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]
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 Summer Contest [3+3]
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 Fall Contest [3]
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.
Geolympiad 2015 Summer [4] dropped problems
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.
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.
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]
Solved in the aops links in their names
day 1
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.
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
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.
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.
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.
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.
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]
Solved in the aops links in their names
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 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.
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.
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.
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.
Fun with Mixtillinear Circles
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.
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.
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.
[Shortlist 2015 spring solution pdf, without problems here]
Geolympiad 2015 Summer [4] dropped problems
G1 (dropped for too easy)
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.
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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