BMAD

where does 3w − 1 fit?

A bent is the union of the interior and boundary of a simple quadrilateral such that an interior angle formed by two adjacent edges exceeds 180◦ and the interior angle formed by the other two edges exceeds 90◦. Prove that the union of the interior and boundary of an acute triangle can not be the union of a finite number of bents which have disjoint interiors.

An airplane flies at constant airspeed c directly above a closed polygonal path in a plane, completing one circuit. Prove or disprove that, compared to no wind, the presence of a wind of constant speed and constant direction will increase the time required.

Let w be the greatest common divisor of m and n. Show that 3w − 1 is the greatest common divisor of 3m − 1 and 3n−1.

Are you asking me or Gavinksong or is this Rhetorical?

I would first focus on the expected number of Martians that would land.

Farmer Brown is standing in the middle of his perfectly circular field feeling very content. It is midnight and there is no moon and unknown to the farmer, Martian zoologists are landing randomly at points on the circumference of his field. They land at one minute intervals, starting at midnight. As soon as there are martians at points A,B,C such that triangle ABC contains the center of the field, Farmer Brown will be teleported to the waiting space-ship and transported to spend the rest of his life as an exhibit in a Martian zoo. What is the expected time until he is abducted?

A long worm crawls into a cosmetic veterinary surgery, complaining of a problem with 1's. A worm can be thought of as a string of 0's and 1's and the most beautiful worm is 00000 ... 0. The worm is a sequence of segments and there is a 0 or a 1 on each segment. The procedure for removing 1's is complicated. If there is a 1 at the right hand end (where the tail is), then the surgeon can remove this 1 (and the segment it lies on) and place a new segment with 0 or a 1 at the left hand end (where the head is). If however there is a 0 at the right hand end then the surgeon can remove it, but he has no control over what appears on the new segment at the left hand end. Indeed, the surgeon has to operate under the assumption that there is an adversary making a choice of what to replace a 0 by. The adversary would like the operation to continue indefinitely. The surgeon claims a 100% success rate. Do you believe him?

How many n are there such that n! + 1 is composite?

Given triangle ABC, with cevians AD, BE, CF dividing the sides in ratio 1:2, a smaller triangle forms in the middle. What is the area of the triangle in the middle?

I've only been able to show it is true for cyclic quadrilaterals

Mack "The Knife", the notorious tough guy, was found murdered one night in an alley behind the nightclub he usually frequented. The police brought in three suspects the morning after. That afternoon a police investigator interrogated the three men. They made the following statements. Bud: 1. I didn't kill Mack. 2. Jack is not my friend. 3. I knew Mack. Jack: 1. I didn't kill Mack. 2. Bud and Tug are friends of mine. 3. Bud didn't kill Mack. Tug: 1. I didn't kill Mack. 2. Bud lied when he said that Jack was not his friend. 3. I don't know who killed Mack. Only one of the three is guilty, and only one of each man's statements is false. who killed Mack the Knife?

If you saw three shadows on three fence posts, one painted white, one painted blue, and one painted red, which shadow would be the darkest?

But, the OP says that the words can be less that 4 characters. It also doesn't say that the same character cannot be repeated inside the word. I disagree. 27x27x27x26 I believe implies that the first letter has 26 choices and the next three has 27 but this is not true. The second letter has 27 choices as it could be empty but the third and fourth would only have a 27th choice if the second one chose to not be the empty letter as a_a is not a word.

yay. I so rarely successfully answer these puzzles.

excellent point. I should have more clearly stated that I meant the modern English Alphabet.

Fifteen children together gathered 100 nuts. Prove that some pair of children gathered the same number of nuts.

Wilma operates a computer with a magnetic tape drive. One day she is given a tape that contains 500,000 ”words” of four or fewer lowercase letters (consecutive words on the tape are separated by a blank character). Could all 500,000 words be different?

ah, i understand. nice job.

i am unsure by what you mean in your second answer.

On the blackboard the mathematics professor wrote a polynomial f(x) with integer coefficients and said, "Today is my son's birthday. When his age A is substituted for x, then f(A) = A. You will note also that f(0) = P and that P is a prime number larger than A. How old is the professor's son?