geometry problems from Japanese Mathematical Olympiads Finals (JMO Final)

with aops links in the names

with aops links in the names

1991 - 2018

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 S

_{BCED }= 2S_{PBC}.
Let P

_{0}be a point in the plane of triangle A_{0}A_{1}A_{2}. Define P_{i}(i = 1,…,6) inductively as the point symmetric to P_{i}_{−1 }with respect to A_{k}, where*k*is the remainder when*i*is divided by 3.
(a) Prove that P

_{6}≡ P_{0}.
(b) Find the locus of points P

_{0}for which P_{i}P_{i}_{+1 }does not meet the interior of △A_{0}A_{1}A_{2}for
0 ≤ I ≤ 5 .

In a triangle ABC, M is the midpoint of BC. Given that ÐMAC = 15

^{o}, 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 = X

_{0}distinct from A,B,C,D. Prove that ÐAX_{0}B = ÐCX_{0}D.
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 C

_{0}. Let P be a point on the arc AB not containing M. Circle C_{1}is internally tangent to C_{0}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 π

_{1}and π_{2}are given in space. Let A and B be points on their intersection and C be a point on π_{2}but not on π_{1}. The bisector of angle BCA meets AB at P. Denote by S the circle on π_{1}with diameter AB. An arbitrary plane π_{3}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 = BM

^{2}.
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 S

_{1}passing through A, C intersects with the circle S_{2}passing through B, D at distinct points X, Y . Prove that the circumcenter of the triangle PXY is the midpoint of the centers of S_{1}, S_{2}.
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.

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 $\omega$.

sources:

artofproblemsolving.com

www.imomath.com

**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 $\omega$.

sources:

artofproblemsolving.com

www.imomath.com

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