This is the booklet containing the problems used for training and selection purposes.
Here I compiled number theory problems from IMO 1959 to 2015. Cheers.
Here is a compilation of all number theory problems from APMO 1989 to 2015.
Floor function of where is a real number, denoted by is the largest integer less than or equal to . For example, , and . Is it true that, for integer , we have ? Do you think is correct? You can proceed if your answers to both questions are yes. It is obvious that , and we say that is the fractional part of . For example, . Clearly, .
Problem 1: Prove that for real and integer , we have .
Problem 2: Prove that for real , we have .
Solution 3: We can prove this easily this way. Write and .
because is an integer. See the second exercise in the link above. This inequality can be treated as the triangle inequality of floor function.
Now, we will discuss Ceiling function. Ceiling of a real number , is the smallest integer greater than or equal to . So, and as well. But whereas . It is clear that if is an integer, then . Which one is true?
The answer should be quite obvious! More precisely, if is not an integer then . Make sense of the following identities:
You should do the following exercises to make sure you understand these functions well.
Problem 4: Is it true that for an integer ? If so, why?
Problem 5: Is it true that for an integer ? If so, why?
Problem 6: Prove that, if is an integer, , otherwise .
Problem 7: If is an integer, , otherwise .
Problem 8: If is an integer, , otherwise .
Here is a tutorial on how to create documents using latex. It just covers the basics.
Euler’s identity is regarded as the most beautiful equation in mathematics, since it combines five very important numbers . The identity is:
The identity is a special case of the following general one:
It has many proofs. Here is an elegant one. Let . Then
Now let’s integrate both sides. As we know and , we have
for some constant (since it is indefinite integration). We get
We need to find the value of , so let’s set , and we get
So and we have
Set and we have
Here is a tutorial on chicken mcnugget theorem.