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BMAD

Member Since 18 Feb 2013
Offline Last Active Today, 03:06 PM
*****

#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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#337644 Circles to make a pentagon

Posted by BMAD on 01 March 2014 - 04:15 PM

Spoiler for pentagon solution

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#337634 Circles to make a pentagon

Posted by BMAD on 28 February 2014 - 04:34 PM

tslf, you have me worried.  For I have done it much the same way.  Let me re-examine my own solution as well.


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#337586 scheme

Posted by BMAD on 27 February 2014 - 03:14 PM

Dear friend,
Do you want to get rich quick? Just follow the instructions carefully below and you
may never need to work again:
1. At the bottom of this email there are 8 names and addresses.
 Send $5 to the name at the top of this list.
2. Delete that name and add your own name and address at the bottom of the list.
3. Send this email to 5 new friends.
 
 
If the process goes as planned, how much money would you make?

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#337544 Complex numbers

Posted by BMAD on 26 February 2014 - 05:26 PM

Consider computing the product of two complex numbers (a + bi) and (c + di). By foiling the polynomials as we learned in grade school. We get:
 
 
 
       a   + bi 
       c   + di  
     ---------- 
       adi - bd 
ca + cbi 
---------------- 
(ca - bd) + (ad + cb)i   
 
 
Note that this standard method uses 4 multiplications and 2 additions to compute the product. (The plus sign in between (ca - bd) and (ad + cb)i does not count as an addition. Think of a complex number as simply a 2-tuple.)
 
It is actually possible to compute this complex product using only 3 multiplications and 3 additions. From a logic design perspective, this is preferable since multiplications are more expensive to implement than additions. Can you figure out how to do this?

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#337502 Spin the bottle

Posted by BMAD on 25 February 2014 - 05:55 PM

6 friends are sitting in a circle whom you have been observing playing the game called 'spin the bottle'.  They invite you to play as well.  Being a good sport, you agree to play.  You happen to notice however that it is HIlga's turn and you would rather not kiss Hilga.  You have also noticed that the bottle when spun with the left hand has a tendency to stop on a person an even distance away (e.g. the second, fourth, sixth person from her) and when spun with the right hand will land on a multiple of 3 distance away from HIlga.  Currently the bottle is pointing directly across from Hilga but if it were to land on herself she would immediately spin it again from that position, switching hands in the process.  You have no idea which hand she will use first nor do you know which way rotation she will place on the bottle [clockwise or counter-clockwise]. 

 

Where should you sit?


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