Pure maths, physics, logic (braingames.ru): non-trade-related brain games - page 71
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(5) The cake is shaped like an arbitrary triangle. Two megabrains divide it in the following way: the first points a point on the cake, the second makes a rectilinear cut through this point and takes the largest part. How much of the cake can the first megabrain get for himself? The cake is thought to have the same thickness everywhere.
It just shows that the kettle is heating up, not cooling down. The curves are different.
(4) At the initial moment, a large number of bodies are launched simultaneously from the same point along differently directed straight chutes. All the chutes are in the same vertical plane. The initial velocity of the bodies is zero. There is no friction. On what curve will these bodies be placed after 1 second of falling? Why?
(5) The cake is shaped like an arbitrary triangle. Two megabrains divide it in the following way: the first points a point on the cake, the second makes a rectilinear cut through this point and takes the largest part. How much of the cake can the first megabrain get for himself? The cake is assumed to be the same thickness everywhere.
On a sphere?
then on a circle, it's all in a plane =)
(5) The crest of an ancient family of megalomaniacs shows four circles of the same radius: three red and one blue. And any two red and blue circles intersect at the same point. Prove that all three red circles also intersect at the same point.
Another TV challenge (not with braingames, quite challenging and interesting).
Mr and Mrs megabrains play coin toss. Mr. megabrain has a fair coin, Mrs. has 0.4 probability of tails (for eagle: 1 - 0.4 = 0.6), and she knows it. Mega-brains flip their coins the same number of times and the one with the most tails at the end of the game wins. Mrs. Megamind realizes that her chances of winning are less than her husband's and she can decide how many times in the game the coin is flipped before the winner is determined.
Question: What number of flips must Mrs. Megamogs set in order to have the maximum chance of winning? Does this number differ from 1?
I'll start solving the problem myself. If interested, join in.
First step. If the megabrains agree to toss the coins once and then determine the winner, then the probability that Mrs. MM will win is equal to the probability of her tails 0.4 multiplied by the probability that Mr. MM will have 0.5 = 0.2.
Second step. The mega-brains have agreed to flip coins twice before the winner is revealed. In this case:
The probability of Mrs. MM winning is 0.24.
From this point we can already answer the second part of the question: the number of flips must not be equal to (greater than) one.
I will also tell you that the function of probability of winning Mrs. MM on the number of throws has an extremum, i.e. the problem is exactly solved.
Ah, I misunderstood. Heating is convex, cooling is concave, where burning is more likely.