MathDB

p1. From the measurement of the height of nine trees obtained data as following. a) There are three different measurement results (in meters) b) All data are positive numbers c) Mean

=

median

=

mode

= 3

d) The sum of the squares of all data is

87.

Determine all possible heights of the nine trees.
p2. If

x

and

y

are integers, find the number of pairs

(x,y)

that satisfy

|x|+|y|\le 50

.
p3. The plane figure

ABCD

on the side is a trapezoid with

AB

parallel to

CD

. Points

E

and

F

lie on

CD

so that

AD

is parallel to

BE

and

AF

is parallel to

BC

. Point

H

is the intersection of

AF

with

BE

and point

G

is the intersection of

AC

with

BE

. If the length of

AB

4

cm and the length of

CD

10

cm, calculate the ratio of the area of the triangle

AGH

to the area of the trapezoid

ABCD

. https://cdn.artofproblemsolving.com/attachments/c/7/e751fa791bce62f091024932c73672a518a240.png
p4. A prospective doctor is required to intern in a hospital for five days in July

2011

. The hospital leadership gave the following rules: a) Internships may not be conducted on two consecutive days. b) The fifth day of internship can only be done after four days counted since the fourth day of internship. Suppose the fourth day of internship is the date

20

, then the fifth day of internship can only be carried out at least the date

24

. Determine the many possible schedule options for the prospective doctor.
p5. Consider the following sequences of natural numbers:

5

55

555

5555

55555

...

\underbrace{\hbox{5555...555555...}}_{\hbox{n\,\,numbers}}

. The above sequence has a rule: the

n

th term consists of

n

numbers (digits)

5

. Show that any of the terms of the sequence is divisible by

2011

Problem Statement