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 EGCPM, 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 EGCPM 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: info.egcpm@gmail.com
Please see Directions before you start
1. Which is bigger, 

or 

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:
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:
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).

5.  Is the sum 1+2+3+4+ ¼+98+99+100 even or odd? 
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):
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? 
 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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