geometry problems from Japanese Mathematical Olympiads Finals (JMO Final)
with aops links in the names
with aops links in the names
1991 - 2022
Let P,Q and R be points on the sides BC,CA and AB of a triangle ABC respectively,
such that $\overrightarrow{BP}:\overrightarrow{PC}=\overrightarrow{CQ}:\overrightarrow{QA}=\overrightarrow{AR}:\overrightarrow{RB}=t:\left(
1-t \right)$ for some real number t.
Prove that there is a triangle ∆ whose side lengths are AP,BQ,CR, and find the ratio of the area of
triangle ABC to that of ∆ in
terms of t.
Let ABC be a given
triangle with the area 1. Let D and
E be points on sides AB and AC respectively. Lines BE
and CD intersect at P. Find the maximum possible value of
SPDE under the condition SBCED = 2SPBC.
Let P0 be a
point in the plane of triangle A0A1A2. Define Pi
(i = 1,…,6) inductively as
the point symmetric to Pi−1
with respect to Ak,
where k is the remainder
when i is divided by 3.
(a) Prove that P6
≡ P0.
(b) Find the locus of points P0
for which PiPi+1
does not meet the interior of △A0A1A2 for
0 ≤ I ≤ 5 .
In a triangle ABC, M is the midpoint of BC. Given that ÐMAC = 15o,
find the maximum value of ÐABC.
Let ABCDE be a convex
pentagon. Diagonal BE meets AC,AD at S,R, BD meets CA,CE at T,P, and CE meets AD at Q,
respectively. Suppose the areas of triangles ASR, BTS, CPT, DQP, ERQ are all
equal to 1.
(a) Determine the area of pentagon PQRST.
(b) Determine the area of pentagon ABCDE.
Let q be the maximum of the six angles between six edges of a regular
tetrahedron in space and a fixed plane. When the tetrahedron is rotated in
space, find the maximum of q .
Let A,B,C,D be points in
space which are not on a plane. Suppose that f (X) = AX +BX +CX +DX attains its minimum at a point X = X0 distinct from A,B,C,D. Prove that ÐAX0B = ÐCX0D.
All side lengths of a convex hexagon ABCDEF are 1. Let M=max{|AD|, |DE|, |CF|} and m = min{|AD|, |DE|, |CF|}. Find possible ranges of M and m.
Given five points A,B,C,P,Q in a plane, no three of which are
collinear, prove the inequality
AB+BC+CA+PQ≤ AP+AQ+BP+BQ+CP+CQ.
Suppose that triangles ABC and
PQR have the following
properties:
(i) A and P are the midpoints of QR and BC respectively,
(ii) QR and BC are the bisectors of ÐBAC and ÐQPR respectively.
Prove that AB+AC = PQ+PR.
Distinct points A, M, B with AM = MB are given on a circle C0. Let P be a point on the arc AB not containing M. Circle C1 is internally tangent to C0 at P and
tangent to AB at Q. Prove that the product MP ·MQ is independent of the position of P.
A point P lies in a
triangle ABC. The lines BP and CP meet AC and AB at Q and R respectively.
Given that AR = RB = CP and CQ = PQ, find ÐBRC.
Two orthogonal planes π1 and π2 are given in space. Let A and B be
points on their intersection and C be
a point on π2 but not on π1. The bisector of angle BCA meets AB at P. Denote by S the circle on π1 with diameter AB.
An arbitrary plane π3 containing CP meets S at
D and E. Prove that CP bisects
ÐDCE.
The tangents to a circle G from a point X meet the circle at points A and B. A line
through X intersects the circle
at C and D with D between X and C so that the lines AC and BD are perpendicular and meet at F. Let CD meet AB at G and let the perpendicular bisector of GX meet the segment BD at
H. Prove that the points X, F, G, and H lie on a circle.
Five distinct points A,M,B,C,D are on a circle Ο in this order with MA = MB. The lines AC and MD meet
at P and the lines BD and MC meet at Q. If
the line PQ meets the circle Ο at X and Y, prove that MX = MY.
Let G be the circumcircle of triangle ABC. Denote the circle tangent to AB,AC and
internally to G by GA. Define GB and GC analogously. Let GA,GB,GC touch G at P,Q,R, respectively. Prove that the lines AP,BQ and CR are concurrent.
Let O be the
circumcenter of the acute-angled triangle ABC. A circle passing through A and O intersects
the lines AB and AC at P and Q respectively.
If the lengths of the segments PQ and
BC are equal, find the length
of the angle between the lines PQ and
BC.
Let G be the circumcircle of △ABC. A circle with center O touches
the segment BC at P and the arc BC of G not containing A
at Q. If ÐBAO= ÐCAO, prove that ÐPAO = ÐQAO.
Given an acute-angled triangle ABC such that AB ≠AC. Draw the
perpendicular AH from A to BC. Suppose that if we take points P, Q in such a
way that three points A, B, P and three points A, C, Q are collinear in this
order respectively, then we have our points B, C, P, Q are concyclic and HP = HQ.
Prove that H is the circumcenter of △APQ.
Given an acute triangle ABC with the midpoint M of BC. Draw the
perpendicular HP from the orthocenter H of ABC to AM. Show that AM·PM = BM2.
Given 4 points on a plane. Suppose radii of 4 incircles of the
triangles, which can be formed by any 3 points taken from the 4 points, are
equal. Prove that all of the triangles are congruent.
Given a triangle ABC, the tangent of the circumcircle at A intersects
with the line BC at P. Let Q, R be the points of symmetry for P across the
lines AB, AC respectively. Prove that the line BC intersects orthogonally with
the line QR.
Given two triangles PAB and PCD such that PA = PB, PC = PD, P, A, C and B,
P, D are collinear in this order respectively. The circle S1 passing
through A, C intersects with the circle S2 passing through B, D at
distinct points X, Y . Prove that the circumcenter of the triangle PXY is the
midpoint of the centers of S1, S2.
Given an acute-angled triangle ABC, let H be the orthocenter. A circle
passing through the points B, C and a circle with a diameter AH intersect at
two distinct points X, Y . Let D be the foot of the perpendicular drawn from A to
line BC, and let K be the foot of the perpendicular drawn from D to line XY . Show
that ÐBKD = ÐCKD.
Let O be the circumcenter of triangle ABC, and let l be the line passing through the midpoint of segment BC which is
also perpendicular to the bisector of angle ÐBAC. Suppose that the midpoint of segment AO lies on l. Find ÐBAC.
Let Γ be the circumcircle of triangle ABC, and let l be the tangent line of Γ passing A. Let D,E be the points each on side AB,AC such
that BD : DA = AE : EC. Line DE meets Γ at
points F,G. The line parallel to AC passing D meets l at H, the line parallel to AB passing E meets l at I. Prove that there exists a circle
passing four points F,G,H, I and tangent to line BC.
Scalene triangle ABC has circumcircle Γ and incenter I. The incircle of triangle ABC touches side AB,AC at D,E respectively.
Circumcircle of triangle BEI intersects Γ again at
P distinct from B, circumcircle of triangle CDI intersects Γ again at Q distinct from C. Prove that the 4 points D,E, P,Q are
concyclic.
Let ABCD be a concyclic quadrilateral such that AB : AD = CD : CB. The line
AD intersects the line BC at X, and the line AB intersects the line CD at Y .
Let E, F, G and H are the midpoints of the edges AB, BC, CD and DA respectively.
The bisector of angle AXB intersects the segment EG at S, and that of angle AY
D intersects the segment FH at T. Prove that the lines ST and BD are pararell.
Let ABC be an acute-angled triangle with circumcenter O. Let D,E and F be
the feet of the altitudes from A,B and C, respectively, and let M be the midpoint
of BC. AD and EF meet at X, AO and BC meet at Y , and let Z be the midpoint of XY
. Prove that A,Z,M are collinear.
Japanese MO Finals 2018 P2
Given a scalene triangle $\triangle ABC$, $D,E$ lie on segments $AB,AC$ respectively such that $CA=CD, BA=BE$. Let $\omega$ be the circumcircle of $\triangle ADE$. $P$ is the reflection of $A$ across $BC$, and $PD,PE$ meets $\omega$ again at $X,Y$ respectively. Prove that $BX$ and $CY$ intersect on $\omega$.
Japanese MO Finals 2019 P4
sources:
artofproblemsolving.com
www.imomath.com
Given a scalene triangle $\triangle ABC$, $D,E$ lie on segments $AB,AC$ respectively such that $CA=CD, BA=BE$. Let $\omega$ be the circumcircle of $\triangle ADE$. $P$ is the reflection of $A$ across $BC$, and $PD,PE$ meets $\omega$ again at $X,Y$ respectively. Prove that $BX$ and $CY$ intersect on $\omega$.
Japanese MO Finals 2019 P4
Let $ABC$ be a triangle with its inceter $I$, incircle $w$, and let $M$ be a midpoint of the side $BC$. A line through the point $A$ perpendicular to the line $BC$ and a line through the point $M$ perpendicular to the line $AI$ meet at $K$. Show that a circle with line segment $AK$ as the diameter touches $w$.
Triangle $ABC$ satisfies $BC<AB$ and $BC<AC$. Points $D,E$ lie on segments $AB,AC$ respectively, satisfying $BD=CE=BC$. Lines $BE$ and $CD$ meet at point $P$, circumcircles of triangle $ABE$ and $ACD$ meet at point $Q$ other than $A$. Prove that lines $PQ$ and $BC$ are perpendicular.
Points $D,E$ on the side $AB,AC$ of an acute-angled triangle $ABC$ respectively satisfy $BD=CE$. Furthermore, points $P$ on the segmet $DE$ and $Q$ on the arc $BC$ of the circle $ABC$ not containing $A$ satisfy $BP:PC=EQ:QD$. Points $A,B,C,D,E,P,Q$ are pairwise distinct.
Prove that $\angle BPC=\angle BAC+\angle EQD$ holds.
In an acute triangle $ABC$, $AB<AC$. The perpendicular bisector of the segment $BC$ intersects the lines $AB,AC$ at the points $D,E$ respectively. Denote the mid-point of $DE$ as $M$. Suppose the circumcircle of $\triangle ABC$ intersects the line $AM$ at points $P$ and $A$, and $M,A,P$ are arranged in order on the line. Prove that $\angle BPE=90^{\circ}$.
artofproblemsolving.com
www.imomath.com
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