International Mathematics Competition

Let be a real matrix such that .

1. Prove that there is a unique real matrix that satisfies the equation
2. Express in terms of .

Let be a positive integer. Compute the number of words (finite sequences of letters) that satisfy the following three properties:

1. consists of letters, all of them are from the alphabet .
2. contains an even number of letters .
3. contains an even number of letters .

(For example, for , there are 6 such words: , , , , and )

Let and be real matrices such that where is the identity matrix. Prove that

Find all twice continuously differentiable functions satisfying for all .

Evaluate the product

A four-digit number is called very good if the system of linear equations in the variables , , and has at least two solutions. Find all very good s in the 21st century (the 21st century starts in 2001 and ends in 2100).

Determine whether there exist an odd positive integer and matrices y with integer entries, that satisfy the following conditions:

1. .
2. .
3. .

Let be the set of composite positive integers. For each , let be the smallest positive integer such that is divisible by . Determine whether the following series converges:

Let and be two sequences of positive numbers. Show that the following statements are equivalent:

1. There is a sequencen of positive numbers such that and both converge.
2. converges.

Let be a continuous function such that exists (it may be finite or infinite). Prove that

Let be a real function. Prove or disprove each of the following statements:

1. If is continuous and then is monotonic.
2. If is monotonic and then is continuous.
3. If is monotonic and is continuous then .

Let and be real symmetric matrices with all eigenvalues strictly greater than . Let be a real eigenvalue of matrix . Prove that .

Let be a complex number with . Prove that .

Let be a fixed positive integer. Determine the smallest possible rank of an matrix that has zeros along the main diagonal and strictly positive real numbers off the main diagonal.

Let . Prove that:

Let be a continuous function. Suppose that for any , the graph of can be moved to the graph of using only a translation or a rotation. Does this imply that , for some real numbers and ?

Let be the matrix whose entry is for all . What is the rank of ?

Let be a function such that is a polynomial for every . Does it follow that is a polynomial?

Let be a real matrix and be a real matrix such that Find .

Let be continuous and non-decreasing functions such that for each we have and . Prove that

and are square complex matrices of the same size and . Show that .

Suppose that in a not necessarily commutative ring the square of any element is . Prove that for any three elements .