geometry problems from China Girls Math Olympiad (CGMO)

with aops links in the names

CGMO 2002.4

Circles O

CGMO 2002.7

An acute triangle ABC has three heights AD, BE and CF respectively. Prove that the perimeter of triangle DEF is not over half of the perimeter of triangle ABC.

CGMO 2011.2

The diagonals AC,BD of the quadrilateral ABCD intersect at E. Let M,N be the midpoints of AB,CD respectively. Let the perpendicular bisectors of the segments AB,CD meet at F. Suppose that EF meets BC,AD at P,Q respectively. If MF·CD =NF·AB and DQ·BP=AQ·CP, prove that PQ ⊥ BC.

CGMO 2011.8

The A-excircle (O) of ∆ABC touches BC at M. The points D,E lie on the sides AB,AC respectively such that DE // BC. The incircle (O

CGMO 2012.2

Circles Q

CGMO 2012.5

As shown in the figure below, the incircle of ABC is tangent to sides AB and AC at D and E respectively, and O is the circumcenter of BCI. Prove that <ODB = <OEC.

with aops links in the names

2002 - 2018

Circles O

_{1}and O

_{2}interest at two points B and C, and BC is the diameter of circle O

_{1}. Construct a tangent line of circle O

_{1}at C and intersecting circle O

_{2}at another point A. We join AB to intersect circle O

_{1 }at point E, then join CE and extend it to intersect circle O

_{2}at point F. Assume H is an arbitrary point on line segment AF. We join HE and extend it to intersect circle O

_{1 }at point G, and then join BG and extend it to intersect the extend line of AC at point D. Prove that$\frac{AH}{HF}=\frac{AC}{CD}$ .

CGMO 2002.7

An acute triangle ABC has three heights AD, BE and CF respectively. Prove that the perimeter of triangle DEF is not over half of the perimeter of triangle ABC.

CGMO 2003.1

Let ABC be a triangle. Points D and E are on sides AB and AC, respectively, and point F is on line segment DE. Let $\frac{AD}{AB}=\text{ }x,\frac{AE}{AC}=\text{ }y,\text{ }\frac{DF}{DE}=\text{ }z$. Prove that

Let ABC be a triangle. Points D and E are on sides AB and AC, respectively, and point F is on line segment DE. Let $\frac{AD}{AB}=\text{ }x,\frac{AE}{AC}=\text{ }y,\text{ }\frac{DF}{DE}=\text{ }z$. Prove that

(1) ${{S}_{\vartriangle
BDF}}=(1-x)zy{{S}_{\vartriangle ABC}}$ and ${{S}_{\vartriangle
CEF}}=x(1-y)(1-z){{S}_{\vartriangle ABC}}$

(2) $\sqrt[3]{{{S}_{\vartriangle
BDF}}}+\sqrt[3]{{{S}_{\vartriangle CEF}}}\le \sqrt[3]{{{S}_{\vartriangle
ABC}}}$

CGMO 2003.3

As shown in the figure, quadrilateral ABCD is inscribed in a circle with AC as its diameter, BD ⊥ AC, and E the intersection of AC and BD. Extend line segment DA and BA through A to F and G respectively, such that DG // BF. Extend GF to H such that CH ⊥ GH. Prove that points B,E, F and H lie on one circle.

As shown in the figure, quadrilateral ABCD is inscribed in a circle with AC as its diameter, BD ⊥ AC, and E the intersection of AC and BD. Extend line segment DA and BA through A to F and G respectively, such that DG // BF. Extend GF to H such that CH ⊥ GH. Prove that points B,E, F and H lie on one circle.

CGMO 2003.7

Let the sides of a scalene triangle △ABC be AB = c, BC = a, CA = b, and D,E, F be points on BC,CA,AB such that AD,BE,CF are angle bisectors of the triangle, respectively. Assume that DE = DF. Prove that

Let the sides of a scalene triangle △ABC be AB = c, BC = a, CA = b, and D,E, F be points on BC,CA,AB such that AD,BE,CF are angle bisectors of the triangle, respectively. Assume that DE = DF. Prove that

(1) $\frac{a}{b+c}=\frac{b}{c+a}+\frac{c}{a+b}$

(2) < BAC
> 90þ

CGMO 2004.3

Let ABC be an obtuse inscribed in a circle of radius 1. Prove that ∆ABC can be covered by an isosceles right-angled triangle with hypotenuse of length $\sqrt{2}+1$ .

Let ABC be an obtuse inscribed in a circle of radius 1. Prove that ∆ABC can be covered by an isosceles right-angled triangle with hypotenuse of length $\sqrt{2}+1$ .

CGMO 2004.6

Given an acute triangle ABC with O as its circumcenter. Line AO intersects BC at D. F are on AB, AC respectively such that A, E, D, F are concyclic. Prove that the length of the projection of line segment EF on side BC does not depend on the positions of E and F.

Given an acute triangle ABC with O as its circumcenter. Line AO intersects BC at D. F are on AB, AC respectively such that A, E, D, F are concyclic. Prove that the length of the projection of line segment EF on side BC does not depend on the positions of E and F.

CGMO 2005.1

As shown in the following figure, point P lies on the circumcircle of triangle ABC. Lines AB and CP meet at E, and lines AC and BP meet at F. The perpendicular bisector of line segment AB meets line segment AC at K, and the perpendicular bisector of line segment AC meets line segment AB at J. Prove that ${{\left( \frac{CE}{BF} \right)}^{2}}=\frac{AJ\cdot JE}{AK\cdot KF}$

As shown in the following figure, point P lies on the circumcircle of triangle ABC. Lines AB and CP meet at E, and lines AC and BP meet at F. The perpendicular bisector of line segment AB meets line segment AC at K, and the perpendicular bisector of line segment AC meets line segment AB at J. Prove that ${{\left( \frac{CE}{BF} \right)}^{2}}=\frac{AJ\cdot JE}{AK\cdot KF}$

(1) it has 12 edges, 6 faces and 8 vertices and

(2) it has 4 faces with each pair of them sharing a common edge of the
polyhedron.

CGMO 2005.8

Given an a × b rectangle with a > b > 0, determine the minimum side of a square that covers the rectangle. (A square covers the rectangle if each point in the rectangle lies inside the square.)

Given an a × b rectangle with a > b > 0, determine the minimum side of a square that covers the rectangle. (A square covers the rectangle if each point in the rectangle lies inside the square.)

CGMO 2006.2

Let O be the intersection of the diagonals of convex quadrilateral ABCD. The circumcircles of ∆OAD and ∆OBC meet at O and M. Line OM meets the circumcircles of ∆OAB and ∆OCD at T and S respectively. Prove that M is the midpoint of ST.

Let O be the intersection of the diagonals of convex quadrilateral ABCD. The circumcircles of ∆OAD and ∆OBC meet at O and M. Line OM meets the circumcircles of ∆OAB and ∆OCD at T and S respectively. Prove that M is the midpoint of ST.

CGMO 2007.2

Let ABC be an acute triangle. Points D, E, and F lie on segments BC, CA, and AB, respectively, and each of the three segments AD, BE, and CF contains the circumcenter of ABC. Prove that if any two of the ratios ${\frac{BD}{DC},\frac{CE}{EA},\frac{AF}{FB},\frac{BF}{FA},\frac{AE}{EC},\frac{CD}{DB}}$ are integers, then triangle ABC is isosceles.

Let ABC be an acute triangle. Points D, E, and F lie on segments BC, CA, and AB, respectively, and each of the three segments AD, BE, and CF contains the circumcenter of ABC. Prove that if any two of the ratios ${\frac{BD}{DC},\frac{CE}{EA},\frac{AF}{FB},\frac{BF}{FA},\frac{AE}{EC},\frac{CD}{DB}}$ are integers, then triangle ABC is isosceles.

CGMO 2007.5

Point D lies inside triangle ABC such that <DAC = <DCA = 30þ and <DBA = 60þ. Point E is the midpoint of segment BC. Point F lies on segment AC with AF = 2 FC. Prove that DE ⊥ EF .

Point D lies inside triangle ABC such that <DAC = <DCA = 30þ and <DBA = 60þ. Point E is the midpoint of segment BC. Point F lies on segment AC with AF = 2 FC. Prove that DE ⊥ EF .

CGMO 2008.3

Determine the least real number a greater than 1 such that for any point P in the interior of the square ABCD, the area ratio between two of the triangles PAB, PBC, PCD, PDA lies in the interval $\left[ \frac{1}{a},a \right]$ .

Determine the least real number a greater than 1 such that for any point P in the interior of the square ABCD, the area ratio between two of the triangles PAB, PBC, PCD, PDA lies in the interval $\left[ \frac{1}{a},a \right]$ .

CGMO 2008.4

Equilateral triangles ABQ, BCR, CDS, DAP are erected outside of the convex quadrilateral ABCD. Let X, Y , Z, W be the midpoints of the segments PQ, QR, RS, SP, respectively. Determine the maximum value of ${\frac{XZ\text{ }+\text{ }YW}{AC\text{ }+\text{ }BD}}$

Equilateral triangles ABQ, BCR, CDS, DAP are erected outside of the convex quadrilateral ABCD. Let X, Y , Z, W be the midpoints of the segments PQ, QR, RS, SP, respectively. Determine the maximum value of ${\frac{XZ\text{ }+\text{ }YW}{AC\text{ }+\text{ }BD}}$

CGMO 2008.5

In convex quadrilateral ABCD, AB = BC and AD = DC. Point E lies on segment AB and point F lies on segment AD such that B, E, F, D lie on a circle. Point P is such that triangles DPE and ADC are similar and the corresponding vertices are in the same orientation (clockwise or counterclockwise). Point Q is such that triangles BQF and ABC are similar and the corresponding vertices are in the same orientation. Prove that points A, P, Q are collinear.

In convex quadrilateral ABCD, AB = BC and AD = DC. Point E lies on segment AB and point F lies on segment AD such that B, E, F, D lie on a circle. Point P is such that triangles DPE and ADC are similar and the corresponding vertices are in the same orientation (clockwise or counterclockwise). Point Q is such that triangles BQF and ABC are similar and the corresponding vertices are in the same orientation. Prove that points A, P, Q are collinear.

CGMO 2009.2

Right triangle ABC, with <A = 90þ, is inscribed in circle Γ. Point E lies on the interior of arc BC (not containing A) with EA > EC. Point F lies on ray EC with < EAC = < CAF. Segment BF meets Γ again at D (other than B). Let O denote the circumcenter of triangle DEF. Prove that A, C, O are collinear.

Right triangle ABC, with <A = 90þ, is inscribed in circle Γ. Point E lies on the interior of arc BC (not containing A) with EA > EC. Point F lies on ray EC with < EAC = < CAF. Segment BF meets Γ again at D (other than B). Let O denote the circumcenter of triangle DEF. Prove that A, C, O are collinear.

CGMO 2009.6

Circle Γ

Circle Γ

_{1}, with radius r, is internally tangent to circle Γ_{2}at S. Chord AB of Γ_{2}is tangent to Γ_{1}at C. Let M be the midpoint of arc AB (not containing S), and let N be the foot of the perpendicular from M to line AB. Prove that AC · CB = 2r · MN.
CGMO 2010.2

In triangle ABC, AB = AC. Point D is the midpoint of side BC. Point E lies outside the triangle ABC such that CE ⊥ AB and BE = BD. Let M be the midpoint of segment BE. Point F lies on the minor arc d AD of the circumcircle of triangle ABD such that MF ⊥ BE. Prove that ED ⊥ FD.

In triangle ABC, AB = AC. Point D is the midpoint of side BC. Point E lies outside the triangle ABC such that CE ⊥ AB and BE = BD. Let M be the midpoint of segment BE. Point F lies on the minor arc d AD of the circumcircle of triangle ABD such that MF ⊥ BE. Prove that ED ⊥ FD.

CGMO 2010.6

In acute triangle ABC, AB > AC. Let M be the midpoint of side BC. The exterior angle bisector of <BAC meet ray BC at P. Point K and F lie on line PA such that MF ⊥ BC and MK ⊥ PA. Prove that BC

In acute triangle ABC, AB > AC. Let M be the midpoint of side BC. The exterior angle bisector of <BAC meet ray BC at P. Point K and F lie on line PA such that MF ⊥ BC and MK ⊥ PA. Prove that BC

^{2}= 4 PF · AK.CGMO 2011.2

The diagonals AC,BD of the quadrilateral ABCD intersect at E. Let M,N be the midpoints of AB,CD respectively. Let the perpendicular bisectors of the segments AB,CD meet at F. Suppose that EF meets BC,AD at P,Q respectively. If MF·CD =NF·AB and DQ·BP=AQ·CP, prove that PQ ⊥ BC.

CGMO 2011.8

The A-excircle (O) of ∆ABC touches BC at M. The points D,E lie on the sides AB,AC respectively such that DE // BC. The incircle (O

_{1}) of ∆ADE touches DE at N. If BO

_{1}∩DO = F and CO

_{1}∩EO = G, prove that the midpoint of FG lies on MN.

CGMO 2012.2

Circles Q

_{1}and Q

_{2}are tangent to each other externally at T. Points A and E are on Q

_{1}, lines AB and DE are tangent to Q

_{2}at B and D, respectively, lines AE and BD meet at point P. Prove that

(1) $\frac{AB}{AT}=\frac{ED}{ET}$ (2) <ATP + <ETP = 180þ.

CGMO 2012.5

As shown in the figure below, the incircle of ABC is tangent to sides AB and AC at D and E respectively, and O is the circumcenter of BCI. Prove that <ODB = <OEC.

CGMO 2013.2

As shown in the figure below, ABCD is a trapezoid, AB // CD. The sides DA, AB, BC are tangent to circle (O

As shown in the figure below, ABCD is a trapezoid, AB // CD. The sides DA, AB, BC are tangent to circle (O

_{1}) and AB touches circle (O_{1}) at P. The sides BC, CD, DA are tangent to circle (O_{2}), and CD touches circle (O_{2}) at Q. Prove that the lines AC, BD, PQ meet at the same point.
CGMO 2013.7

As shown in the figure, circles (O

As shown in the figure, circles (O

_{1}) and (O_{2}) touches each other externally at a point T, quadrilateral ABCD is inscribed in circle (O_{1}), and the lines DA, CB are tangent to circle (O_{2}) at points E and F respectively. Line BN bisects <ABF and meets segment EF at N. Line FT meets the arc AT (not passing through the point B) at another point M different from A. Prove that M is the circumcenter of ∆BCN.
CGMO 2014.1

In the figure , circles (O

In the figure , circles (O

_{1}) and (O_{2}) intersect at two points A, B. The extension of O_{1}A meets circle (O_{2}) at C, and the extension of O_{2}A meets circle (O_{1}) at D, and through B draw BE // O_{2}A intersecting circle (O_{1}) again at E. If DE // O_{1}A, prove that DC ⊥ CO_{2}.
CGMO 2014.6

In acute triangle ABC, AB > AC. D and E are the midpoints of AB, AC respectively. The circumcircle of ADE intersects the circumcircle of BCE again at P. The circumcircle of ADE intersects the circumcircle BCD again at Q. Prove that AP = AQ.

In acute triangle ABC, AB > AC. D and E are the midpoints of AB, AC respectively. The circumcircle of ADE intersects the circumcircle of BCE again at P. The circumcircle of ADE intersects the circumcircle BCD again at Q. Prove that AP = AQ.

CGMO 2015.1

Let ∆ABC be an acute-angled triangle with AB > AC, O be its circumcenter and D the midpoint of side BC. The circle with diameter AD meets sides AB,AC again at points E, F respectively. The line passing through D parallel to AO meets EF at M. Show that EM = MF.

Let ∆ABC be an acute-angled triangle with AB > AC, O be its circumcenter and D the midpoint of side BC. The circle with diameter AD meets sides AB,AC again at points E, F respectively. The line passing through D parallel to AO meets EF at M. Show that EM = MF.

CGMO 2015.6

Let Γ

Let Γ

_{1}and Γ_{2}be two non-overlapping circles. A,C are on Γ_{1}and B,D are on Γ_{2}such that AB is an external common tangent to the two circles, and CD is an internal common tangent to the two circles. AC and BD meet at E. F is a point on Γ_{1}, the tangent line to Γ_{1}at F meets the perpendicular bisector of EF at M. MG is a line tangent to Γ_{2}at G. Prove that MF = MG.
(1)
Determine a necessary and sufficient condition on a, b and c if there exists a
unique point P (P ≠ B, P ≠C) on the arc BC of Γ not passing through point A such that PA = PB + PC.

(2) Let P
be the unique point stated in (1). If AP bisects BC, prove that <BAC < 60þ.

In acute
triangle ABC, AB < AC, I is its incenter, D is the foot of perpendicular
from I to BC, altitude AH meets BI,CI at P,Q respectively. Let O be the
circumcenter of ∆IPQ, extend AO to meet BC at L. Circumcircle of ∆AIL meets BC
again at N. Prove that ${\frac{BD}{CD}=\frac{BN}{CN}}$ .

CGMO 2017.2

Given quadrilateral ABCD such that <BAD + 2<BCD = 180þ. Let E be the intersection of BD and the internal bisector of <BAD. The perpendicular bisector of AE intersects CB,CD at X, Y, respectively. Prove that A,C,X, Y are concyclic.

Given quadrilateral ABCD such that <BAD + 2<BCD = 180þ. Let E be the intersection of BD and the internal bisector of <BAD. The perpendicular bisector of AE intersects CB,CD at X, Y, respectively. Prove that A,C,X, Y are concyclic.

CGMO 2017.7

Let the ABCD be a cyclic quadrilateral with circumcircle ω

Let the ABCD be a cyclic quadrilateral with circumcircle ω

_{1}.Lines AC and BD intersect at point E, and lines AD,BC intersect at point F .Circle ω_{2 }is tangent to segments EB, EC at points M, N respectively, and intersects with circle ω_{1 }at points Q, R. Lines BC, AD intersect line MN at S, T respectively. Show that Q, R, S, T are concyclic.
CGMO 2018.2

Points $D,E$ lie on segments $AB,AC$ of $\triangle ABC$ such that $DE\parallel BC$. Let $O_1,O_2$ be the circumcenters of $\triangle ABE, \triangle ACD$ respectively. Line $O_1O _2$ meets $AC$ at $P$, and $AB$ at $Q$. Let $O$ be the circumcenter of $\triangle APQ$, and $M$ be the intersection of $AO$ extended and $BC$. Prove that $M$ is the midpoint of $BC$.

CGMO 2018.8

Let $I$ be the incenter of triangle $ABC$. The tangent point of $\odot I$ on $AB,AC$ is $D,E$, respectively. Let $BI \cap AC = F$, $CI \cap AB = G$, $DE \cap BI = M$, $DE \cap CI = N$, $DE \cap FG = P$, $BC \cap IP = Q$. Prove that $BC = 2MN$ is equivalent to $IQ = 2IP$.

Points $D,E$ lie on segments $AB,AC$ of $\triangle ABC$ such that $DE\parallel BC$. Let $O_1,O_2$ be the circumcenters of $\triangle ABE, \triangle ACD$ respectively. Line $O_1O _2$ meets $AC$ at $P$, and $AB$ at $Q$. Let $O$ be the circumcenter of $\triangle APQ$, and $M$ be the intersection of $AO$ extended and $BC$. Prove that $M$ is the midpoint of $BC$.

CGMO 2018.8

Let $I$ be the incenter of triangle $ABC$. The tangent point of $\odot I$ on $AB,AC$ is $D,E$, respectively. Let $BI \cap AC = F$, $CI \cap AB = G$, $DE \cap BI = M$, $DE \cap CI = N$, $DE \cap FG = P$, $BC \cap IP = Q$. Prove that $BC = 2MN$ is equivalent to $IQ = 2IP$.

## Δεν υπάρχουν σχόλια:

## Δημοσίευση σχολίου