# Math Circle

The main objectives of our circle are

• Introduce students to the kind of math that is both important and fun,
• Encourage students to think logically and solve challenging problems;

Every meeting of the circle will consist of two parts. In the lecture part, students will learn interesting math subjects; in the recitation part, student  will solve math problems. Prerequisites are what’s taught in the middle school math course plus interest in math.

Here are some examples of the topics covered in the lectures:

• How to cut an apple into five parts, rearrange the parts and produce two apples of the same size?
• Can we solve all math problems?
• What is non-Euclidean geometry (can the sum of angles in a triangle be not 180 degrees)?
• What is non-Archimedean arithmetic (can there be a number p such that 1+1+1+… is never greater than p)?

Here are some types of problem that  will be discussed during   recitations:

• Three mice found a cake. How should they arrange dividing the cake so that each mouse would be sure to get at least 1/3 of the cake (for two mice the solution is easy: one mouse divides the cake, the other mouse chooses the bigger part).
• 24 cannibals attended a cannibal party. It is known that among every three cannibals at the party, one ate another. Show that after the party there were 8 nested cannibals (cannibal 1 ate cannibal 2, who ate cannibal 3, etc. ).
• Take the number 100! = 1*2*3*…*100 (product of all integers from 1 to 100), and take the sum of all it’s digits (for example, sum of all digits of the number 2015 is 2+0+1+5=8). Now, take the sum of all digits of the result. Repeat this procedure until you get a one-digit number. What is this number?
• There are 10 red, 8 blue, 8 green and 4 yellow crayons in a box. At most how many crayons can one take out of the box, so that no more than 6 blue crayons will be surely left in the box?

There is  some beautiful math behind each of these problems! Solving these problems you will learn combinatorics, measure theory, Dilworth theorem, the Dirichlet (pigeonhall) principle and more.

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