MathDB

2013 HMMT Algebra #10: Minimum Value of Rational Expression

Source:

2/17/2013

Let

N

be a positive integer whose decimal representation contains

11235

as a contiguous substring, and let

k

be a positive integer such that

10^k>N

. Find the minimum possible value of

\dfrac{10^k-1}{\gcd(N,10^k-1)}.

HMMTmodular arithmetic

2013 Team #10: Chim Tu

Source:

3/1/2013

Chim Tu has a large rectangular table. On it, there are ﬁnitely many pieces of paper with nonoverlapping interiors, each one in the shape of a convex polygon. At each step, Chim Tu is allowed to slide one piece of paper in a straight line such that its interior does not touch any other piece of paper during the slide. Can Chim Tu always slide all the pieces of paper off the table in ﬁnitely many steps?

geometryrectangleanalytic geometry

2013 HMMT Guts #10: Wesyu and her Cao Pasture

Source:

3/26/2013

Wesyu is a farmer, and she's building a cao (a relative of the cow) pasture. Shw starts with a triangle

A_0A_1A_2

where angle

A_0

is

90^\circ

, angle

A_1

is

60^\circ

, and

A_0A_1

is

1

. She then extends the pasture. FIrst, she extends

A_2A_0

to

A_3

such that

A_3A_0=\dfrac12A_2A_0

and the new pasture is triangle

A_1A_2A_3

. Next, she extends

A_3A_1

to

A_4

such that

A_4A_1=\dfrac16A_3A_1

. She continues, each time extending

A_nA_{n-2}

to

A_{n+1}

such that

A_{n+1}A_{n-2}=\dfrac1{2^n-2}A_nA_{n-2}

. What is the smallest

K

such that her pasture never exceeds an area of

K

?

HMMTgeometry

2013 HMMT Geometry # 10

Source:

3/3/2024

Triangle

ABC

is inscribed in a circle

\omega

. Let the bisector of angle

A

meet

\omega

at

D

and

BC

at

E

. Let the reflections of

A

across

D

and

C

be

D'

and

C'

, respectively. Suppose that

\angle A = 60^o

,

AB = 3

, and

AE = 4

. If the tangent to

\omega

at

A

meets line

BC

at

P

, and the circumcircle of APD' meets line

BC

at

F

(other than

P

), compute

FC'

.

geometry

10

Problems(4)