Students in the Library

Five students visited the library on a certain day. The library is open from 8 AM - 8 PM on all days. Each went for a lunch at some time and returned after it. Any two of them met in the library. Prove that at some moment of time three of them were in the library. Show also that this is not always true for four students.

Hint (posted on July 27, 2005): Draw interval lines for each student for the period during which he was present.

Girlie company

25 boys and 25 girls are to be seated in a round table with 50 seats. Prove that no matter what the arrangement is, there would be a person who is seated in-between two girls.

The hole principle

Let A = (a_1, a_2,...,a_n) be a sequence on real numbers with n > k m (k and m - integers). Prove that there is a subsequence of length k +1 of A of which is increasing OR a subsequence of length m +1 of A of which is decreasing.

A non-rational path

Consider R^2 - the Euclidean plane consisting of 2-tuples of real numbers. Remove all the points that have rational co-ordinates from it. For example (1/4, 5/100) should be removed, (1/2, sqrt(7)) should NOT be removed. We are left with what is denoted R^2 - Q^2; Q being the rational points. Pick any two points a and b in R^2 - Q^2. Is it possible to find a path (a line - not nessesarily straight) from a and b in R^2 - Q^2?

If the answer is yes, give a precise path and if the answer is no, why?

Count the sum of the determinants

To each positive integer with n^2 number of digits, we can associate a corresponding determinant by writing the digits in order across the rows.

That is for n = 2, we have n^2 = 4. Thus take all 4-digit positive integers. For 7615, we have the determinant
| 7 6 |
| 1 5 |
which is equal to 35 - 6 = 29.

Can you find the sum of all determinants associated with n^2-digit integers as a function of n?

Just as a motivating example - for n = 1, we have only 9 determinants and their sum is 45. For n = 2, we have 9000 determinants.

A computational Problem

Let x = .1234567891011121314...998999, where the digits are obtained by writing the integers 1 through 999 in order. Find the 1983rd digit to the right of the decimal point.