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BMAD

Member Since 18 Feb 2013
Offline Last Active Yesterday, 04:31 PM
*****

#337841 Santa Claus's speed

Posted by BMAD on 19 March 2014 - 03:57 AM

Assume that the earth is a perfect sphere with a circumference of 40Mm. Santa needs to travel from the North Pole to the South Pole while avoiding daylight. Assuming that he can go faster than the speed of light, what path would be the best path to take and what is the slowest he can travel?
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#337840 gun with unlimited bullets

Posted by BMAD on 19 March 2014 - 03:48 AM

There's a gun located on an infinite line. It starts shooting bullets along that line at the rate of one bullet per second. Each bullet has a velocity between 0 and 1 m/s randomly chosen from a uniform distribution. If two bullets collide, they explode and disappear. What is the probability that at least one of the bullets will infinitely fly without colliding with another bullet?
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#337790 Majority Rules

Posted by BMAD on 14 March 2014 - 05:45 PM

In an presidential election between Bipa and Viktor, the winning candidate Bipa received n+k votes, whereas Viktor has received n votes. (n and k are positive integers.)  If ballots are counted in a random order, what is the probability that Bipa's accumulating count will always lead his opponent's, and why?


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#337789 Play on a classic: computers

Posted by BMAD on 14 March 2014 - 05:42 PM

Alright, I'll add a puzzle to the bunch... 
 
You have N computers on a space station. An accident happens, and some of the computers are damaged, but you know the number of good (undamaged) computers is greater than the number of bad (damaged) ones. 
 
Your goal is to find *one* computer that's still good. 
 
Your only method of testing is the following: Use one computer (say, X) to test another (Y). If X is a good computer, it tells you correctly the status of Y. If X is bad, it may or may not give the correct status of Y; assume it will give whatever answer is least useful to your testing strategy. 
 
In worst-case, how many tests must you use to find one computer that's still good? (in terms of N) 
 
You're permitted any combination of tests, though keep in mind the bad machines may not be consistent in the results they give you.

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#337788 Penny Game

Posted by BMAD on 14 March 2014 - 05:39 PM

There are three one-dimensional tracks, of length 12, 7, and 5 spaces respectively. You start with pennies in the first space of each track; your opponent starts with pennies in the last space of each track. On your turn, you may move any one of your pennies any number of spaces in either direction along a track (as a chess rook), however you are not permitted to bypass the other player's penny or occupy its space. If a player has no legal move, he loses. 
 
What should your first move be? 

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#337787 SBT Fraction

Posted by BMAD on 14 March 2014 - 05:30 PM

There are two fractions, 34/55 and 55/89. We are looking for a third fraction of positive integers a/b, where 34/55>a/b>55/89 and 55<b<89. What is the smallest b where this is possible?


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#337786 semi-regular polygons

Posted by BMAD on 14 March 2014 - 05:19 PM

Define semiregular polygon as a polygon which has all of its' edges of the same length. Also, all of its' interior or exterior angles must be equal (meaning that any interior angle must be x or 360-x). It must be concave and simple (it should not self-intersect) and only two of its' edges are allowed to meet in each corner. 
Find the semiregular polygon that has the minimum number of edges.

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#337766 Smallest Consecutive Integer Triangle Possible

Posted by BMAD on 11 March 2014 - 02:41 PM

Given a triangle whose three sides are consecutive integer values, and the area of which is divisible by 20, find the smallest possible side for which these  conditions hold true:
 
  two sides are odd numbers
  at least one side is a prime number.
 
 
 
The added condition to my original problem gets into some cool number theory (if you go that route)

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#337748 checker chase game

Posted by BMAD on 10 March 2014 - 02:12 AM

Player A begins by placing a checker in the lower left-hand corner of a checkerboard (8 by 8 squares). Player B places a checker one square to the right or one square up or one square diagonally up and to the right of Player A's checker. Then A places a checker one square to the right or one square up or one square diagonally up and to the right of Player B. The players continue alternating moves in this way. The winner is the player who places a checker in the upper right corner. Would you rather be Player A or Player B?
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#337747 average birthdays

Posted by BMAD on 10 March 2014 - 02:08 AM

At the classroom costume party the average age of the (b) boys is g and the average age of the (g) girls is b. If the average age of everyone, including the 42-year-old teacher, is b+g, what is the value
of b+g?
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#337746 antifirst

Posted by BMAD on 10 March 2014 - 01:55 AM

An antifirst number is a natural number that has more divisors than any preceding number before it.
E.g.
1 has 1 divisor,
2 has 2 divisors, (skip 3 since it only has 2 divisors)
4 has 3 divisors,
6 has 4 divisors, and so on...
So the first four numbers are (1,2,4, and 6).
Your tasks,
find the biggest antifirst number under 1,000,000.
Prove or provide a counter example to the following conjecture, all antifirst numbers greater than 6 are abundant or perfect.
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#337731 Sheep Mowing service

Posted by BMAD on 07 March 2014 - 04:47 PM

If we tie a sheep to one peg it eats out a circle in grass. If we put a rope through a ring on its neck and tie both ends of the rope to two pegs it eats out an ellipse. If we want an oval we tighten one rope between two pegs put a ring with a rope on it and tie the sheep to its other end.

How to tie a sheep so that it eats out a square in grass? We have one sheep ropes pegs and rings.

 

sheep.png


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#337678 reversed coefficients

Posted by BMAD on 03 March 2014 - 06:27 PM

If P(x) and Q(x) have 'reversed' coefficients,

 

for example: 

P(x) = x5+3x4+9x3+11x2+6x+2 

Q(x) = 2x5+6x4+11x3+9x2+3x+1

 

What can you say about the roots of P(x) and Q(x)?


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#337677 Simplified American Football

Posted by BMAD on 03 March 2014 - 06:22 PM

In a game of "simplified football," a team can score 3 points for a field goal and 7 points for a touchdown. Notice a team can score 7 but not 8 points. What is the largest score a team cannot have?
 

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#337676 number theory: 7 conjectures

Posted by BMAD on 03 March 2014 - 06:12 PM

All the numbers below should be assumed to be positive integers. 
 
Definition. An abundant number is an integer n whose divisors add up to more than 
In. 
Definition. A perfect number is an integer n whose divisors add up to exactly In. 
Definition. A deficient number is an integer n whose divisors add up to less than In. 
 
Example. 12 is an abundant number, because 1 + 2 + 3+ 4 + 6+12 = 28 and 28 > 
2x12. However, 14 is a deficient number, because 1 + 2 + 7 + 14 = 24, and 24 < 
2 x 14. 
 
Your task is to consider the following conjectures and determine, with proofs, 
whether they are true or false. 
 
Conjecture 1. A number is abundant if and only if it is a multiple of 6. 
Conjecture 2. If n is perfect, then kn is abundant for any k in N. 
Conjecture 3. If p1 and p2 are primes, then p1/p2 is abundant. 
Conjecture 4. If n is deficient, then every divisor of n is deficient. 
Conjecture 5. If n and m are abundant, then n + m is abundant. 
Conjecture 6. If n and m are abundant, then nm is abundant. 
Conjecture 7. If n is abundant, then n is not of the form pm for some natural m and 
prime p.

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