Jokes aside:

The probability that no car passes through a tollbooth in a 20 min time-window is .5. What is the probability that no car passes thru the tollbooth in a 5 minute time-window?
Answer should be only in decimal notation.

CourtseyDPS

Let

A=1 + x^3/3! + x^6/6! + x^9/9!..........
B=x + x^4/4! + x^7/7! + x^10/10!.........
C=x^2/2! + x^5/5! + x^8/8! + x^11/11! ............

Find the value of A^3 + B^3 + C^3 -3*A*B*C.

P.S. The judge has been enabled for this problem.

Note: a^b stands for a raised to the power b.

This could be too difficult:

You are given an elliptical-shaped land and you are asked to choose n arbitrary points on its boundary. Then you connect each point with every other point using straight lines, forming n(n - 1)/2 connections.

What is the maximum number of pieces of land you will get by choosing 'n' points on the boundary carefully?

Here are some sample answers:
n=1 Pieces=1
n=2 Pieces=2
n=3 Pieces=4
n=4 Pieces=8


The diagram shows the division of land at n=6,

Practice:

Here are some simple mechanics problems for 'practice':

The diagrams are self-explanatory.

In the first problem, find the force required to keep the platform moving at the constant velocity 1m/s.

In the second, assume no slipping at contacts, and find the accn. of the block in red.

In the third, find the velocity of the wedge after the collision. Assume e=.5.

Facebook Hacker Cup

This is one problem which I could not solve during the contest. (Well, I could not solve any problem)...
You have to command an army of Guards and Warriors in a RTS game to beat Sauron.
Each Guard adds '1' second of game-play to your army, it costs 'G' coins.
Each Warrior deals '1' damage for every second of its existence and costs 'W' coins. All Warriors die when the time given by your Guards runs out.

You have 'M' coins.
Give an approximate answer stating how many Warriors will you buy. Write the answer for the following triplets of(G,W,M).

(15,12,123124423423)
(23,43,234234232489)
(34,24,234879)
(432,25,435354)

Guts:

A charge is kept coaxial with a thin ring, which is made of material with resistance 1 ohm/m. Then it is allowed to fall, under the force of gravity.

In the process that follows, what is the maximum value of current that flows through the ring?

Submit the value of this current, correct upto 3 places of decimal.


From now on:
If you get A/C, do not write your answer as a comment.

Lifespan

A thin tube is inserted into a spherical bubble. Find the lifespan of the bubble.


Assume:
1) Length of pipe=1m.
2) Radius of pipe=1cm.
3) Initial radius of bubble=1m.
4) Surface tension of soap water=1 SI units.
5) Viscosity of air=1 SI units.

Repose

We plan to keep the block of mass 'm' stationary wrt. the wedge by accelerating the latter to the left.

This is possible for a significantly wide range of accelerations. Infact, the acceleration can be anywhere between g*tan(A) and g*tan(B). However, a certain value of acceleration, equal to g*tan(C) is of significant interest.

Your task is to report the values of A,B,C; in degrees. Your answer should be 3 comma separated values, correct upto 2 places of decimal.

Note:
Angle of the incline: 60*.
Coeff. of friction b/w wedge and mass: 1.
PS. : we've done a very similar problem before.