BMAD

Given a triangle whose three sides are 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.

i found two positive integer answers

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?

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?

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.

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.

Emperor Akbar once ruled over India. He was a wise and intelligent ruler, and he had in his court the Nine Gems, his nine advisors, who were each known for a particular skill. One of these Gems was Birbal, known for his wit and wisdom. A farmer and his neighbour once went to Emperor Akbar's court with a complaint. 'Your Majesty, I bought a well from him,' said the farmer pointing to his neighbour, 'and now he wants me to pay for the water.' 'That's right, your Majesty,' said the neighbour. 'I sold him the well but not the water!' The Emperor asked Birbal to settle the dispute. How did Birbal solve the dispute in favor of the farmer?

Suppose you have a unit square and a equilateral triangle of the same area. A vertex of the triangle shares location with the center of the square. What is the maximum possible overlap? What is the minimum possible overlap?

The definitions should be in comparison of 2n not in.

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)?

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?

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.

tongue in cheek cook an additional cake that completely encapsulates the pentagon with the exact same type of icing as the other cake. place the cake around the cake to make a square. cut the cake into 7 equal strips. Op solved tongue removed from cheeck

I was under the same impression Phil regarding the Heptagon, which is why i asked in the op (a possible and impossible solution). Thanks for the link and thank you for explaining the construction of linear points. I missed that part .

Can you construct a heptagon without assuming you can make several linear points?

lol

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

well done. Now about that heptagon.

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?

No you do not.

In the sense of needing to bisect a line....yes.

Fair point. Shoot. Thought I solved one.

There are 6 points in a rectangle with the sides, 3 and 4. Prove that the distance between at least two of these points is smaller than the square root of 5.