Japan Junior Finals 2009-22 (JJMO) 21p

geometry problems from Japanese Junior Mathematical Olympiads Finals (JJMO Final)
with aops links in the names

 [proof problems only]
inside aops

it didn't take place in 2021

2009- 2022 (-2021)

2009 Japanese JMO p2

There are points $P$ and $Q$ on sides $AB$ and $AC$ of triangle $ABC$, respectively, satisfying $BP+CQ=PQ$. Let $R$ be the intersection point, other than $A$, between the bisector of $\angle BAC$ and the circumscribed circle of triangle $ABC$. If $\angle BAC=\alpha$, express $\angle PQR$ in terms of $\alpha$.

2009 Japanese JMO p5

The angle formed by any two diagonals $AD, BE$, and $CF$ of the convex hexagon $ABCDEF$ is $60^o$. Prove that $$AE + BC + CD + DE + EF + FA \ge  AD + BE + CF,$$A polygon is convex if all of its interior angles are less than $180^o$.

2010 Japanese JMO p1

Let $n$ be an integer with $  n\ge 3$. Find all $n$ such that there exist a $n$-gon with every interior angle $120^o$ or $240^o$ and with all sides of equal length.

2010 Japanese JMO p4

Given a triangle $ABC$. A circle $\omega$ passing through points $B$ and $C$, intersects line segments $AB$ and $AC$ (not including endpoints) at points $D$ and $E$, respectively, and $AD+AE=BC$ holds. Let $I$ be the incenter of the triangle $ABC$. Let $P$ and $Q$ be the points, other than $B,C$ , where the straight lines $BI$ and $CI$ intersect the circle $\omega$ . Show that $A, I, P$ and $Q$ lie on the same circle.

2011 Japanese JMO p1

There is a hexagon $ABCDEF$ inscribed in a circle. Show that if sides $AB$ and $DE$ are parallel, and sides $BC$ and $EF$ are parallel, then sides $CD$ and $FA$ are also parallel.

2011 Japanese JMO p3

Let $H$ be the orthocenter of acute triangle $ABC$, and let $D$ be the intersection point of two straight lines $AH$ and $BC$. Let $E$ be the intersection point of the circumscribed circle of triangle $ABD$ with straight line $CH$, that is outside triangle $ABC$. Let $F$ be the intersection point of the circumscribed circle of triangle $ACD$ and straight line $BH$, that is outside triangle $ABC$. Prove that the lengths of the two segments $AE$ and $AF$ are equal.

2012 Japanese JMO p1

There is a convex pentagon $ABCDE$, and the quadrilateral $ABCD$ is a square. Show that when $\angle AEC+\angle BED=180^o$, the pentagon $ABCDE$ is cyclic.

A pentagon whose interior angles are all less than $180^o$ is called a convex pentagon.

2012 Japanese JMO p4

Given an acute-angled triangle $ABC$ with $\angle BAC=30^o$.Inside the triangle $ABC$, take point $X$ so that $\angle XBC=\angle XCB=30^o$.On the straight lines $BX$ and $CX$, points $P$ and $Q$ are taken $AP=BP$ and $AQ=CQ$, respectively. Let $M$ be the midpoint of $BC$. In this case, show that $\angle PMQ=90^o$.

2013 Japanese JMO p4

Given an acute triangle $ABC$ such that $AB<AC$. Let $H$ and $I$ be the orthocenter and the incenter, respectively. Point $J$ on side $AB$ and point $K$ on side $BC$ satisfy $\angle HIK=90^o$ and $AC=AJ+CK$. In this case, express $\angle HJK$ in terms of $\angle BCA$.

2014 Japanese JMO p4

Triangle $ABC$ has $\angle BAC=60^o$. Let $P$ be the intersection point of the bisector of $\angle ABC$ with side $AC$, and let $Q$ be the intersection point of the bisector of $\angle ACB$ with side $AB$. Show that the symmetric point to $A$ with respect to the straight line $PQ$ lies on the straight line $BC$.

2015 Japanese JMO p1
Let $ABCD$ be a quadrilateral inscribed in a circle. Point $P(\neq A,B)$ satisfies $\angle PAC=\angle PBD=90^{\circ}$. Prove that the line through $P$ and perpendicular to $CD$ passes through the circumcenter of triangle $PAB$.

2015 Japanese JMO p4
Given  circle $ C $ and two points $ A,B $, which $ A $ is in circle. Let $ l $ the lie which passes through $ A $ and don't pass through $ B $.  $ l $ intersects $ C $ in two points $ P, Q $. Prove that, there exists a line $ m $ such that the circumcenter of  $ \triangle BPQ $ lie on $ m $ for every $ l $.

2016 Japanese JMO p1

Let $D, E,F$ be the feet of perpendiculars drawn on sides $BC$, $CA$, $AB$ from $A, B,C$ respectively . Let $G$ be the intersection point of straight lines $AD$ and $EF$. Let $P$ be the intersection point of the circumscribed circle of triangle $DFG$ with side $AB$, that is not $F$. Let $Q$ be the intersection point of the circumscribed circle of triangle $DEG$ with side $AC$ that is not $E$. In this case, show that straight line $PQ$ passes through the midpoint of line segment $DG$.

2016 Japanese JMO p4

Let $O$ be the circumcenter and $H$ be the orthocenter of an acute-angled triangle $ABC$. Let $P$ and $Q$ be the intersection points of a straight line passing through $O$ and parallel to the straight line $BC$, with sides $AB$ and $AC$, respectively. Let $M$ be the midpoint of the line segment $AH$. In this case, show $\angle BMP = \angle  CMQ$.

2017 Japanese JMO p5
Let $ABC$ be an acute-angled triangle with orthocenter $H$. Let $D,E$ and $F$ be the feet of the altitudes from $A,B$ and $C$, respectively. The circumcircle of $ACD$ and $BE$ meet at $P$, the circumcircle of $ABD$ and $CF$ meet at $Q$, the circumcircle of $ABH$ and $DF$ meet at $S$, and the circumcircle of $ACH$ and $DE$ meet at $T$. Prove that $P,Q,S,T$ are concyclic.

2018 Japanese JMO p3
Let $ABC$ be a triangle with $AB\neq AC$. Points $D,E$ on segments $AB,AC$, respectively, are chosen so that $B,C,D,E$ are concyclic. Lines $DE$ and $BC$ intersect at $F$, circle $ABC$ and line $AF$ intersect at $G(\neq A)$. Let $H$ be an arbitrary point on segment $DE$, and circle $ABC$ and line $AH$ intersect at $I(\neq A)$. Prove that $F,G,H,I$ are concyclic.

2018 Japanese JMO p5

Let $ABC$ be a triangle with its incircle $\omega$. Let the midpoints of segments $BC,CA,AB$ be $A',B',C'$, respectively. Segment $B_{a}C_{a}$ is a radius of $\omega$ which is parallel (with direction) to $BC$,  and lines $B'C_{a}$ and $C'B_{a}$ intersect at $X$. Take $Y$ and $Z$ similary. Prove that lines $AX,BY,CZ$ have a common point.

2019 Japanese JMO p1
Let $ABC$ be a triangle with $AB=AC\neq BC$. Let $D$ be an interior point of the triangle $ABC$ with $\angle ABD=\angle ACD=30^{\circ}$. Prove that the angle bisectors of $\angle ACB$ and $\angle ADB$ meets on the segment $AB$.

2019 Japanese JMO p5
Let $ABC$ be a triangle with $AB\neq AC$. Let $M$ be the midpoint of $BC$, and $N$ be the midpoint of the arc $BC$ of the circumcircle of the triangle $ABC$ containing $A$. Let $H$ be the foot of the perpendicular from $N$ to $AC$. The circumcircle of the triangle $AMC$ and the line $CN$ meet at $K(\neq C)$. Prove that $\angle AKH = \angle CAM$.

2020 Japanese JMO p2
Let $ABC$ be a triangle with $AB=AC$, and $H$ be the foot of the perpendicular to the line $AC$ from $B$. A point $D$ on the segment $BH$ satisfies $AB=2BD$ and $BC=2CD$. Find $\angle BCD$.

2022 Japanese JMO p4

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 $\angle BPE=90^{\circ}$.

source: https://www.imojp.org/