gianlino

In the following question

http://answers.yahoo.com/question/index?qid=201305...

just replace a cone C with a pyramide P with square base.

Again the question is to determine the probability that by picking 3 points randomly in P, the corresponding squares have pairwise non-empty intersections.

I expect the answer to be slightly smaller than in the cone version, which has been estimated to be around 0.115

I am interested in theoretical or numerical answers.

Suppose you take the spades out of a game of cards and pick them at random while calling "Ace, King" etc down to 2 in that order. The probability that your call never agrees with the drawn card is a number P close to 1/e.

http://mathworld.wolfram.com/Derangement.html

Suppose you take the Spades and the Clubs and you draw them while calling "Ace King"... down to two, twice in a row. Then the probability Q of having only miscalls should be close to P^2.

Can you find some heuristic argument to guess whether Q will be greater, less than or equal to P^2?

N,p,q are nonnegative integers. N = 30 is the smallest solution. Is there a way to find all others?

Consider the cone C over the unit disk D defined in R^3 by

z > 0 and x^2+y^2 <= (1-z)^2.

Each point (u,v,r) in C defines a disc of center (u,v) and radius r included in D.

The question is to determine the probability that by picking 3 points randomly in C, the corresponding disks have pairwise non-empty intersections.

"Randomly in C" means "uniformly with respect to the Lebesgue measure" normalised by the volume of the cone namely pi / 3.

Let S consist of any set of 10 distinct positive integers that are all less or equal than N. Prove that there will always exist at least two subsets of S whose elements sum to the same number.

Inspired by

http://answers.yahoo.com/question/index;%E2%80%A6

where it is shown that N is at least 100.

That is.... without a calculator....

This is related to the length of the royal cubit.

Let A be an even integer such that A^2 + 1 be composite. Can you always find (a,b) integers with b odd, such that a^2+b^2 = A^2+1, and 10*b > A.

Related to

http://answers.yahoo.com/question/index;_ylt=AqKV8...

If not, counterexamples welcome, smallest wins...

This q originates from

http://answers.yahoo.com/question/index;_ylt=Ap4YM...

Let PQRS a quadrilateral inscribed in a circle C. Let I be the middle of PR and J be the middle of QS.

Suppose the line QI intersects C at Q and Q' and RJ intersects C at R and R'.

Show that if SQ' is parallel to PR then PR' is parallel to QS.

This is a rewording of :

http://answers.yahoo.com/question/index;_ylt=AiuOk...

Here ABC is an acute triangle. A similar inequality was asked not long ago.

http://answers.yahoo.com/question/index;_ylt=AoNIl...

This one seems harder. I only have computer evidence of its validity.

If you set a_0 = 1, a_1 = 2 etc, then a_k is the number of integer solutions of

x + 2y + 3v + .......+ <= k.

The sum of the all the squares up to 81^2 is 81*82*163/6 = 180441 so that the sum in each group would be 20049.

Algebraic solution preferred. Thx

Circles are in geometric progression if their radii R_n are in geometric progression R_n = a r^n with r not equal to 1, and their centers can be isometrically mapped on the complex plane so that their images z_n, are also in geometric progression z_n = b z^n with b in C and |z| = r so that the sequence is self-similar.

The question is: what is the maximum length of a geometric sub-sequence of circles such that there exists a circle (not in the sequence) tangent to all of them?

Follow up from

http://in.answers.yahoo.com/question/index?qid=201...

Circles are in geometric progression if their radii R_n are in geometric progression R_n =r^n with r not equal to 1, and their centers can be isometrically mapped on the complex plane so that their images z_n, are also in geometric progression z_n = z^n.

The question is: what is the maximum length of a geometric sequence of circles such that there exists a circle (not in the sequence) tangent to all of them.

Start with E a set with N elements. You want N subsets A_1,...,A_N of same cardinal k, whose union is E and such that pairwise intersections consist of exactly one element of E for all A_i, A_j,

For N = 3: E = {a,b,c} you can take {a,b} {b,c} {c,a}

For which integers N is this possible?

http://in.answers.yahoo.com/question/index;_ylt=Am...

For which values of N can you find a permutation (a0, a1, a2.....aN) of numbers 0-N such that (a0, |a1-1|,|a2-2|,...|aN-N|) is also a permutation?

http://in.answers.yahoo.com/question/index;_ylt=Ai...

Consider quadrilaterals ABCD with integer sides all different, area and perimeters both square integers.

ABCD is convex with 2 right angles in B and D.

How many such quadrilaterals are there assuming that the smallest side has length 1: none, finitely many or infinitely many?

From the answers in

http://in.answers.yahoo.com/question/index;_ylt=Ai...

you see that the smallest side can be 5.