2006 JHMT

Part of JHMT problems

Subcontests

(9)

1

2006 JHMT Geometry #8

P

Q

R

are externally tangent to one another. The external tangent of

P

Q

that does not intersect

R

P

Q

P_Q

Q_P

, respectively.

Q_R

R_Q

R_P

P_R

are defined similarly. If the radius of

Q

4

\overline{Q_PP_Q} \parallel \overline{R_QQ_R}

R_PP_R

1

2006 JHMT Geometry #7

AD

is the angle bisector of the right triangle

ABC

\angle ABC = 60^o

\angle BCA = 90^o

E

\overline{AB}

so that the line parallel to

\overline{DE}

C

\overline{AE}

\angle EDB

1

2006 JHMT Geometry # 6

A right cylinder is inscribed in a right circular cone with height

2

2

so that the cylinder’s bottom base sits on the cone’s base. What is the maximum possible surface area of the cylinder?

1

2006 JHMT Geometry # 5

An ant is on the bottom edge of a right circular cone with base area

\pi

and slant length

6

. What is the shortest distance that the ant has to travel to loop around the cone and come back to its starting position?

1

2006 JHMT Geometry # 4

DEFG

is contained in equilateral triangle

ABC

E

\overline{AC}

G

\overline{AD}

F

as the midpoint of

\overline{BC}

AD

DE = 6

1

2006 JHMT Geometry # 3

ABCD

is folded in half so that the vertices

D

B

coincide, creating the crease

\overline{EF}

E

\overline{AD}

F

\overline{BC}

O

be the midpoint of

\overline{EF}

DOC

DCF

are congruent, what is the ratio

BC : CD

1

2006 JHMT Geometry # 2

If two altitudes of a triangle have length

12

4

, what integral lengths can the third altitude attain?

1

2006 JHMT Geometry # 1

ZINC

is constructed in the interior of hexagon

CARBON

. What is the area of triangle

BIO

1

2006 JHMT Team Round - Johns Hopkins Mathematics Tournament

S

S =\frac{10000^2 - 1}{\sqrt{10000^2 - 19999}}

p2. Starting on a triangular face of a right triangular prism and allowing moves to only adjacent faces, how many ways can you pass through each of the other four faces and return to the first face in five moves?
p3. Given that

(a + b) + (b + c) + (c + a) = 18

\frac{1}{a + b}+\frac{1}{b + c}+ \frac{1}{c + a}=\frac59,

\frac{c}{a + b}+\frac{a}{b + c}+\frac{b}{c + a}.

p4. Find all primes

p

2^{p+1} + p^3 - p^2 - p

is prime.
p5. In right triangle

ABC

with the right angle at

A

AF

AH

is the altitude, and

AE

is the angle bisector. If

\angle EAF = 30^o

\angle BAH

in degrees.
p6. For which integers

a

does the equation

(1 - a)(a - x)(x- 1) = ax

not have two distinct real roots of

x

?
p7. Given that

a^2 + b^2 - ab - b +\frac13 = 0

, solve for all

(a, b)

E

\overline{AB}

of the unit square

ABCD

F

\overline{BC}

AE = BF

G

is the intersection of

\overline{DE}

\overline{AF}

. As the location of

E

varies along side

\overline{AB}

, what is the minimum length of

\overline{BG}

?
p9. Sam and Susan are taking turns shooting a basketball. Sam goes first and has probability

P

of missing any shot, while Susan has probability

P

of making any shot. What must

P

be so that Susan has a

50\%

chance of making the first shot?
p10. Quadrilateral

ABCD

AB = BC = CD = 7

AD = 13

\angle BCD = 2\angle DAB

\angle ABC = 2\angle CDA

. Find its area.
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.