MathDB

2021 Algebra/NT #2: raised to the log power

Source:

5/30/2021

Compute the number of ordered pairs of integers

(a, b),

with

2 \le a, b \le 2021,

that satisfy the equation

a^{\log_b \left(a^{-4}\right)} = b^{\log_a \left(ba^{-3}\right)}.

algebra

2021 Combo #2: The Knockout Torny

Source:

5/30/2021

Ava and Tiffany participate in a knockout tournament consisting of a total of

32

players. In each of

5

rounds, the remaining players are paired uniformly at random. In each pair, both players are equally likely to win, and the loser is knocked out of the tournament. The probability that Ava and Tiffany play each other during the tournament is

\tfrac{a}{b},

where

a

and

b

are relatively prime positive integers. Compute

100a + b.

Combo

2021 Geo #2: Recursive Area

Source:

5/30/2021

Let

X_0

be the interior of a triangle with side lengths

3, 4,

and

5

. For all positive integers

n

, define

X_n

to be the set of points within

1

unit of some point in

X_{n-1}

. The area of the region outside

X_{20}

but inside

X_{21}

can be written as

a\pi + b

, for integers

a

and

b

. Compute

100a + b

.

geometry

2021 Team #2

Source:

6/27/2021

Let

ABC

be a right triangle with

\angle A= 90^{\circ}

. A circle

\omega

centered on

BC

is tangent to

AB

at

D

and

AC

at

E

. Let

F

and

G

be the intersections of

\omega

and

BC

so that

F

lies between

B

and

G

. If lines

DG

and

EF

intersect at

X

, show that

AX=AD.

geometryright triangle

2

Problems(4)