MathDB

Speed P10

Source:

10/16/2021

Let

ABCD

be a unit square in the plane. Points

X

and

Y

are chosen independently and uniformly at random on the perimeter of

ABCD

. If the expected value of the area of triangle

\triangle AXY

can be expressed as

\frac{m}{n}

for relatively prime positive integers

m

and

n

, compute

m+n

.
Proposed by Nathan Xiong

MOAA 2021speed

Accuracy P10

Source:

10/16/2021

In

\triangle ABC

, let

X

and

Y

be points on segment

BC

such that

AX=XB=20

and

AY=YC=21

. Let

J

be the

A

-excenter of triangle

\triangle AXY

. Given that

J

lies on the circumcircle of

\triangle ABC

, the length of

BC

can be expressed as

\frac{m}{n}

for relatively prime positive integers

m

and

n

. Compute

m+n

.
Proposed by Andrew Wen

MOAA 2021Accuracy

Team Round P10

Source:

10/16/2021

For how many nonempty subsets

S \subseteq \{1, 2, \ldots , 10\}

is the sum of all elements in

S

even?
Proposed by Andrew Wen

MOAA 2021team

Gunga P10

Source:

10/16/2021

We say that an ordered pair

(a,b)

of positive integers with

a>b

is square-ish if both

a+b

and

a-b

are perfect squares. For example,

(17,8)

is square-ish because

17+8=25

and

17-8=9

are both perfect squares. How many square-ish pairs

(a,b)

with

a+b<100

are there?
Proposed by Nathan Xiong

MOAA 2021Gunga

10

Problems(4)