2005 JHMT

Part of JHMT problems

Subcontests

(11)

1

2005 JHMT Geometry #10

ABCD

has sides in the ratio of

\sqrt2

1

DEC

is an isosceles right triangle, with

E

inside the rectangle, find angle

\angle AEB

1

2005 JHMT Geometry #9

A square with side length

1

is inscribed in a hemisphere such that one side of the square is on the hemisphere’s diameter. What is the semicircle’s perimeter?

1

2005 JHMT Geometry #8

DEAF

is constructed inside the

30^o-60^o-90^o

ABC

, with the hypotenuse

BC = 4

D

BC

AC

, and F on side

AB

. What is the side length of the square?

1

2005 JHMT Geometry #7

Equilateral triangle

ABC

is inscribed in a circle with radius

6

. Find the area of the region enclosed by

AB

AC

, and the minor arc

BC

1

2005 JHMT Geometry #6

DE

cuts through triangle

ABC

DF

BE

BD =DF = 10

AD = BE = 25

BC

. https://cdn.artofproblemsolving.com/attachments/0/e/d6e3d7c1f9bd15f4573ccd5fc67c190b9cf7e9.png

1

2005 JHMT Geometry #5

Equilateral triangle

ABC

AD = DB = FG = AE = EC = 4

BF = GC = 2

D

G

are drawn perpendiculars to

EF

intersecting at

H

I

, respectively. The three polygons

ECGI

FGI

BFHD

are rearranged to

EANL

MNK

AMJD

so that the rectangle

HLKJ

is formed. Find its area. https://cdn.artofproblemsolving.com/attachments/d/4/7e6667f0f0544b6fbc860f8d86c8ceaaf85cc1.png

1

2005 JHMT Geometry #4

Given an isosceles trapezoid

ABCD

AB = 6

CD = 12

36

BC

1

2005 JHMT Geometry #3

Isosceles triangle

ABC

\angle BAC = 135^o

AB = 2

. What is its area?

1

2005 JHMT Geometry #2

Regular hexagon

ABCDEF

is inscribed in rectangle

PQRS

AB = 1

B

PQ

C

QR

D

E

RS

F

SP

. What is the area of

PQRS

1

2005 JHMT Geometry #1

A circle with diameter

23

is cut by a chord

AC

. Two different circles can be inscribed between the large circle and

AC

. Find the sum of the two radii.

1

2005 JHMT Team Round - Johns Hopkins Mathematics Tournament

p1. Consider the following function

f(x) = \left(\frac12 \right)^x - \left(\frac12 \right)^{x+1}

. Evaluate the infinite sum

f(1) + f(2) + f(3) + f(4) +...

p2. Find the area of the shape bounded by the following relations

y \le |x| -2

y \ge |x| - 4

y \le 0

where |x| denotes the absolute value of

x

.
p3. An equilateral triangle with side length

6

is inscribed inside a circle. What is the diameter of the largest circle that can fit in the circle but outside of the triangle?
p4. Given

\sin x - \tan x = \sin x \tan x

x

in the interval

(0, 2\pi)

, exclusive.
p5. How many rectangles are there in a

6

6

square grid?
p6. Find the lateral surface area of a cone with radius

3

4

9

positive integer scores on a

10

-point quiz, the mean is

8

, the median is

8

, and the mode is

7

. Determine the maximum number of perfect scores possible on this test.
p8. If

i =\sqrt{-1}

, evaluate the following continued fraction:

2i +\frac{1}{2i +\frac{1}{2i+ \frac{1}{2i+...}}}

p9. The cubic polynomial

x^3-px^2+px-6

p, q

r

(1-p)(1-q)(1-r)

?
p10. (Variant on a Classic.) Gilnor is a merchant from Cutlass, a town where

10\%

of the merchants are thieves. The police utilize a lie detector that is

90\%

accurate to see if Gilnor is one of the thieves. According to the device, Gilnor is a thief. What is the probability that he really is one?
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.