Hungary 1947 - 2022 (Kürschák) 55p

geometry problems from Hungarian Mathematical Olympiads (Kürschák Competition)
with aops links in the names

(Hungarian) Eötvös - Kürschák Competitions 1894-2003 

(by Ercole Suppa)

collected inside aops here

Kürschák started in 1947

1947 - 2020 

1948 Kürschák Competition P2

A convex polyhedron has no diagonals (every pair of vertices are connected by an edge). Prove that it is a tetrahedron.

1949 Kürschák Competition P2

$P$ is a point on the base of an isosceles triangle. Lines parallel to the sides through $P$ meet the sides at $Q$ and $R$. Show that the reflection of $P$ in the line $QR$ lies on the circumcircle of the triangle.

1950 Kürschák Competition P2

Three circles $C_1$, $C_2$, $C_3$ in the plane touch each other (in three different points). Connect the common point of $C_1$ and $C_2$ with the other two common points by straight lines. Show that these lines meet $C_3$ in diametrically opposite points.

1951 Kürschák Competition P1

$ABCD$ is a square. $E$ is a point on the side $BC$ such that $BE =1/3 BC$, and $F$ is a point on the ray $DC$ such that $CF =1/2 DC$. Prove that the lines $AE$ and $BF$ intersect on the circumcircle of the square.

1952 Kürschák Competition P1

A circle $C$ touches three pairwise disjoint circles whose centers are collinear and none of which contains any of the others. Show that its radius must be larger than the radius of the middle of the three circles.

1952 Kürschák Competition P3

$ABC$ is a triangle. The point A' lies on the side opposite to $A$ and $BA'/BC = k$, where $1/2 < k < 1$. Similarly, $B'$ lies on the side opposite to $B$ with $CB'/CA = k$, and $C'$ lies on the side opposite to $C$ with $AC'/AB = k$. Show that the perimeter of $A'B'C'$ is less than $k$ times the perimeter of $ABC$.

1953 Kürschák Competition P3

$ABCDEF$ is a convex hexagon with all its sides equal. Also $\angle A +  \angle C +  \angle E =  \angle B +  \angle D +  \angle F$. Show that $\angle A =  \angle D$, $\angle B =  \angle E$ and $\angle C =  \angle F$.

1954 Kürschák Competition P1

$ABCD$ is a convex quadrilateral with $AB + BD = AC + CD$. Prove that $AB < AC$.

1955 Kürschák Competition P1

Prove that if the two angles on the base of a trapezoid are different, then the diagonal starting from the smaller angle is longer than the other diagonal.

1957 Kürschák Competition P1

$ABC$ is an acute-angled triangle. $D$ is a variable point in space such that all faces of the tetrahedron $ABCD$ are acute-angled. $P$ is the foot of the perpendicular from $D$ to the plane $ABC$. Find the locus of $P$ as $D$ varies.

1958 Kürschák Competition P3

The hexagon $ABCDEF$ is convex and opposite sides are parallel. Show that the triangles $ACE$ and $BDF$ have equal area

1959 Kürschák Competition P2

The angles subtended by a tower at distances $100$, $200$ and $300$ from its foot sum to $90^o$. What is its height?

1960 Kürschák Competition P3

$E$ is the midpoint of the side $AB$ of the square $ABCD$, and $F, G$ are any points on the sides $BC$, $CD$ such that $EF$ is parallel to $AG$. Show that $FG$ touches the inscribed circle of the square.

1961 Kürschák Competition P3

Two circles centers $O$ and $O'$ are disjoint. $PP'$ is an outer tangent (with $P$ on the circle center O, and P' on the circle center $O'$). Similarly, $QQ'$ is an inner tangent (with $Q$ on the circle center $O$, and $Q'$ on the circle center $O'$). Show that the lines $PQ$ and $P'Q'$ meet on the line $OO'$.

1962 Kürschák Competition P3

$P$ is any point of the tetrahedron $ABCD$ except $D$. Show that at least one of the three distances $DA$, $DB$, $DC$ exceeds at least one of the distances $PA$, $PB$ and $PC$.

1963 Kürschák Competition P3

A triangle has no angle greater than $90^o$. Show that the sum of the medians is greater than four times the circumradius.

1964 Kürschák Competition P1

$ABC$ is an equilateral triangle. $D$ and$D'$ are points on opposite sides of the plane $ABC$ such that the two tetrahedra $ABCD$ and $ABCD'$ are congruent (but not necessarily with the vertices in that order). If the polyhedron with the five vertices $A, B, C, D, D'$ is such that the angle between any two adjacent faces is the same, find $DD'/AB$ .

1965 Kürschák Competition P3

A pyramid has square base and equal sides. It is cut into two parts by a plane parallel to the base. The lower part (which has square top and square base) is such that the circumcircle of the base is smaller than the circumcircles of the lateral faces. Show that the shortest path on the surface joining the two endpoints of a spatial diagonal lies entirely on the lateral faces.

1967 Kürschák Competition P3

For a vertex $X$ of a quadrilateral, let $h(X)$ be the sum of the distances from $X$ to the two sides not containing $X$. Show that if a convex quadrilateral $ABCD$ satisfies $h(A) = h(B) = h(C) = h(D)$, then it must be a parallelogram.

1969 Kürschák Competition P2

A triangle has side lengths $a, b, c$ and angles $A, B, C$ as usual (with $b$ opposite $B$ etc). Show that if $$a(1 - 2 \cos A) + b(1 - 2 \cos B) + c(1 - 2 \cos C) = 0$$ then the triangle is equilateral.

1971 Kürschák Competition P1

A straight line cuts the side $AB$ of the triangle $ABC$ at $C_1$, the side $AC$ at $B_1$ and the line $BC$ at $A_1$. $C_2$ is the reflection of $C_1$ in the midpoint of $AB$, and $B_2$ is the reflection of $B_1$ in the midpoint of $AC$. The lines $B_2C_2$ and $BC$ intersect at $A_2$. Prove that $$\frac{sen \, \, B_1A_1C}{sen\, \,  C_2A_2B} = \frac{B_2C_2}{B_1C_1}$$  

1975 Kürschák Competition P2

Prove or disprove: given any quadrilateral inscribed in a convex polygon, we can find a rhombus inscribed in the polygon with side not less than the shortest side of the quadrilateral.

1976 Kürschák Competition P1

$ABCD$ is a parallelogram. $P$ is a point outside the parallelogram such that angles $\angle PAB$ and $\angle  PCB$ have the same value but opposite orientation. Show that $\angle APB = \angle DPC$.

1977 Kürschák Competition P2

$ABC$ is a triangle with orthocenter $H$. The median from $A$ meets the circumcircle again at $A_1$, and $A_2$ is the reflection of $A_1$ in the midpoint of $BC$. The points$B_2$ and $C_2$ are defined similarly. Show that $H$, $A_2$, $B_2$ and $C_2$ lie on a circle.

1978 Kürschák Competition P3

A triangle has inradius $r$ and circumradius $R$. Its longest altitude has length $H$. Show that if the triangle does not have an obtuse angle, then $H \ge r+R$. When does equality hold?

1979 Kürschák Competition P1

The base of a convex pyramid has an odd number of edges. The lateral edges of the pyramid are all equal, and the angles between neighbouring faces are all equal. Show that the base must be a regular polygon.

1981 Kürschák Competition P1

Prove that $$AB + PQ + QR + RP \le  AP + AQ + AR + BP + BQ + BR$$ where $A, B, P, Q$ and $R$ are any five points in a plane.

1984 Kürschák Competition P3

$A_1B_1A_2$, $B_1A_2B_2$, $A_2B_2A_3$,...,$B_{13}A_{14}B_{14}$, $A_{14}B_{14}A_1$ and $B_{14}A_1B_1$ are equilateral rigid plates that can be folded along the edges $A_1B_1$,$B_1A_2$, ..., $A_{14}B_{14}$ and $B_{14}A_1$ respectively. Can they be folded so that all $28$ plates lie in the same plane?

1985 Kürschák Competition P3
We reflected each vertex of a triangle on the opposite side. Prove that the area of the triangle formed by these three reflection points is smaller than the area of the initial triangle multiplied by five.

1988 Kürschák Competition P1
Prove that if there exists a point $P$ inside the convex quadrilateral $ABCD$ such that the triangles $PAB$, $PBC$, $PCD$, $PDA$ have the same area, then one of the diagonals of $ABCD$ bisects the area of the quadrilateral.

1988 Kürschák Competition P3

Consider the convex lattice quadrilateral $PQRS$ whose diagonals intersect at $E$. Prove that if $\angle P+\angle Q<180^\circ$, then the $\triangle PQE$ contains inside it or on one of its sides a lattice point other than $P$ and $Q$.

1989 Kürschák Competition P1
In the plane, two intersecting lines $a$ and $b$ are given, along with a circle $\omega$ that has no common points with these lines. For any line $\ell||b$, define $A=\ell\cap a$, and $\{B,C\}=\ell\cap \omega$ such that $B$ is on segment $AC$. Construct the line $\ell$ such that the ratio $\frac{|BC|}{|AB|}$ is maximal.

1990 Kürschák Competition P2
The incenter of $\triangle A_1A_2A_3$ is $I$, and the center of the $A_i$-excircle is $J_i$ ($i=1,2,3$). Let $B_i$ be the intersection point of side $A_{i+1}A_{i+2}$ and the bisector of $\angle A_{i+1}IA_{i+2}$ ($A_{i+3}:=A_i$ $\forall i$). Prove that the three lines $B_iJ_i$ are concurrent.

1991 Kürschák Competition P2
A convex polyhedron has two triangle and three quadrilateral faces. Connect every vertex of one of the triangle faces with the intersection point of the diagonals in the quadrilateral face opposite to it. Show that the resulting three lines are concurrent.

1993 Kürschák Competition P2
Triangle $ABC$ is not isosceles. The incircle of $\triangle ABC$ touches the sides $BC$, $CA$, $AB$ in the points $K$, $L$, $M$. The parallel with $LM$ through $B$ meets $KL$ at $D$, the parallel with $LM$ through $C$ meets $KM$ at $E$.
Prove that $DE$ passes through the midpoint of $\overline{LM}$.

1994 Kürschák Competition P1
The ratio of the sides of a parallelogram is $\lambda>1$. Given $\lambda$, determine the maximum of the acute angle subtended by the diagonals of the parallelogram.

1995 Kürschák Competition P3

Points $A$, $B$, $C$, $D$ are such that no three of them are collinear. Let $E=AB\cap CD$ and $F=BC\cap DA$. Let $k_1$, $k_2$ and $k_3$ denote the circles with diameter $\overline{AC}$, $\overline{BD}$ and $\overline{EF}$, respectively. Prove that either $k_1,k_2,k_3$ pass through one point, or no two of them intersect.

1996 Kürschák Competition P1
Prove that in a trapezoid with perpendicular diagonals, the product of the legs is at least as much as the product of the bases.

1997 Kürschák Competition P2
The center of the circumcircle of $\triangle ABC$ is $O$. The incenter of the triangle is $I$, and the intouch triangle is $A_1B_1C_1$. Let $H_1$ be the orthocenter of $\triangle A_1B_1C_1$. Prove that $O$, $I$, and $H_1$ are collinear.

1999 Kürschák Competition P2
Given a triangle on the plane, construct inside the triangle the point $P$ for which the centroid of the triangle formed by the three projections of $P$ onto the sides of the triangle happens to be $P$.

2000 Kürschák Competition P2
Let $ABC$ be a non-equilateral triangle in the plane, and let $T$ be a point different from its vertices. Define $A_T$, $B_T$ and $C_T$ as the points where lines $AT$, $BT$, and $CT$ meet the circumcircle of $ABC$. Prove that there are exactly two points $P$ and $Q$ in the plane for which the triangles $A_PB_PC_P$ and $A_QB_QC_Q$ are equilateral. Prove furthermore that line $PQ$ contains the circumcenter of $\triangle ABC$.

2001 Kürschák Competition P3

In a square lattice let us take a lattice triangle that has the smallest area among all the lattice triangles similar to it. Prove that the circumcenter of this triangle is not a lattice point.

2002 Kürschák Competition P1
We have an acute-angled triangle which is not isosceles. We denote the orthocenter, the circumcenter and the incenter of it by $H$, $O$, $I$ respectively. Prove that if a vertex of the triangle lies on the circle $HOI$, then there must be another vertex on this circle as well.

2003 Kürschák Competition P1
Draw a circle $k$ with diameter $\overline{EF}$, and let its tangent in $E$ be $e$. Consider all possible pairs $A,B\in e$ for which $E\in \overline{AB}$ and $AE\cdot EB$ is a fixed constant. Define $(A_1,B_1)=(AF\cap k,BF\cap k)$. Prove that the segments $\overline{A_1B_1}$ all concur in one point.

2004 Kürschák Competition P1
Given is a triangle $ABC$, its circumcircle $\omega$, and a circle $k$ that touches $\omega$ from the outside, and also touches rays  $AB$ and $AC$ in $P$ and $Q$, respectively. Prove that the $A$-excenter of $\triangle ABC$ is the midpoint of $\overline{PQ}$.

2010 Kürschák Competition P2
Consider a triangle $ABC$, with the points $A_1$, $A_2$ on side  $BC$, $B_1,B_2\in\overline{AC}$, $C_1,C_2\in\overline{AB}$ such that $AC_1<AC_2$, $BA_1<BA_2$, $CB_1<CB_2$. Let the circles $AB_1C_1$ and $AB_2C_2$ meet at $A$ and $A^*$. Similarly, let the circles $BC_1A_1$ and $BC_2A_2$ intersect at $B^*\neq B$, let $CA_1B_1$ and $CA_2B_2$ intersect at $C^*\neq C$. Prove that the lines $AA^*$, $BB^*$, $CC^*$ are concurrent.

2012 Kürschák Competition P1
Let $J_A$ and $J_B$ be the $A$-excenter and $B$-excenter of $\triangle ABC$. Consider a chord $\overline{PQ}$ of circle $ABC$ which is parallel to $AB$ and intersects segments $\overline{AC}$ and $\overline{BC}$. If lines $AB$ and $CP$ intersect at $R$, prove that
$$\angle J_AQJ_B+\angle J_ARJ_B=180^\circ.$$

2013 Kürschák Competition P2
Consider the closed polygonal discs $P_1$, $P_2$, $P_3$ with the property that for any three points $A\in P_1$, $B\in P_2$, $C\in P_3$, we have $[\triangle ABC]\le 1$. (Here $[X]$ denotes the area of polygon $X$.)
(a) Prove that $\min\{[P_1],[P_2],[P_3]\}<4$.
(b) Give an example of polygons $P_1,P_2,P_3$ with the above property such that $[P_1]>4$ and $[P_2]>4$.

2014 Kürschák Competition P2
We are given an acute triangle $ABC$, and inside it a point $P$, which is not on any of the heights $AA_1$, $BB_1$, $CC_1$. The rays $AP$, $BP$, $CP$ intersect the circumcircle of $ABC$ at points $A_2$, $B_2$, $C_2$. Prove that the circles $AA_1A_2$, $BB_1B_2$ and $CC_1C_2$ are concurrent.

2014 Kürschák Competition P3

Let $K$ be a closed convex polygonal region, and let $X$ be a point in the plane of $K$. Show thatthere exists a finite sequence of reflections in the sides of $K$, such that $K$ contains the image of

$X$ after these reflections.

2015 Kürschák Competition P2
Consider a triangle $ABC$ and a point $D$ on its side $\overline{AB}$. Let $I$ be a point inside $\triangle ABC$ on the angle bisector of $ACB$. The second intersections of lines $AI$ and $CI$ with circle $ACD$ are $P$ and $Q$, respectively. Similarly, the second intersection of lines $BI$ and $CI$ with circle $BCD$ are $R$ and $S$, respectively. Show that if $P\neq Q$ and $R\neq S$, then lines $AB$, $PQ$ and $RS$ pass through a point or are parallel.

2017 Kürschák Competition P1
Let $ABC$ be a triangle. Choose points $A'$, $B'$ and $C'$ independently on side segments $BC$, $CA$ and $AB$ respectively with a uniform distribution. For a point $Z$ in the plane, let $p(Z)$ denote the probability that $Z$ is contained in the triangle enclosed by lines $AA'$, $BB'$ and $CC'$. For which interior point $Z$ in triangle $ABC$ is $p(Z)$ maximised?

2018 Kürschák Competition P1
Given a triangle $ABC$ with its incircle touching sides $BC,CA,AB$ at $A_1,B_1,C_1$, respectively. Let the median from $A$ intersects $B_1C_1$ at $M$. Show that $A_1M\perp BC$.

2019 Kürschák Competition P1
In an acute triangle $\bigtriangleup ABC$, $AB<AC<BC$, and $A_1,B_1,C_1$ are the projections of $A,B,C$ to the corresponding sides. Let the reflection of $B_1$ wrt $CC_1$ be $Q$, and the reflection of $C_1$ wrt $BB_1$ be $P$. Prove that the circumcirle of $A_1PQ$ passes through the midpoint of $BC$.

2021 Kürschák Competition P3

Let $A_1B_3A_2B_1A_3B_2$ be a cyclic hexagon such that $A_1B_1,A_2B_2,A_3B_3$ intersect at one point. Let $C_1=A_1B_1\cap A_2A_3,C_2=A_2B_2\cap A_1A_3,C_3=A_3B_3\cap A_1A_2$. Let $D_1$ be the point on the circumcircle of the hexagon such that $C_1B_1D_1$ touches $A_2A_3$. Define $D_2,D_3$ analogously. Show that $A_1D_1,A_2D_2,A_3D_3$ meet at one point.