BMAD

The tourist has come to the Moscow by train. All-day-long he wandered randomly through the streets. Than he had a supper in the cafe on the square and decided to return to the station only through the streets that he has passed an odd number of times. Prove that he is always able to do that.

Given 120 unit squares arbitrarily situated in the 20x25 rectangle. Prove that You can place a circle with the unit diameter without intersecting any of the squares.

Does their exist four distinct natural numbers (a,b,c,d) where a*b, c*d is equivalent to the sum of all four numbers? What if i add the condition that a*d also equals the sum of all four numbers?

find a formula for the perimeter of a regular polygon as a function of its area. It needs to be shown that as the number of sides of a polygon of given area increases, the perimeter of the figure decreases.

Put a knight on a 4x4 chessboard that is able to make standard moves only. How many moves must it take until it is able to land on every square? Does it matter where it begins? What if the board was 5x5?

Here is a calculus problem from a class i teach. The problem itself illustrates the benefits of recognizing the fluidity and openness one can take in mathematics as the direct approach is ugly and messy but there is a simpler and elegant indirect way of solving this one too. Enjoy. Find the equation of the line tangent to the ellipse b^2*x^2 + a^2*y^2 = a^2*b^2 in the first quadrant that forms with the coordinate axes the triangle of smallest possible area (a & b are positive constants)

My apologies on the repost

Consider an infinite "tower of powers" of x, defined by x^x^x^... = x^(x^(x^...)) Can we find a value of x so that this tower is equal to 2?

Each of these true statements refers to seven out of eight of the living members of the Jones and King families, either by their personal name or by their relationship to another member. 1. John's wife's sister's husband's mother's son is Peter's cousin's father. 2. Sally's grand-daughter's cousin's father's sister-in-law is Gregory's son-in-law's wife. Question If Anita has no children, what relationship is she to Mike? (Yes, all the information is here. There are two possible solutions.)

Something = nothing? Let S= 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + . . . . . . . Then taking 1/2 of each term 1/2 S= 1/4 + 1/6 + 1/8 + 1/10 + 1/12 + 1/14 + . . . . . . . Subtract 1/2 S from S, the result is 1/2 S= 1/2 + 1/3 + 1/5 + 1/7 + 1/9 + 1/11 + . . . . . . . . (By removing 3rd, 5th, 7th, and so on from the original terms set) Subtracting the first expression for 1/2 S from the second expression for 1/2 S, we get 1/2 + (1/3 - 1/4) + (1/5 - 1/6) + . . . . = 0 Or 1/2 + 1/12 + 1/30 + 1/56 + 1/90 = 0 Proving that something equals nothing.

Both of these proofs presented appear to argue specific cases for the op but do they extend to prove that no matter what orientation of six squares that meet the op requirements have to share 1 point?

any five squares you pick, all of them will share precisely two points. This does not preclude the selection of a fewer amount having more than two points in common.

I have to choose whether to go to work or stay at home on a particular day. If I go to work, I will earn $500 for the day and if I stay home, I earn nothing. If I go to work, there is a chance that I will be killed in a car accident, and if I stay home this risk is 0. Finally, I am expecting a shipment from UPS and this is their last attempted delivery. If I go to work, there is a 80% chance that UPS will attempt delivery while I am at work and I will lose the package, and there is a 20% chance that UPS will attempt delivery after I get home and I will get it. If I stay home, I am certain to get the package. My preference ordering is: Living > dying $500 > Package My most preferred outcome is to live, and get both $500 and the package. Your task, find that chance, percent of getting killed, that would make me indifferent about going to work.

my apologies. Yes. every five squares share exactly two points. and to your question: they are all in the same Euclidean plane

At a tug-o-war contest two teams are pitted against each other. Team 1 has five players: Alice, Bob, Carol, Douglas, and Eric. Team 2 also has five players: Frank, George, Harry, Ian, Jessica. When both teams used all of their members, team 1 won. When Team 1 and Team 2 each lost two players, it was a draw. When two new mini-teams were formed with Alice and Frank competing against Bob and George, Bob and George won the contest. Besides what was already given, what other claims could be made about individual or team strengths and of outcomes of other tug-o-war contests?

and the squares are not solid

Assume the squares are congruent

As a follow up to my circles problem: Given six squares where any group of five have two common points. Does there exist a common point for all 6?

feel free to extend to disks

My spacing went away: Nicholas and Peter are dividing (2n+1) nuts. Each wants to get more. Three ways for that were suggested. (Each consist of three stages.) First two stages are common. 1 stage: Peter divides nuts onto 2 heaps, each contain not less than 2 nuts. 2 stage: Nicholas divides both heaps onto 2 heaps, each contain not less than 1 nut. 3 stage: The 3rd stage will be one of the following ways: 1 way: Nicholas takes the biggest and the least heaps. 2 way: Nicholas takes two middle size heaps. 3 way: Nicholas takes either the biggest and the least heaps or two middle size heaps, but gives one nut to the Peter for the right of choice. Find the most and the least profitable method for the Nicholas.

Eight men had participated in the chess tournament. (Each meets each; draws are allowed, giving 1/2 of a point; winner gets 1.) Everyone has a different number of points. The second one has as many points as the four weakest participants together. What was the result of the play between the third prizer and the chess-player that have occupied the seventh place?

Given five circles where any group of 4 have a common point. Does there exist a common point for all 5?

What is the maximal area of a triangle if its sides a,b,c satisfying the inequality: 0<=a<=1<=b<=2<=c<=3?

Given 1962 -digit number that is divisible by 9. Let x be the sum of its digits. Let the sum of the digits of x be y. Let the sum of the digits of y be z. Find z.

Nicholas and Peter are dividing (2n+1) nuts. Each wants to get more. Three ways for that were suggested. (Each consist of three stages.) First two stages are common. 1 stage: Peter divides nuts onto 2 heaps, each contain not less than 2 nuts. 2 stage: Nicholas divides both heaps onto 2 heaps, each contain not less than 1 nut. 3 stage: The 3rd stage will be one of the following ways: 1 way: Nicholas takes the biggest and the least heaps. 2 way: Nicholas takes two middle size heaps. 3 way: Nicholas takes either the biggest and the least heaps or two middle size heaps, but gives one nut to the Peter for the right of choice. Find the most and the least profitable method for the Nicholas.