2015 JHMT

Part of JHMT problems

Subcontests

(10)

1

2015 JHMT Geometry #5

ABCDEF

are evenly spaced on a unit circle and line segments

AD

DF

FB

BE

EC

CA

are drawn. The line segments intersect each other at seven points inside the circle. Denote these intersections

p_1

p_2

...

p_7

p_7

is the center of the circle. What is the area of the

12

A_{p_1}B_{p_2}C_{p_3}D_{p_4}E_{p_5}F_{p_6}

? https://cdn.artofproblemsolving.com/attachments/9/2/645b3c31b0bb7f81d0c0b86e13b27ec5b32864.png

1

2015 JHMT Geometry #9

ABC

, two angle trisectors of

A

BC

D

E

respectively so that

B,D,E,C

comes in order. If we have

BD = 3

DE = 1

EC = 2

\angle DAE

1

2015 JHMT Geometry #7

ABC

is isoceles with

AB = AC

D

AB

such that the inradius of

ADC

and the inradius of

BDC

\frac{3-\sqrt3}{2}

. The inradius of

ABC

1

. What is the length of

BD

1

2015 JHMT Geometry #4

Suppose you have

15

circles of radius

1

. Compute the side length of the smallest equilateral triangle that could possibly contain all the circles, if you are free to arrange them in any shape, provided they don’t overlap.

1

RMT 2015 Geometry #8

Consider the parallelogram

ABCD

CD = 8

BC = 14

. The diagonals

\overline{AC}

\overline{BD}

E

AC = 16

. Consider a point

F

\overline{ED}

F D =\frac{\sqrt{66}}{3}

CF

1

RMT 2015 Geometry #7

A unit sphere is centered at

(0, 0, 1)

. There is a point light source located at

(1, 0, 4)

that sends out light uniformly in every direction but is blocked by the sphere. What is the area of the sphere’s shadow on the

x-y

plane? (A point

(a, b, c)

denotes the point in three dimensions with

x

a

y

b

z

c

1

RMT 2015 Geometry #6

ABC

D

E

BC

BD = DE = EC

F

be the point on

AB

\frac{AF}{F B}= 2

G

AC

\frac{AG}{GC} =\frac12

P

be the intersection of

DG

EF

AP

BC

X

\frac{BX}{XC}

1

RMT 2015 Geometry #3

Consider a triangular pyramid

ABCD

with equilateral base

ABC

1

AD = BD =CD

\angle ADB = \angle BDC = \angle ADC = 90^o

. Find the volume of

ABCD

1

RMT 2015 Geometry #2

In a certain right triangle, dropping an altitude to the hypotenuse divides the hypotenuse into two segments of length

2

3

respectively. What is the area of the triangle?

1

RMT 2015 Geometry #1

Clyde is making a Pacman sticker to put on his laptop. A Pacman sticker is a circular sticker of radius

3

inches with a sector of

120^o

cut out. What is the perimeter of the Pacman sticker in inches?