geometry problems from Real IMO Shortlist (a.k.a. International Monster's Olympiad)
with aops links in the names
2018-19
2018 Real IMO Shortlist G1
The bus outlined in the figure is transferring the members of the Problem Selection Committee and their $1024$ gigapists from Bucharest to Cluj. In which direction is Cluj? To the left, or to the right? Justify your answer.
“IMO cheese” is a brand name of processed cheese that comes in a round box containing eight individually wrapped sectors that just fit in the box. Johnny ate two sectors for breakfast, and arranged the remaining ones in the box as shown in the figure. His Mom Pegasus did not like that. She said that Johnny was squeezing the cheese pieces. Johnny insisted that there was no squeezing, the pieces just fit. Who was right?
2019 Real IMO Shortlist G2
Let $F$ be the midpoint of circle arc $AB$, and let $M$ be a point on the arc such that $AM <MB$. The perpendicular dropped from point $F$ to $AM$ intersects $AM$ at point $T$. Show that $T$ bisects the broken line $AMB$, that is $AT =TM+MB$.
2019 Real IMO Shortlist G3 (IMO Shortlist 2019/G3, IMO 2019/2)
In triangle $ABC$, point $A_1$ lies on side $BC$ and point $B_1$ lies on side $AC$. Let $P$ and $Q$ be points on segments $AA_1$ and $BB_1$, respectively, such that $PQ$ is parallel to $AB$. Let $P_1$ be a point on line $PB_1$, such that $B_1$ lies strictly between $P$ and $P_1$, and $\angle PP_1C=\angle BAC$. Similarly, let $Q_1$ be the point on line $QA_1$, such that $A_1$ lies strictly between $Q$ and $Q_1$, and $\angle CQ_1Q=\angle CBA$.
Prove that points $P,Q,P_1$, and $Q_1$ are concyclic.
2019 Real IMO Shortlist G4
2019 Real IMO Shortlist G5
The circle $k$ touches the circle $\ell$ internally at point $P$. A line $p$ passing through $P$ intersects the circles again at the points $K$ and $L$, respectively. $u$ is the tangent drawn at a point $U$ of circle $k$. One intersection of $u$ with the circle $\ell$ is $V$ , and the intersection of lines $KU$ and $LV$ is $T$. Determine the locus of the point $T$ as $U$ traverses the circle $k$. (Consider both intersections of $u$ and $\ell$, if $U =K$, the line $KU$ is the tangent to $k$ at $K$. Analogously, if $V = L$ then $LV$ is the tangent drawn to $\ell$ at $L$.)
2019 Real IMO Shortlist G6
A circle centred at $I$ is tangent to the sides $BC, CA$, and $AB$ of an acute-angled triangle $ABC$ at $A_1, B_1$, and $C_1$, respectively. Let $K$ and $L$ be the incenters of the quadrilaterals $AB_1IC_1$ and $BA_1IC_1$, respectively. Let $CH$ be an altitude of triangle $ABC$. Let the internal angle bisectors of angles $AHC$ and $BHC$ meet the lines $A_1C_1$ and $B_1C_1$ at $P$ and $Q$, respectively. Prove that $Q$ is the orthocenter of the triangle $KLP$.
2019 Real IMO Shortlist G7 (IMO Shortlist 2019/G7, IMO2019/6)
Let $I$ be the incentre of acute triangle $ABC$ with $AB\neq AC$. The incircle $\omega$ of $ABC$ is tangent to sides $BC, CA$, and $AB$ at $D, E,$ and $F$, respectively. The line through $D$ perpendicular to $EF$ meets $\omega$ at $R$. Line $AR$ meets $\omega$ again at $P$. The circumcircles of triangle $PCE$ and $PBF$ meet again at $Q$.
Prove that lines $DI$ and $PQ$ meet on the line through $A$ perpendicular to $AI$.
2019 Real IMO Shortlist G8
Let $k$ and $K$ be concentric circles on the plane, and let $k$ be contained inside $K$. Assume that $k$ is covered by a finite system of convex angular domains with vertices on $K$. Prove that the sum of the angles of the domains is not less than the angle under which $k$ can be seen from a point of $K$.
2019 Real IMO Shortlist G9
Do there exist incongruent convex quadrangular pyramids $SA_1A_2A_3A_4$ and $SB_1B_2B_3B_4$ such that, for every $i = 1, 2, 3, 4$, the triangles $SA_iA_{i+1}$ and $SB_iB_{i+1}$ are congruent (with correspondence of vertices)? Numeration of vertices is cyclic.
2019 Real IMO Shortlist G10
Determine all positive integers $n \ge 3$ satisfying the following property:
Let $X$ be a set of $n$ points in general position in the plane. Then there exist several convex polygons $P_1, . . . , P_k$ such that no two of them have a common point (neither on the boundary, not in the interior), and the set of all vertices of the $P_i$ coincides with $X$.
with aops links in the names
sources: Real Shorltist, 2018, 2019
collected inside aops here
[in the sum, SL problems do not count]
collected inside aops here
[in the sum, SL problems do not count]
2018-19
The bus outlined in the figure is transferring the members of the Problem Selection Committee and their $1024$ gigapists from Bucharest to Cluj. In which direction is Cluj? To the left, or to the right? Justify your answer.
2018 Real IMO Shortlist G2
Let $\Omega$ be the circumcircle of a triangle $ABC$. Let $A_0, B_0$, and $C_0$ be the midpoints of the arcs $BAC, CBA$, and $ACB$, respectively. Let $A_1, B_1$, and $C_1$ be the Feuerbach points in the triangles $AB_0C_0, A_0BC_0$, and $A_0B_0C$, respectively. Prove that the triangles $A_0B_0C_0$ and $A_1B_1C_1$ are similar.
2019 Real IMO Shortlist G1Let $\Omega$ be the circumcircle of a triangle $ABC$. Let $A_0, B_0$, and $C_0$ be the midpoints of the arcs $BAC, CBA$, and $ACB$, respectively. Let $A_1, B_1$, and $C_1$ be the Feuerbach points in the triangles $AB_0C_0, A_0BC_0$, and $A_0B_0C$, respectively. Prove that the triangles $A_0B_0C_0$ and $A_1B_1C_1$ are similar.
Kolmogorov Cup, 2014,
based on a problem by A. Yakubov
2018 Real IMO Shortlist G3
Given a convex quadrilateral $ABCD$ such that $AB= CD, \angle ABC=90^o$ and $\angle BCD=100^o$. The perpendicular bisectors of the segments $AD$ and $BC$ meet at $S$. Compute the angle $\angle ASD$.
Given a convex quadrilateral $ABCD$ such that $AB= CD, \angle ABC=90^o$ and $\angle BCD=100^o$. The perpendicular bisectors of the segments $AD$ and $BC$ meet at $S$. Compute the angle $\angle ASD$.
KöMaL Gy.2938. (October 1994),
based on the folklore fake proof for $90^o= 100^o$
2018 Real IMO Shortlist G4
The circle $\omega_1$ is internally tangent to the circle $\Omega$ which is externally tangent to $\omega_2$. The common external tangents of $\omega_1$ and $\omega_2$ are $u$ and $v$. The line $u$ is tangent to $\omega_1$ and $\omega_2$ at $P$ and $Q$, respectively, and meets $\Omega$ at $A$ and $B$ in such a way that $B$ lies between $P$ and $Q$. Analogously, the line $v$ is tangent to $\omega_1$ and $\omega_2$ at $R$ and $S$, respectively, and meets $\Omega$ at $C$ and $D$ in such a way that $D$ lies between $R$ and $S$ and $\omega_1$ is tangent to that arc $BD$ of $\Omega$ which does not contain $A$ and $C$. Show that $$\frac{ AB \cdot AD}{AP \cdot AQ} =\frac{ CB \cdot CD}{CR \cdot CS }$$
The circle $\omega_1$ is internally tangent to the circle $\Omega$ which is externally tangent to $\omega_2$. The common external tangents of $\omega_1$ and $\omega_2$ are $u$ and $v$. The line $u$ is tangent to $\omega_1$ and $\omega_2$ at $P$ and $Q$, respectively, and meets $\Omega$ at $A$ and $B$ in such a way that $B$ lies between $P$ and $Q$. Analogously, the line $v$ is tangent to $\omega_1$ and $\omega_2$ at $R$ and $S$, respectively, and meets $\Omega$ at $C$ and $D$ in such a way that $D$ lies between $R$ and $S$ and $\omega_1$ is tangent to that arc $BD$ of $\Omega$ which does not contain $A$ and $C$. Show that $$\frac{ AB \cdot AD}{AP \cdot AQ} =\frac{ CB \cdot CD}{CR \cdot CS }$$
G4: KöMaL A.579. (January 2013), G. Kós
2018 Real IMO Shortlist G5
A sphere $S$ lies within tetrahedron $ABCD$, touching faces $ABD, ACD$, and $BCD$, but having no point in common with plane $ABC$. Let $E$ be the point in the interior of the tetrahedron for which $S$ touches planes $ABE$, $ACE$, and $BCE$ as well. Suppose the line $DE$ meets face $ABC$ at $F$, and let $L$ be the point of $S$ nearest to plane $ABC$. Show that segment $FL$ passes through the centre of the inscribed sphere of tetrahedron $ABCE$.
A sphere $S$ lies within tetrahedron $ABCD$, touching faces $ABD, ACD$, and $BCD$, but having no point in common with plane $ABC$. Let $E$ be the point in the interior of the tetrahedron for which $S$ touches planes $ABE$, $ACE$, and $BCE$ as well. Suppose the line $DE$ meets face $ABC$ at $F$, and let $L$ be the point of $S$ nearest to plane $ABC$. Show that segment $FL$ passes through the centre of the inscribed sphere of tetrahedron $ABCE$.
KöMaL A.723. (April 2018), G. Kós
2018 Real IMO Shortlist G6
Let $ABC$ be a scalene acute-angled triangle. The tangents to its circumcircle at points $A$ and $B$ meet the opposite sidelines at $A_1$ and $B_1$, respectively. Let $L$ be Lemoine point of the triangle (where the symmedians meet), and let $P$ be a point inside the triangle such that its projections onto the sides form an equilateral triangle. Prove that $LP \perp A_1B_1$
Let $ABC$ be a scalene acute-angled triangle. The tangents to its circumcircle at points $A$ and $B$ meet the opposite sidelines at $A_1$ and $B_1$, respectively. Let $L$ be Lemoine point of the triangle (where the symmedians meet), and let $P$ be a point inside the triangle such that its projections onto the sides form an equilateral triangle. Prove that $LP \perp A_1B_1$
Kolmogorov Cup, 2002,
S. Berlov, D. Shiryaev, A. Smirnov
2018 Real IMO Shortlist G7
In the Cartesian plane call those regions as strips which are bounded by two par allel lines. Define the width of a strip as the side length of the inscribed squares, with sides being parallel to the coordinate axes. Prove that if finitely many strips cover the unit square $0 \le x, y \le 1$ then the sum of widths of those strips is at least $1$.
2018 Real IMO Shortlist G8
In the Cartesian plane call those regions as strips which are bounded by two par allel lines. Define the width of a strip as the side length of the inscribed squares, with sides being parallel to the coordinate axes. Prove that if finitely many strips cover the unit square $0 \le x, y \le 1$ then the sum of widths of those strips is at least $1$.
KöMaL A.526. (January 2011),
based on Keith Ball: The plank problem for symmetric bodies
2018 Real IMO Shortlist G8
Let $O$ be an arbitrary point inside a tetrahedron $ABCD$.
Prove that $[AOC] \cdot [BOD] \le [AOB] \cdot [COD] + [AOD]\cdot [BOC]$.
(Here $[XYZ]$ stands for the area of triangle $XYZ$.)
Prove that $[AOC] \cdot [BOD] \le [AOB] \cdot [COD] + [AOD]\cdot [BOC]$.
(Here $[XYZ]$ stands for the area of triangle $XYZ$.)
Kolmogorov Cup, 2011, I. Bogdanov
“IMO cheese” is a brand name of processed cheese that comes in a round box containing eight individually wrapped sectors that just fit in the box. Johnny ate two sectors for breakfast, and arranged the remaining ones in the box as shown in the figure. His Mom Pegasus did not like that. She said that Johnny was squeezing the cheese pieces. Johnny insisted that there was no squeezing, the pieces just fit. Who was right?
KöMaL K. 124. (March 2007),G. Kós
Let $F$ be the midpoint of circle arc $AB$, and let $M$ be a point on the arc such that $AM <MB$. The perpendicular dropped from point $F$ to $AM$ intersects $AM$ at point $T$. Show that $T$ bisects the broken line $AMB$, that is $AT =TM+MB$.
KöMaL Gy. 2404. (March 1987),
Archimedes of Syracuse
2019 Real IMO Shortlist G3 (IMO Shortlist 2019/G3, IMO 2019/2)
In triangle $ABC$, point $A_1$ lies on side $BC$ and point $B_1$ lies on side $AC$. Let $P$ and $Q$ be points on segments $AA_1$ and $BB_1$, respectively, such that $PQ$ is parallel to $AB$. Let $P_1$ be a point on line $PB_1$, such that $B_1$ lies strictly between $P$ and $P_1$, and $\angle PP_1C=\angle BAC$. Similarly, let $Q_1$ be the point on line $QA_1$, such that $A_1$ lies strictly between $Q$ and $Q_1$, and $\angle CQ_1Q=\angle CBA$.
Prove that points $P,Q,P_1$, and $Q_1$ are concyclic.
Anton Trygub, Ukraine
2019 Real IMO Shortlist G4
Show that every closed curve $c$ of length less than $ 2\pi $ on the surface of the unit sphere lies entirely on the surface of some hemisphere of the unit sphere.
Miklós Schweitzer 1953/4
2019 Real IMO Shortlist G5
The circle $k$ touches the circle $\ell$ internally at point $P$. A line $p$ passing through $P$ intersects the circles again at the points $K$ and $L$, respectively. $u$ is the tangent drawn at a point $U$ of circle $k$. One intersection of $u$ with the circle $\ell$ is $V$ , and the intersection of lines $KU$ and $LV$ is $T$. Determine the locus of the point $T$ as $U$ traverses the circle $k$. (Consider both intersections of $u$ and $\ell$, if $U =K$, the line $KU$ is the tangent to $k$ at $K$. Analogously, if $V = L$ then $LV$ is the tangent drawn to $\ell$ at $L$.)
KöMaL B. 4511. (January 2013),András Hraskó
2019 Real IMO Shortlist G6
A circle centred at $I$ is tangent to the sides $BC, CA$, and $AB$ of an acute-angled triangle $ABC$ at $A_1, B_1$, and $C_1$, respectively. Let $K$ and $L$ be the incenters of the quadrilaterals $AB_1IC_1$ and $BA_1IC_1$, respectively. Let $CH$ be an altitude of triangle $ABC$. Let the internal angle bisectors of angles $AHC$ and $BHC$ meet the lines $A_1C_1$ and $B_1C_1$ at $P$ and $Q$, respectively. Prove that $Q$ is the orthocenter of the triangle $KLP$.
Kolmogorov Cup 2018, Major League,
Day 3, Problem 1; A. Zaslavsky
2019 Real IMO Shortlist G7 (IMO Shortlist 2019/G7, IMO2019/6)
Let $I$ be the incentre of acute triangle $ABC$ with $AB\neq AC$. The incircle $\omega$ of $ABC$ is tangent to sides $BC, CA$, and $AB$ at $D, E,$ and $F$, respectively. The line through $D$ perpendicular to $EF$ meets $\omega$ at $R$. Line $AR$ meets $\omega$ again at $P$. The circumcircles of triangle $PCE$ and $PBF$ meet again at $Q$.
Prove that lines $DI$ and $PQ$ meet on the line through $A$ perpendicular to $AI$.
Anant Mudgal, India
2019 Real IMO Shortlist G8
Let $k$ and $K$ be concentric circles on the plane, and let $k$ be contained inside $K$. Assume that $k$ is covered by a finite system of convex angular domains with vertices on $K$. Prove that the sum of the angles of the domains is not less than the angle under which $k$ can be seen from a point of $K$.
Miklós Schweitzer 1985/3, Zsolt Páles
2019 Real IMO Shortlist G9
Do there exist incongruent convex quadrangular pyramids $SA_1A_2A_3A_4$ and $SB_1B_2B_3B_4$ such that, for every $i = 1, 2, 3, 4$, the triangles $SA_iA_{i+1}$ and $SB_iB_{i+1}$ are congruent (with correspondence of vertices)? Numeration of vertices is cyclic.
Kolmogorov Cup 2018, Major League,
Day 4, Problem 9, L. Emelyanov, I. Bogdanov
2019 Real IMO Shortlist G10
Determine all positive integers $n \ge 3$ satisfying the following property:
Let $X$ be a set of $n$ points in general position in the plane. Then there exist several convex polygons $P_1, . . . , P_k$ such that no two of them have a common point (neither on the boundary, not in the interior), and the set of all vertices of the $P_i$ coincides with $X$.
Kolmogorov Cup 2018, Major League,
Day 2, Problem 8, I. Bogdanov
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