content of the triangle $DEF$.

2020 Czech & Slovak MO, III A p5

Given an isosceles triangle $ABC$ with base $BC$. Inside the side $BC$ is given a point $D$. Let $E, F$ be respectively points on the sides $AB, AC$ that $|\angle BED | = |\angle  DF C| > 90^o$ . Prove that the circles circumscribed around the triangles $ABF$ and $AEC$ intersect on the line $AD$ at a point different from point $A$.

2021 Czech & Slovak MO, III A p2

Let $I$ denote the center of the circle inscribed in the right triangle $ABC$ with right angle at the vertex $A$. Next, denote by $M$ and $N$ the midpoints of the lines $AB$ and $BI$. Prove that the line $CI$ is tangent to the circumscribed circle of triangle $BMN$.

2021 Czech & Slovak MO, III A p6

An acute triangle $ABC$ is given. Let us denote $X$ for each of its inner points $X_a, X_b, X_c$ its images in axial symmetries sequentially along the lines $BC, CA, AB$. Prove that all $X_aX_bX_c$ triangles have a common point.

source: https://skmo.sk/ , http://www.matematickaolympiada.cz/