China 2nd Round T2 1983 - 2021 43p

geometry problems from 2nd Round of China Mathematical Competitions (Test 2)
with aops links in the names

also known as (China) National High School Mathematics League

Exam 2 / Test 2 started in 1983

1983- 2021

1983 China Second Round Test 2 p3

In quadrilateral $ABCD$, $S_{\triangle ABD}:S_{\triangle BCD}:S_{\triangle ABC}=3:4:1$. $M\in AC,N\in CD$, satisfying that $\frac{AM}{AC}=\frac{CN}{CD}$. If $B,M,N$ are collinear, prove that $M,N$ are mid points of $AC,CD$.

1983 China Second Round Test 2 p4

In a tetrahedron, lengths of six edges are $2,3,3,4,5,5$. Find its largest volume.

1984 China Second Round Test 2 p2

$a,b$ are two skew lines, the angle they form is $\theta$. Length of their common perpendicular $AA'$ is $d$($A'\in a,A\in b)$. $E\in a,F\in b,|A'E|=m,|AF|=n$. Calculate $|EF|$.

1984 China Second Round Test 2 p3

In $\triangle ABC$, $P$ is a point on $BC$. $F\in AB,E\in AC,PF//CA,PE//BA$. If $S_{\triangle ABC}=1$, prove that at least one of $S_{\triangle BPF},S_{\triangle PCE},S_{PEAF}$ is not less than $\frac{4}{9}$.

1985 China Second Round Test 2 p2

In cube $ABCD-A_1B_1C_1D_1$, $E$ is midpoint of $BC$, $F\in AA_1$, and $A_1F:FA=1:2$. Calculate the dihedral angle between plane $B_1EF$ and plane $A_1B_1C_1D_1$.

1986 China Second Round Test 2 p2

In acute triangle $ABC$, $D\in BC,E\in CA,F\in AB$. Prove that the necessary and sufficient condition of $AD,BE,CF$ are heights of $\triangle ABC$ is that $S=\frac{R}{2}(EF+FD+DE)$. 

Note: $S$ is the area of $\triangle ABC$, $R$ is the circumradius of $\triangle ABC$.

1987 China Second Round Test 2 p1

$\triangle ABC$ and $\triangle ADE$ $(\angle ABC=\angle ADE=\frac{\pi}{2})$ are two isosceles right triangle that are not congruent. Fix $\triangle ABC$, but rotate $\triangle ADE$ on the plane. Prove that there exists point $M\in BC$, satisfying that $\triangle BMD$ is an isosceles right triangle.

1988 China Second Round Test 2 p2

In $\triangle ABC$, $P,Q,R$ divides the perimeter of $\triangle ABC$ into three equal parts. $P,Q\in AB$. Prove that $\frac{S_{\triangle PQR}}{S_{\triangle ABC}}>\frac{2}{9}$.

1989 China Second Round Test 2 p1

In $\triangle ABC$, $AB>AC$, bisector of outer angle $\angle A$ intersects circumcircle of $\triangle ABC$ at $E$. Projection of $E$ on $AB$ is $F$. Prove that $2AF=AB-AC$.

1990 China Second Round Test 2 p1

Quadrilateral $ABCD$ is inscribed on circle $O$. $AC\cap BD=P$. Circumcenters of $\triangle ABP,\triangle BCP,\triangle CDP,\triangle DAP$ are $O_1,O_2,O_3,O_4$. Prove that $OP,O_1O_3,O_2O_4$ share one point.

1991 China Second Round Test 2 p2

Area of convex quadrilateral $ABCD$ is $1$. Prove that we can find four points on its side (vertex included) or inside, satisfying: area of triangles comprised of any three points of the four points is larger than $\frac{1}{4}$.

1992 China Second Round Test 2 p1

$A_1A_2A_3A_4$ is cyclic quadrilateral of $\odot O$. $H_1,H_2,H_3,H_4$ are orthocentres of $\triangle A_2A_3A_4,\triangle A_3A_4A_1,\triangle A_4A_1A_2,\triangle A_1A_2A_3$. Prove that $H_1,H_2,H_3,H_4$ are concyclic, and determine its center.

1993 China Second Round Test 2 p3

Horizontal line $m$ passes the center of circle $\odot O$. Line $l\perp m$, $l$ and $m$ intersect at $M$, and $M$ is on the right side of $O$. Three points $A,B,C$ ($B$ is in the middle) lie on line $l$, which are outside the circle, above line $m$. $AP,BQ,CR$ are tangent to $\odot O$ at $P,Q,R$. Prove:

(a) If $l$ is tangent to $\odot O$, then $AB\cdot CR+BC\cdot AP=AC\cdot BQ$.

(b) If $l$ and $\odot O$ intersect, then $AB\cdot CR+BC\cdot AP<AC\cdot BQ$.

(c) If $l$ and $\odot O$ are apart, then $AB\cdot CR+BC\cdot AP>AC\cdot BQ$.

1994 China Second Round Test 2 p3

Circumcircle of $\triangle ABC$ is $\odot O$, incentre of $\triangle ABC$ is $I$. $\angle B=60^{\circ}.\angle A<\angle C$. Bisector of outer angle $\angle A$ intersects $\odot O$ at $E$. Prove:

(a) $IO=AE$.

(b) The radius of $\odot O$ is $R$, then $2R<IO+IA+IC<(1+\sqrt3)R$.

1995 China Second Round Test 2 p3

Inscribed Circle of rhombus $ABCD$ touches $AB,BC,CD,DA$ at $E,F,G,H$. $l_1,l_2$ are two lines that are tangent to the circle. $l_1\cap AB=M,l_1\cap BC=N,l_2\cap CD=P,l_2\cap DA=Q$. Prove that $MQ/\! /NP$.

1996 China Second Round Test 2 p3

$\odot O_1$ and $\odot O_2$ are escribed circles of $\triangle ABC$ ($\odot O_1$ is in $\angle ACB$, $\odot O_2$ is in $\angle ABC$). $\odot O_1$ touches $CB,CA$ at $E,G$; $\odot O_1$ touches $BC,BA$ at $F,H$. $EG\cap FG=P$, prove that $AP\perp BC$.

1997 China Second Round Test 2 p1

Two circles with different radius $O_1$ and $O_2$ are both tangent to a larger circle $O$, tangent points are $S,T$. Note that intersections of $O_1$ and $O_2$ are $M,N$, prove that the sufficient and necessary condition of $OM\perp MN$ is $S,N,T$ are colinear.

1998 China Second Round Test 2 p1

Circumcenter and incentre of $\triangle ABC$ are $O,I$. $AD$ is the height on side $BC$. If $I$ is on line $OC$, prove that the radius of circumcircle and escribed circle (in $\angle BAC$) are equal.

1999 China Second Round Test 2 p1

In convex quadrilateral $ABCD$, $\angle BAC=\angle CAD$. $E$ lies on segment $CD$, $BE$ and $AC$ intersect at $F,$ $DF$ and $BC$ intersect at $G.$ Prove that $\angle GAC=\angle EAC$.

2000 China Second Round Test 2 p1

In acute-angled triangle $ABC,$ $E,F$ are on the side $BC,$ such that $\angle BAE=\angle CAF,$ and let $M,N$ be the projections of $F$ onto $AB,AC,$ respectively. The line $AE$ intersects $\odot (ABC)$ at $D$(different from point $A$). Prove that $S_{AMDN}=S_{\triangle ABC}.$

2001 China Second Round Test 2 p1
Let $O,H$ be the circumcenter and orthocenter of $\triangle ABC,$ respectively. Line $AH$ and $BC$ intersect at $D,$ Line $BH$ and $AC$ intersect at $E,$ Line $CH$ and $AB$ intersect at $F,$ Line $AB$ and $ED$ intersect at $M,$ $AC$ and $FD$ intersect at $N.$ Prove that
a) $OB\perp DF,OC\perp DE;$
b) $OH\perp MN.$

2002 China Second Round Test 2 p1
In $\triangle ABC$, $\angle A = 60$, $AB>AC$, point $O$ is the circumcenter and $H$ is the intersection point of two altitudes $BE$ and $CF$. Points $M$ and $N$ are on the line segments $BH$ and $HF$ respectively, and satisfy $BM=CN$. Determine the value of $\frac{MH+NH}{OH}$.

2003 China Second Round Test 2 p1
From point $P$ outside a circle draw two tangents to the circle touching at $A$ and $B$. Draw a secant line intersecting the circle at points $C$ and $D$, with $C$ between $P$ and $D$. Choose point $Q$ on the chord $CD$ such that $\angle DAQ=\angle PBC$. Prove that $\angle DBQ=\angle PAC$.

2004 China Second Round Test 2 p1
In an acute triangle $ABC$, point $H$ is the intersection point of altitude $CE$ to $AB$ and altitude $BD$ to $AC$. A circle with $DE$ as its diameter intersects $AB$ and $AC$ at $F$ and $G$, respectively. $FG$ and $AH$ intersect at point $K$. If $BC=25$, $BD=20$, and $BE=7$, find the length of $AK$.

2005 China Second Round Test 2 p1
In $\triangle ABC$, $AB>AC$, $l$ is a tangent line of the circumscribed circle of $\triangle ABC$, passing through $A$. The circle, centered at $A$ with radius $AC$, intersects $AB$ at $D$, and line $l$ at $E, F$. Prove that lines $DE, DF$ pass through the incenter and an excenter of $\triangle ABC$ respectively.

2006 China Second Round Test 2 p1
An ellipse with foci $B_0,B_1$ intersects $AB_i$ at $C_i$ $(i=0,1)$. Let $P_0$ be a point on ray $AB_0$. $Q_0$ is a point on ray $C_1B_0$ such that $B_0P_0=B_0Q_0$; $P_1$ is on ray $B_1A$ such that $C_1Q_0=C_1P_1$; $Q_1$ is on ray $B_1C_0$ such that $B_1P_1=B_1Q_1$; $P_2$ is on ray $AB_0$ such that $C_0Q_1=C_0Q_2$. Prove that $P_0=P_2$ and that the four points $P_0,Q_0,Q_1,P_1$ are concyclic.

2007 China Second Round Test 2 p1
In an acute triangle $ABC$, $AB<AC$. $AD$ is the altitude dropped onto $BC$ and $P$ is a point on $AD$. Let $PE\perp AC$ at $E$, $PF\perp AB$ at $F$ and let $J,K$ be the circumcentres of triangles $BDF, CDE$ respectively. Prove that $J,K,E,F$ are concyclic if and only if $P$ is the orthocentre of triangle $ABC$.

2008 China Second Round Test 2 p1
Given a convex quadrilateral with $\angle B+\angle D<180$.Let $P$ be an arbitrary point on the plane,define $f(P)=PA \cdot BC+PD \cdot CA+PC  \cdot AB$.
a) Prove that $P,A,B,C$ are concyclic when $f(P)$ attains its minimum.
b) Suppose that $E$ is a point on the minor arc $AB$ of the circumcircle $O$ of $ABC$, such that $AE=\frac{\sqrt 3}{2}AB,BC=(\sqrt 3-1)EC,\angle ECA=2\angle ECB$. Knowing that $DA,DC$ are tangent to circle $O$,$AC=\sqrt 2$,find the minimum of $f(P)$.

2009 China Second Round Test 2 p1
Let $\omega$ be the circumcircle of acute triangle $ABC$ where $\angle A<\angle B$ and $M,N$ be the midpoints of minor arcs $BC,AC$ of $\omega$ respectively. The line $PC$ is parallel to $MN$, intersecting $\omega$ at $P$ (different from $C$). Let $I$ be the incentre of $ABC$ and let $PI$ intersect $\omega$ again at the point $T$.
a) Prove that $MP\cdot MT=NP\cdot NT$;
b) Let $Q$ be an arbitrary point on minor arc $AB$ and $I,J$ be the incentres of triangles $AQC,BCQ$. Prove that $Q,I,J,T$ are concyclic.

2010 China Second Round Test 2 p1
Given an acute triangle whose circumcenter is $O$.let $K$ be a point on $BC$,different from its midpoint.$D$ is on the extension of segment $AK,BD$ and $AC$,$CD$and$AB$intersect at $N,M$ respectively.prove that $A,B,D,C$ are concyclic.

2011 China Second Round Test 2 p1
Let $P,Q$ be the midpoints of diagonals $AC,BD$ in cyclic quadrilateral $ABCD$. If $\angle BPA=\angle DPA$, prove that $\angle AQB=\angle CQB$.

2012 China Second Round Test 2 p1
In an acute-angled triangle $ABC$, $AB>AC$. $M,N$ are distinct points on side $BC$ such that $\angle BAM=\angle CAN$. Let $O_1,O_2$ be the circumcentres of $\triangle ABC, \triangle AMN$, respectively. Prove that $O_1,O_2,A$ are collinear.

2013 China Second Round Test 2 p1
$AB$ is a chord of circle $\omega$, $P$ is a point on minor arc $AB$, $E,F$ are on segment $AB$ such that $AE=EF=FB$. $PE,PF$ meets $\omega$ at $C,D$ respectively. Prove that $EF\cdot CD=AC\cdot BD$.

2014 China Second Round Test 2 p2

Let $ABC$ be an acute triangle such that $\angle BAC \neq 60^\circ$. Let $D,E$ be points such that $BD,CE$ are tangent to the circumcircle of $ABC$ and $BD=CE=BC$ ($A$ is on one side of line $BC$ and $D,E$ are on the other side). Let $F,G$ be intersections of line $DE$ and lines $AB,AC$. Let $M$ be intersection of $CF$ and $BD$, and $N$ be intersection of $CE$ and $BG$. Prove that $AM=AN$.

2015 China Second Round Test 2 p2

In isoceles $\triangle ABC$, $AB=AC$, $I$ is its incenter, $D$ is a point inside $\triangle ABC$ such that $I,B,C,D$ are concyclic. The line through $C$ parallel to $BD$ meets $AD$ at $E$. Prove that $CD^2=BD\cdot CE$.

2016 China Second Round Test 2 p2

Let $X,Y$ be two points which lies on the line $BC$ of $\triangle ABC(X,B,C,Y\text{lies in sequence})$ such that $BX\cdot AC=CY\cdot AB$, $O_1,O_2$ are the circumcenters of $\triangle ACX,\triangle ABY$, $O_1O_2\cap AB=U,O_1O_2\cap AC=V$. Prove that $\triangle AUV$ is a isosceles triangle.

2017 China Second Round Test 2 p1

Given an isocleos triangle $ABC$ with equal sides $AB=AC$ and incenter $I$.Let $\Gamma_1$be the circle centered at $A$ with radius $AB$,$\Gamma_2$ be the circle centered at $I$ with radius $BI$.A circle $\Gamma_3$ passing through $B,I$ intersects $\Gamma_1$,$\Gamma_2$ again at $P,Q$ (different from $B$) respectively.Let $R$ be the intersection of $PI$ and $BQ$.Show that $BR \perp CR$.

2018 China Second Round Test 2 A p2

In triangle $\triangle ABC$, $AB<AC$, $M,D,E$ are the midpoints of $BC$, the arcs $BAC$ and $BC$ of the circumcircle of $\triangle ABC$ respectively. The incircle of $\triangle ABC$ touches $AB$ at $F$, $AE$ meets $BC$ at $G$, and the perpendicular to $AB$ at $B$ meets segment $EF$ at $N$. If $BN=EM$, prove that $DF$ is perpendicular to $FG$.

2018 China Second Round Test 2 B p3

In triangle $\triangle ABC, AB=AC.$ Let $D$ be on segment $AC$ and $E$ be on be a point on the extended line $BC$ such that $C$ is located between $B$ and $E$ and $\frac{AD}{DC}=\frac{BC}{2CE}$. Let $\omega$ be the circle with diameter $AB,$ and $\omega$ touches segment $DE$ at $F.$ Prove that $B,C,F,D$ are concyclic.

2019 China Second Round Test 2 A p1
In acute triangle $\triangle ABC$, $M$ is the midpoint of segment $BC$. Point $P$ lies in the interior of $\triangle ABC$ such that $AP$ bisects $\angle BAC$. Line $MP$ intersects the circumcircles of $\triangle ABP,\triangle ACP$ at $D,E$ respectively. Prove that if $DE=MP$, then $BC=2BP$.

2019 China Second Round Test 2 B p3
Point $A,B,C,D,E$ lie on a line in this order, such that $BC=CD=\sqrt{AB\cdot DE},$ $P$ doesn't lie on the line, and satisfys that $PB=PD.$ Point $K,L$ lie on the segment $PB,PD,$ respectively, such that $KC$ bisects $\angle BKE,$ and $LC$ bisects $\angle ALD.$ Prove that $A,K,L,E$ are concyclic.

2020 China Second Round Test 2 p1

In triangle $ABC,$ $AB=BC,$ and let $I$ be the incentre of $\triangle ABC.$ $M$ is the midpoint of segment $BI.$ $P$ lies on segment $AC,$ such that $AP=3PC.$ $H$ lies on line $PI,$ such that $MH\perp PH.$ $Q$ is the midpoint of the arc $AB$ of the circumcircle of $\triangle ABC$. Prove that $BH\perp QH.$

2021 China Second Round Test 2 p1

In $\triangle ABC$, point $M$ is the middle point of $AC$. $MD//AB$ and meet the tangent of $A$

to $\odot(ABC)$ at point $D$. Point $E$ is in $AD$ and point $A$ is the middle point of $DE$.

$\{P\}=\odot(ABE)\cap AC,\{Q\}=\odot(ADP)\cap DM$. Prove that $\angle QCB=\angle BAC$.