BMAD

ah, thank you.

Alas, English.... But I am confused. From your interviewing you know for a fact that this family has A boy. So regardless of their birth order all we don't know is the probability that the other child is a boy. And the other child could be a boy or girl, so I still believe it is 1/2.

If it is given that they already have one boy then... I get 1/2. Since the second child could either be a boy or a girl. Note: The spoiler buttons seems to be missing on mobile again.

Assume we have a circle and a rectangle. The circumference of the circle is the same as the perimeter of the rectangle and the areas of each shape is also the same. Write the ratio of the circle's circumference to the diagonal of the rectangle as a function in terms of the width of the rectangle.

I hope there is a nuanced way of solving this besides brute force.

The first case

Find the limit of x^(x/2)^(x/4)^(x/8)^(x/16)^(x/32).... (a) as x goes to infinity (b) as x goes to zero

but if you place 3,4 on one side it would not be balanced.

To answer your second part: the machine would scan the candies and intentionally pick out a butterscotch and randomly select four candies from the remaining four. So in this case, if there is a butterscotch candy left then you are guaranteed that the first one chosen was a butterscotch.

n is not known

Alice and Bob are playing the following game: Alice has a secret polynomial P(x) = a_0 + a_1 x + a_2 x^2 + … + a_n x^n, with non-negative integer coefficients a_0, a_1, …, a_n. At each turn, Bob picks an integer k and Alice tells Bob the value of P(k). Find, as a function of the degree n, the minimum number of turns Bob needs to completely determine Alice’s polynomial P(x).

Consider the set {1,11,111, …, ((10^2007) – 1)/9}. At least one of these numbers is divisible by 2007. Is the same true for 2008 (replacing 10^2007 with 10^2008, of course)?

You have 7 generals and a safe with many locks. You assign the generals keys in such a way that EVERY set of four generals has enough keys between them to open ALL the locks; however, NO set of three generals is able to open ALL the locks. How many locks do you need, and list how many keys does the first general get, the second, … Is there more than one way that works?

Yes, the machine only requires that one of its candies be returned it need not be one purchased with the 75 cents.

There is a machine with 20 pieces of candy. Five of those candies are butterscotch. If you put in a 25 cents, one candy is provided at random. If you put in 75 cents, two candies are dropped at random but you may give the machine back one candy in exchange for a 25 cents. And if you put in $1.50 you receive 5 pieces of candy at random but are guaranteed at least one butterscotch. How much should I expect to spend to get all of the butterscotch?

multiples of your six seem to work too

This contradicts the solution below. As your a > b > c > d > e > f > g solution does not follow the condition that a + b must equal the sum of the rest

A balance and a set of metal weights are given, with no two the same. If any pair of these weights is placed in the left pan of the balance, then it is always possible to counterbalance them with one or several of the remaining weights placed in the right pan. What is the smallest possible number of weights in the set?

let a = ln(ln(x))/(ln(x) let b= xa let y = eb find dy/dx

Find all continuous positive functions that possess the following property: integral from 0 to 1 of f(x) dx = 1 and integral from 0 to 1 of x*f(x) dx = a and integral from 0 to 1 of x2*f(x) dx = a2

I am not disputing either argument, I am curious though on your thoughts that something like the pistol would be better than the shotgun because 2/3's of the time it is equal or better than the shotgun and only 1/3 of the time is it not.

I was playing a mobile video game based on the fall out series. In this game, you need to arm your citizens with weapons to protect themselves. When the citizens use the weapons they will hit their target for an amount of damaged indicated within the range given. Currently, I have a surplus of weapons and would like to allocate the best weapons to my people. Place the weapons in order of best to worst: Knife: 0 - 9 damage Shotgun: 6 damage Pistol: 5 - 7 damage Rail gun: 4 - 7 damage Rifle: 3 - 7 damage Grenade: 0, 11 (only 2 possible outcomes)

so far, so good

Working with lengths that are whole units from 1 - 100, how many obtuse triangles can be formed?

At a circus there was a stunt completed by six bears, 3 brown, and 3 black. Initially, the 3 brown bears were each standing in a square to the left of a center square and the 3 black bears were located right of center. The bears either scooted left/right one square or one of the bears jumped over another until each group of bears were on the other group's set of squares. If the bears needed 15 moves (scoots, jumps) to completely switch sides, how many scoots and how many jumps were used? Bonus points: what is the largest total amount of bears (assuming equal amounts of brown and black bears) that could complete this task in fifteen moves? Start: br br br ___ bl bl bl End: bl bl bl ___ br br br