Geolympiad 2015 (Aops) 28p

Geolympiad is a Geometry Olympiad test  that took place in Aops in 2015.
It happened 3 times, in Spring, in Summer and in Fall.
Below are the 6+6+3=15 contest's problems  and the 6+4+3=13 dropped problems respectively.



Geolympiad 2015 Spring Contest [3+3]


Contest pdf  here.  [source]
Solved in the aops links in their names    

day 1

Geolympiad 2015 Spring Contest P1
Let $ABC$ be a triangle. Suppose $P,Q$ are on lines $AB, AC$ (on the same side of A) with $AP=AC$ and $AB=AQ$.  Now suppose points $X,Y$ move along the sides $AB, AC$ of $ABC$ so that $XY || PQ$. Determine the locus of the circumcenters of the variable triangle $AXY$.

Geolympiad 2015 Spring Contest P2
Let $ABC$ be a triangle and $w$ its incircle.  $w$ touches $BC,CA$ at $A_1,B_1$ respectively.  The second intersection of $AA_1$ and $w$ is $A_2$, similarly define $B_2$.  Then $AB,A_1B_1,A_2B_2$ concur at a point $C_3$.

Geolympiad 2015 Spring Contest P3
Let $ABC$ be an acute triangle with orthocenter $H$, incenter $I$, and excenters $I_A, I_B, I_C$. Show that $II_A \cdot  II_B \cdot  II_C \ge 8 AH \cdot  BH  \cdot  CH$.

day 2

Geolympiad 2015 Spring Contest P4
Let $ABC$ be an acute triangle with $\angle A = 60$ and altitudes $BE, CF$. Suppose $BE, CF$ are reflected across the perpendicular bisector of $BC$ and the two new segments $B'E', C'F'$ intersect at a point $X$. If $A$ is reflected across $BC$ to form $A'$, show that $AX$ is bisected by the internal angle bisector of $A$.

Geolympiad 2015 Spring Contest P5
Let $ABC$ be a triangle with circumcircle $w_1$ and incenter $I$. Suppose $w_2$ is a circle tangent to $AB,AC$ at $X,Y$, and internally tangent to $w$ at $D$. Let the parallel to the exterior angle bisector of $A$ through $D$ meet $w_2$ at $P$. Show that $AP, DI$ intersect on $w_2$.

Geolympiad 2015 Spring Contest P6
Let $ABC$ be a triangle, $X$ the midpoint of arc $BC$ on the circumcircle.  The tangents from $X$ to the incircle meet the circumcircle again at $X_1,X_2$, and $X_1X_2$ intersects the incircle at $P,Q$.  Let $M$ be the midpoint of $PQ$, and let $A_1$ be the tangency point of the $A$-mixtillinear incircle with the circumcircle.  Show that $A,M,A_1$ are collinear.


Geolympiad 2015 Summer Contest [3+3]


Contest pdf  here.   [source]     
Solved in the aops links in their names    

day 1

Geolympiad 2015 Summer Contest P1
Show in an acute triangle $ABC$ that $\cot A + \cot B + \cot C \ge \dfrac{12[ABC]}{a^2+b^2+c^2}$.

Geolympiad 2015 Summer Contest P2
Let $ABC$ be a triangle. Let line $\ell$ be the line through the tangency points that are formed when the tangents from $A$ to the circle with diameter $BC$ are drawn. Let line $m$ be the line through the tangency points that are formed when the tangents from $B$ to the circle with diameter $AC$ are drawn. Show that the $\ell$, $m$, and the $C$-altitude concur.

Let $ABC$ be an acute scalene triangle with incenter $I$, circumcircle $w_1$, and denote the circumcircle of $BIC$ as $w_2$. Suppose point $P$ lies on $w_2$ and is inside $w_1$. Let $X,Y$ lie on $BC$ with $XP \perp BP, YP \perp PC$. Circles $O_1, O_2$ are drawn tangent to $w_1$ at points on the same side of $BC$ as $A$ and tangent to $BC$ at $X,Y$ respectively. Let the centers of those two circles be $Z_1, Z_2$. Let $D$ be the point on $w_2$ opposite to $P$ and let $E$ be the foot of the altitude from $P$ to $BC$. Show that $DE \perp Z_1Z_2$

day 2

Geolympiad 2015 Summer Contest P4
Let $ABC$ be a triangle and $I$ be its incenter. Let $D$ be the intersection of the exterior bisectors of $\angle BAC$ and $\angle BIC$, $E$ be the intersection of the exterior bisectors of $\angle ABC$ and $\angle AIC$, and $F$ be the intersection of the exterior bisectors of $\angle ACB$ and $\angle AIB$. Prove that $D$, $E$, $F$ are collinear.

Geolympiad 2015 Summer Contest P5
Let $ABC$ be a triangle and $P$ be in its interior. Let $Q$  be the isogonal conjugate of $P$. Show that $BCPQ$ is cyclic if and only if $AP=AQ$.

Geolympiad 2015 Summer Contest P6
Let $w_1, w_2$ be non-intersecting, congruent circles with centers $O_1, O_2$ and let $P$ be in the exterior of both of them. The tangents from $P$ to $w_1$ meet $w_1$ at $A_1, B_1$ and define $A_2, B_2$ similarly. If lines $A_1B_1, A_2B_2$ meet at $Q$ show that the midpoint of $PQ$ is equidistant from $O_1, O_2$.



Geolympiad 2015 Fall Contest [3]

Fun with Mixtillinear Circles

Contest pdf  here. [source]     

1. Let $ABC$ be an acute triangle with $\angle B > \angle C$. The $B,C$ mixtillinear incircles of ABC are tangent to the circumcircle of $ABC$ at $P,Q$, and $M$ is the  midpoint of minor arc $BC$ on the circumcircle of $ABC$. Let $PQ$ meet $BC$ at $Z$, and suppose $PM,QM$ meet $BC$ at $X, Y$ . Show that  $ZX / ZY <  cosC / cosB.$

2. Acute triangle $ABC$ is given with incenter$ I$, circumcenter $O$, and circumcircle $w$. A variable point $D$ is given on segment $BC$. Then a circle may be drawntangent to $AD,BC$, and internally tangent to $w$ at a point $X_1$ on minor arc $AB$, while a circle may be drawn tangent to $AD,BC$, and internally tangent to $w$ at a point $X_2$ on minor arc $AC$. Let $w_1$ be the circumcircle of $IX_1X_2$. As $D$ varies, determine the locus of the intersection of the tangents from $X_1,X_2$ with respect to $w_1$.

3. Let $ABC$ be a triangle and let $PA$ be the point where the circle passing through $B$ and $C$ di erent from the circumcircle of $ABC$ that is tangent to the A-mixtilinear-incircle is tangent to the $A$-mixtilinear-incircle, and defi ne $PB$ and $PC$ similarly. Prove that $APA, BPB$, and $CPC$ concur.


                                                            Solved in Aops here.



Geolympiad 2015 Spring Shortlist [6+6] 

The proposed and the dropped problems.
                                                              

G1: (vincenthuang75025, rejected for being too easy)
Let $w$ be a circle and let $A,B $ be points on it. Suppose another circle $T$ is drawn which is internally tangent to $w$ at $P$. Let $U$ be the point on $T$ further from $B$ such that $AU$ is tangent to the $T$, and define $V$ similarly. Let the angle bisector of $APB$ meet $AB$ at $C$. Suppose that $\angle UAB = \angle VBA$. Show that $\frac{CU}{CV} = \frac{AP}{BP}$.

G2: (vincenthuang75025, rejected for being too easy)
Let $ABCD$ be a trapezoid with $AD||BC$. Suppose $E$ is the intersection of $AC, BD$ and $F$ is the intersection of $AB, CD$. A line $l$ parallel to $BC$ is given and $AC, BD$ meet it at $X, Y$ respectively. $BX, CY $ meet at $P$. Show that $FP$ bisects the area of the original trapezoid.

G3: (vincenthuang75025, #1 on test) 
Let $ABC$ be a triangle. Suppose $P,Q$ are on lines $AB, AC$ with $AP=AC$ and $AB=AQ$. Now suppose points $X,Y$ move along the sides $AB, AC$ of $ABC$ so that $XY || PQ$. Determine the locus of the circumcenter of the variable triangle $AXY$.

G4: (infiniteturtle, rejected for being troll)
Let $ABC$ be a triangle, and let $A_1,A_2$ be points on segment $BC$ that trisect segment $BC$.  Similarly define $B_1,B_2,C_1,C_2$.  Prove that there exists a ellipse tangent to $AA_1,AA_2,BB_1,BB_2,CC_1,CC_2$

G5: (vincenthuang75025, #4 on test) 
Let $ABC$ be an acute triangle with $\angle A = 60$ and altitudes $BE, CF$. Suppose $BE, CF$ are reflected across the perpendicular bisector of $BC$ and the two new segments $B'E', C'F'$ intersect at a point $X$. If $A$ is reflected across $BC$ to form $A'$, show that $AX$ is bisected by the internal angle bisector of $A$.

G6: (vincenthuang75025, rejected for being too easy)
 Let $ABC$ be a given acute triangle in the plane. Suppose for each point $P$, the quantity $X_P $ denotes the value of $AP^2+BP^2+CP^2$. If $H,G$ are the orthocenter and centroid of $ABC$ respectively, show that $X_G \le X_H$.

G7: (vincenthuang75025, #5 on test)
 Let $ABC$ be a triangle with circumcircle $w_1$ and incenter $I$. Suppose $w_2$ is a circle tangent to $AB,AC$ at $X,Y$, and internally tangent to $w$ at $D$. Let the parallel to the exterior angle bisector of $A$ through $D$ meet $w_2$ at $P$. Show that $AP, DI$ intersect on $w_2$.

G8: (infiniteturtle, thkim1011, rejected for being too simple)
 In triangle ABC, let $A_1,B_1,C_1$ be the midpoints of BC,CA,AB respectively.  Let the orthocenter of ABC be H, and define the feet from H to BC,CA,AB respectively as D,E,F.  Let $A_2,B_2,C_2$ be the respective midpoints of AD,BE,CF.  Prove that $A_1A_2, B_1B_2, C_1C_2$ concur.

G9: (infinteturtle, #2 on test)
 Let $ABC$ be a triangle and $w$ its incircle.  $w$ touches $BC,CA$ at $A_1,B_1$ respectively.  The second intersection of $AA_1$ and $w$ is $A_2$, similarly define $B_2$.  Then $AB,A_1B_1,A_2B_2$ concur at a point $C_3$

G10: (vincenthuang75025, #3 on test)
Let $ABC$ be an acute triangle with orthocenter $H$, incenter $I$, and excenters $I_A, I_B, I_C$. Show that $II_A \cdot  II_B \cdot  II_C \ge 8 AH \cdot  BH  \cdot  CH$.

G11: (thkim1011, rejected for having too advanced projective vocabulary)
Let $ABC$ be a triangle. Now let $PQR$ be the triangle such that $PQ$ is the tangent to the circumcircle at $C$, $QR$ is the tangent to the circumcircle at $A$, and $PR$ is the tangent to the circumcircle at $B$. Now let $XYZ$ be the nagel triangle of $PQR$.
Prove that $ABC$ is homothetic to $XYZ$.
Also show the center of homothety is the projective conjugate of the circumcenter with respect to the tangential quadrangle

G12: (infinteturtle, #6 on the test)
Let $ABC$ be a triangle, $X$ the midpoint of arc $BC$ on the circumcircle.  The tangents from $X$ to the incircle meet the circumcircle again at $X_1,X_2$, and $X_1X_2$ intersects the incircle at $P,Q$.  Let $M$ be the midpoint of $PQ$, and let $A_1$ be the tangency point of the $A$-mixtillinear incircle with the circumcircle.  Show that $A,M,A_1$ are collinear.

  [all solved online here]  
[Shortlist 2015 spring solution pdf, without problems here]



Geolympiad 2015 Summer  [4] dropped problems
                   


G1 (dropped for too easy)    
Given triangle $ABC$ and let $O$ be circumcircle. let the reflections of $O$ across $BC, AC, AB$ be $O_A, O_B, O_C$. Then $ABCO_AO_BO_C$ lie on the same conic, and the midpoint of the foci of the conic  is the nine-point center of $ABC$.

G2 (dropped for similarity to another problem)
Let $ABC$ be a triangle with side lengths $a,b,c$ and inradius $r$. Show that
$\frac{1}{b^2+c^2-a^2} + \frac{1}{c^2+a^2-b^2} + \frac{1}{a^2+b^2-c^2} \ge \frac{1}{4r^2}$.

G3 (too contrived)
 Let ABC be a triangle. Suppose $D,E,F$ are the feet of the angle bisectors from angles $A,B,C$ and $I, I_A$ are the incenter, A-excenter of triangle $ABC$. Suppose $DF$ and $AC$ meet at $X$.  Let $w$ be the circle through $A, I_A$ and tangent to $CI_A$ and suppose it meets the circumcircle of $AIE$ at $Y$. Show that $XYEB$ is cyclic.

G4 (probably too hard)
Let $ABC$ be a triangle and $I$ be the incenter. Let $\omega_A$ be the circle through $I$ tangent to $AB$ and $AC$. Define $\omega_B$ and $\omega_C$ similarly. Let $\ell_A$ be the line through the two tangency points on $\omega_A$, and define $\ell_B$ and $\ell_C$ similarly. Suppose that the vertices of the triangle formed by the three lines $\ell_A\ell_B\ell_C$ is $XYZ$. Prove that $X,Y,Z,A,B,C$ lie on the same conic.
 [the  first 3 have been solved here]


Geolympiad 2015 Fall [3] dropped problems

These are still mixtillinear-themed.     

1. Let $ABC$ be an acute triangle with circumcircle $\omega$ and incenter $I$. Suppose a circle $\omega_1$ is drawn internally tangent to $\omega$ at $T$, and tangent to $AB$, $AC$ at $D$, $E$. Finally, let $TI$ meet $BC$ at $X$. Show that $\frac{BX}{CX}=\frac{BD}{CE}$.

2. Triangle $ABC$ is drawn along with its circumcircle $w$, which has radius $1$. Point $D$ is selected on side $BC$, and distinct circles $w_1,w_2$ are drawn each tangent to $AD,BC$, and internally arc $BAC$ of $w$ at $X,Y$ respectively. The incenter $I$ of $ABC$ is drawn. Suppose that $AB,BC,CA,A,B,C,D,AD,w_1,w_2$ are all erased so that only $w, X,Y,I$ remain. With a straightedge and compass, construct $P,Q$ on $w$ so that $Y$ is the point that the $X$-mixtillinear circle of $XPQ$ meets $w$.

3. An acute triangle $ABC$ is given with incenter $I$, circumcenter $O$ and the midpoint of minor arc $BC$ is $M$. A point $R$ is chosen on $BC$ so that $IR \perp AI$. $RM$ meets the circumcircle of $ABC$ at $P$ and $AR$ meets the circumcircle of $ABC$ at $Q$. If $PQ$ and $AI$ meet at $X$, show that $XR \perp OI$.

[all 3 have been solved here]

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