Sports action

Ticketamerica.com has chicago blackhawks tickets, united center tickets and nhl tickets in all cities. 

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Country music concerts

Ticketamerica.com has Blake Shelton tickets and Dierks Bentley concerts and tours as well as Trace Adkins events and seating charts.

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Sudoku puzzle

Okay, puzzle addition: Sudoku puzzle. Available in lot many variations, in easy and medium difficulty levels. Enjoy!

Divisors problem (one more)

Here is a problem w.r.t divisors: For a given positive integers B and X find the number of positive integers N such that number N*X has at least one divisor D such that N < D <= B (Courtesy: www.codechef.com).

The number N, can be represented as a function of B and X, namely f(B, X). Prove the following:

f(B, 1) = 0
f(B, 2) = Floor(B / 2)
f(B, 3) = 2 * Floor(B / 3) - Floor(B / 6)
f(B, 4) = Floor(B / 2) + Floor(B / 4) - Floor(B / 6)
f(B, 5) = 4 * Floor(B / 5) - Floor(B / 10) - Floor((B + Floor(B / 15)) / 8) - Floor((B + Floor(B / 20)) / 7)

Spanish and French architectures

I was looking at some pictures and magazines, where photos of some spanish villas, and french houses and articles are there. From those, I think spanish and french architecture styles are along these lines.

French use hues of pink, olive green and brown heavily. French windows are long and sleek. Spanish villas typically use terracotta bricks heavily. Windows are characterized by moorish arches, less-height, fat (stubbed) windows called Matador windows. (I will upload photos for all these soon).

Also, one can say that spanish villas have stairs from inside the houses, and a patio set on ground level, slightly elevated, which leads to the staircase. Ceilings are also of varied heights in spanish villas.

Overall, french villas used the purity of colors whereas, spanish villas showed the moslem influences they got over centuries.

About perfect numbers

Prove the following about perfect numbers:

1. Any odd perfect number can't be a square number.
2. The sum of reciprocals of factors of a perfect number is equal to that perfect number.
3. Prove that odd perfect numbers are of the form:

N=p^(4x+1)* Q^2, where p is of the form (4y+1).

4. Also prove that no odd perfect number is a multiple of 105.
5. The smallest factor of an odd perfect number <= (2k + 6)/3, where k is the number of it's prime factors.
6. Prove that, if p is of the form (12k + 1), corresponding odd perfect number will always be of the form (12k + 1).

Math problems for High-school kids

Here are some interesting math problems for high-school grade.

1. When written in decimal notation, how many trailing zeroes will be there in 32! In general, what is the number of zeroes for n! (any positive n).
2. How many numbers from 1 to 100 (inclusive) have odd number of factors? What type of numbers are those? Why? (Factors include one and that number as well).
3. Which number(s) from 100 to 200 (inclusive) has the highest number of factors? How many does it have? (It is better not to use brute-force, try to use "some" method).
4. Prove that the number of factors of a number n, is always not greater than 2 * floor(sqrt(n)).
5. What is the biggest number n, which has number of factors equal to 2 * floor(sqrt(n))?
6. Prove that, x^4n + y^4n = z^4n is not possible whenever x mod 6 = 1 or -1 (5), and y mod 6 = 3, and z mod 6 = 2 or -2 (4)?
7. Prove that the only possible cases for x, y and z above are:
x mod 6 = 0, y mod 6 = 1 or -1, z mod 6 = 1 or -1.
x mod 6 = 3, y mod 6 = 2 or -2, z mod 6 = 1 or -1.
This reduces the cases for the theorem (fermat's last theorem's special case), from a possible 36 combinations to a mere 4.
8. Prove that the powers of 4, always end with either 4 or 6 (in decimal notation). With what digit does 3^1000 end? What number do powers of 6, end with? What about powers of 5?
9. Can you extend the above observation to two/three digit numbers and beyond? For example, with what digits do powers of 3456 end?
10. Prove that for any number a (greater than 2), a^n - 1 (n greater than 1), is a composite number always. Also prove that a^(2*n+1) + 1 is always a composite number?

11. Prove that in x^4 + y^4 = z^4, at least one of x or y is divisible by 5. (If you take the irreducible forms of x and y, then it follows that exactly one of them is divisible by 5).