BMAD

Inspired by the witty code of BobbyGo A random number generator generates integers in the range 1...n, where n is a parameter passed into the generator. The output from the generator is repeatedly passed back in as the input. If the initial input parameter is one googol (10100), find, to the nearest integer, the expected value of the number of iterations by which the generator first outputs the number 1. That is, what is the expected value of x, after running the following pseudo-code? n = 10100 x = 0 do while (n > 1) n = random(n) // Generates random integer in the range 1...n x = x + 1 end-do

A's double exceeds its half by 99. B is twice the product of its digits. C is thrice the sum of its digits. Half of D exceeds its third by the sum of its digits. E is increased by 20% if its digits are reversed. F can be squared by sandwiching in two more digits between its original two. G differs from its reverse by twice the product of its digits. The product of H's digits is twice their sum. And turn I upside down and you'll increase it by 12. If each one of these are whole numbers with two digits, which one is the greatest?

The four squares connect at corners defining an arbitrary quadrilateral. Connecting the centers of opposite squares, show that the segments are perpendicular.

Anything special about these numbers?

my example though shows 3 steps

yes, your example is 2 steps.

What's the maximum product of any whole number of real numbers whose sum is 100? What are they?

Inside an orange unit circle we fit two blue circles of radius 1/2. A yellow circle is tangent to the two blue circles and the inner edge of the orange circle. A green circle inside the orange circle is tangent to one of the blue circles, the yellow circle, and the orange circle. If the orange circle is centered at (0,0) what are the coordinates for the centers of the other four circles? The blue ones should be easy. Presume the circles are placed in the circle as shown in the picture.

Put four different whole numbers from 1 to 49, in the corners, then subtract the smaller from the larger and put the answers in between. Keep doing this until all four numbers become equal. How many steps can you make it last? The demo is "(14, 30, 18, 37) lasts 3 steps (don't count the step where they produce the same number)." Find the longest lasting case. DEMO: stage 0 14 30 ----> 30-14 37-14 30-18 37 18 37-18 stage 1 16 16-13 16-12 13 12 19 19-13 19-12 stage 2 3 4 6 7 stage 3 1 3 3 1 Stage 4 is stasis so it doesn't count

I meant the ceiling as a wall. of course they are on the floor too

yes

yes.

nope

yes!

elite

yes!

NEW GAME:

NEW GAME:

Oh, the things you discuss with your friends when you are supposed to be studying for finals. I don't know how many of you know of the video game called portal (or the sequel). But the premise is pretty straightforward using a 'gun' you can create a portal that lets you travel to locations often out of reach. To create the portal, you shoot two spots. You enter one of the two spots and come out the other much like a wormhole in outer space is supposed to work. So my friends and I got into a pretty heated debate. We wanted to know what would happened if you shot one spot on a wall and the other on a movable object and then slid the movable object through the wall portal, where would the movable object go, what would come out of the movable object's portal?

Let A be a list of n integers between 1 and k. Let B be a list of k integers between 1 and n. Prove that there's a non-empty subset of A and a (non-empty) subset of B having the same sum. Example: Say n=3, k=5 and A={3,4,5}, B={1,1,2,3,3}. Then we can find {1,3,3} is contained in B and {3,4} contained in A with the same sum (I know there're are simpler solutions in this example, it's just for demonstration).

A big ball (of diam 7 ft) just touches three walls, and a little ball touches three walls and the big ball. What's the diameter of the little ball? Assume that the balls are spheres and therefore only touch at a single point.