2024 CMIMC Geometry

Part of 2024 CMIMC

Subcontests

(10)

1

2024 Geo Problem 10

\Omega

be a unit circle with diameter

AB

O

C

D

\Omega

and lie on the same side of

AB

\angle CAB = 50^\circ

\angle DBA = 70^\circ

AD

BC

E

. Let the perpendicular from

O

CD

intersect the perpendicular from

E

AB

F

. Find the length of

OF

.
Proposed by Puhua Cheng

1

2024 Geo Problem 9

ABCD

is inscribed in a circle such that the midpoints of its sides also lie on a (different) circle. Let

M

N

be the midpoints of

\overline{AB}

\overline{CD}

respectively, and let

P

be the foot of the perpendicular from the intersection of

\overline{AC}

\overline{BD}

\overline{BC}

. If the side lengths of

ABCD

1

3

\sqrt 2

2\sqrt 2

in some order, compute the greatest possible area of the circumcircle of triangle

MNP

.
Proposed by Connor Gordon

1

2024 Geo Problem 8

\omega

\Omega

be circles of radius

1

R>1

respectively that are internally tangent at a point

P

. Two tangent lines to

\omega

are drawn such that they meet

\Omega

at only three points

A

B

C

, none of which are equal to

P

ABC

has side lengths in a ratio of

3:4:5

, find the sum of all possible values of

R

.
Proposed by Connor Gordon

1

2024 Geo Problem 7

An irregular octahedron has eight faces that are equilateral triangles of side length

2

. However, instead of each vertex having four "neighbors" (vertices that share an edge with it) like in a regular octahedron, for this octahedron, two of the vertices have exactly three neighbors, two of the vertices have exactly four neighbors, and two of the vertices have exactly five neighbors. Compute the volume of this octahedron.
Proposed by Connor Gordon

1

2024 Geo Problem 6

Andrew Mellon found a piece of melon that is shaped like a octagonal prism where the bases are regular. Upon slicing it in half once, he found that he created a cross-section that is an equilateral hexagon. What is the minimum possible ratio of the height of the melon piece to the side length of the base?
Proposed by Lohith Tummala

1

2024 Geo Problem 5

ABC

AB=13

BC=14

AC=15

P

\overline{BC}

D

E

be the feet of the perpendiculars from

P

\overline{AB}

\overline{AC}

respectively. If

AD=AE

, find this common length.
Proposed by Connor Gordon

1

2024 Geo Problem 4

ABC

be an equilateral triangle with side length

1

D

E

\overline{BC}

\overline{AC}

respectively such that

\triangle BDE

is right isosceles, while points

F

G

\overline{BC}

\overline{AB}

respectively such that

\triangle CFG

is right isosceles. Find the area of the intersection of

\triangle BDE

\triangle CFG

.
Proposed by Ishin Shah

1

2024 Geo Problem 3

C_1

C_2

C_3

are inside a rectangle

WXYZ

C_1

\overline{WX}

\overline{ZW}

\overline{YZ}

C_2

\overline{WX}

\overline{XY}

C_3

\overline{YZ}

C_1

C_2

. If the radii of

C_1

C_2

C_3

1

\tfrac 12

\tfrac 23

respectively, compute the area of the triangle formed by the centers of

C_1

C_2

C_3

.
Proposed by Connor Gordon

1

2024 Geo Problem 2

ABCDEF

be a regular hexagon of side length 1. Compute the area of the intersection of the circle centered at

A

passing through

C

and the circle centered at

D

passing through

E

.
Proposed by Robert Trosten

1

2024 Geo Problem 1

ABCD

be a rectangle with

AB=5

E

\overline{AB}

F

\overline {CD}

AE=CF=4

P

Q

ABCD

such that triangles

AEP

CFQ

are equilateral. If

E

P

Q

F

lie on a single line, find

\overline{BC}

.
Proposed by Connor Gordon