Gelfand Correspondence Program in Mathematics
Introductory Assignment
Thank you for your interest in our Program.
Below are some problems we would like you to try. These problems have
different degrees of difficulty, and you are not expected to be able to
solve all of them.
If you decide to participate in GCPM, please solve these problems
and mail us your solutions. This will only serve us to get acquainted
with your abilities; you will be accepted to GCPM independently of your
results.
When writing down your solutions, please
include all the explanations to your solutions and not only the answers
to problems. If you cannot solve a problem but have some ideas or
a partial solution, write them down. Try to explain your ideas and
answers as clearly as possible. Please keep this page for future
reference and do all your work on separate sheets of paper, numbering
each problem. Send your work to: Gelfand Correspondence Program in
Mathematics, Department of Mathematics,
Rutgers the State University, 110 Frelinghuysen Road, Piscataway,
N.J. 088558019.
Please see directions before you start
Problems 2 and 3 deal with the inhabitants of the city Boole.
Some of the inhabitants of the city Boole are liars and always
lie, all others always tell the truth. 
2.  Once ten inhabitants of the city Boole met in a room and each
one said: "all the rest of you are liars". How many people in the room
were liars? 
3.  Once several inhabitants of the city Boole met in a room. Three
of them made the following statements:
The first one said: 
a) There are no more than three people here. 
b) All of us are liars. 

The second one said: 
a) There are no more than four people here. 
b) Not everybody here is a liar. 

The third one said: 
a) There are five people here. 
b) Three of us are liars. 
How many people were in the room and how many of them were liars?

4.  Each of the equations below is missing a pair of numerators:
a) 
?
7

 
?
5

= 
1
35

, b) 
?
5

 
?
7

= 
1
35

. 

Assume that the numerators are positive integers. Find as many pairs of
numerators as you can. Do not forget that mixed fractions are allowed.
Example: Let us check whether 2 and 4 form a solution for a).

2
7

 
4
5

= 
10
35

 
28
35

=  
18
35

¹ 
1
35

. 

So the pair (2,4) is not a solution for a).

5.  Is the sum 1+2+3+4+ ¼+98+99+100 even or odd?
Note: The "¼" stands for the missing terms, and there are
100 terms in all.

6.  What is the measurement, in degrees, of the angle between the
diagonals, AB and BC, of the adjacent faces of the cube?

7.  Remove the parentheses (multiply out):
(1x)(1+x+x^{2}+x^{3}+x^{4}+x^{5}+ ¼+x^{99}+x^{100}) . 

For example: (1+x)(1+x+x^{2}) = 1+x+x^{2}+x+x^{2}+x^{3} = 1+2x+2x^{2}+x^{3}.

8.  The difference of two numbers is 0.01. Is it possible that
the difference of their squares is more than 1000?

9.  In a box, there are fresh cucumbers which weigh 100
pounds. Each cucumber is composed of 99% water. After some time,
the cucumbers dried out. Now each cucumber is composed of 98%
water. How much do the cucumbers weigh now?

Below are three more challenging problems:
 Four points, A,B,C and D, are the corners of a square.
Each side of the square is 10 feet long. Draw a system of straight
lines connecting the four points so that the total length of the lines
is 28 feet or less.
Below are three examples that do NOT work:



Total length is 30 feet 
Total length is more than 28.2 feet 
Total length is more than 28 feet 
 Remove the parentheses (multiply out):
(1x)^{2}(1+2x+3x^{2}+4x^{3}+5x^{4}+ ¼+99x^{98}+100x^{99}+101x^{100}) . 

 The shaded region in the figure is bounded by three
semicircles. Cut this region into four congruent parts, i.e. parts
of equal size and shape.
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