MathDB

p1. Determine which of the following numbers is greater :

99!

50^{99}

.
p2. Let

x , y

be positive real numbers with

x > y

that satisfy the relation

x^2 +y^2 = axy

where

a

is a real number greater than

2

. Find all possible values of

a

that make

\frac{x + y}{x - y}

an integer.
[url=https://artofproblemsolving.com/community/c1068820h2917784p26063850]p3. In the figure below

ABCD

is a square. https://cdn.artofproblemsolving.com/attachments/f/3/999a7e4eb004edcbe11951fc2adab1ebaa2243.png Segment

AE

3

units and segment

EF

1

unit. The angles

\angle AED

and

\angle BFA

are right. Calculate the length of the segments

FC

[url=https://artofproblemsolving.com/community/c6h2945857p26372014]p4. A design

X

is an array of the digits

1,2,..., 9

in the shape of an

X

, for example, https://cdn.artofproblemsolving.com/attachments/8/e/d371a2cd442cb7a8784e1cc7635344df722e20.png We will say that a design

X

is balanced if the sum of the numbers of each of the diagonals match. Determine the number of designs

X

that are balanced.

Problem Statement