BMAD

It appears that we have disagreement, both of you have provided proofs where the other has provided a counter example

Yes. I found those using the Laurant series

There are two answers to this I believe

I am missing how that proves k=1 and m=2 is the smallest

Yes.

Forgive my grammar here. What I mean is an exclusive "and". They cannot share any common points but may share only one common side and/or one common vertex.

Now to prove your answer is the smallest.

I see the error in my clarification. Yes the result should be natural. That is my mistake.

would player 2 pick a midpoint?

nice. now for b.

it does not need to be natural, only k and M have to.

A point P is inside an equilateral triangle of side length d such that the distances to the vertices are given by a, b, c. Find the formula relating a,b,c,d. Try with the case a,b,c = 3,4,5

A pentagon and its mid points are drawn. The original pentagon is then erased, leaving just its mid points visible. Is it possible to reconstruct the pentagon?

Let us call "big" a triangle with all sides longer than 1. Given a equilateral triangle with all the sides equal to 5. Prove that: a) You can cut 100 big triangles out of the given one. b) You can divide the given triangle onto 100 big nonintersecting ones fully covering the initial one. c) The same as b), but the triangles either do not have common points, or have one common side, or one common vertex.

Given a triangle ABC with the unit area. The first player chooses a point X on the side [AB], than the second -- Y on [bC] side, and, finally, the first chooses a point Z on [AC] side. The first tries to obtain the greatest possible area of the XYZ triangle, the second -- the smallest. What area can the first obtain for sure and how?

Two are playing the game "cats and rats" on the chess-board 8x8. The first has one piece -- a rat, the second -- several pieces -- cats. All the pieces have four available moves -- up, down, left, right -- to the neighbour field, but the rat can also escape from the board if it is on the boarder of the chess-board. If they appear on the same field -- the rat is eaten. The players move in turn, but the second can move all the cats in independent directions. a) Let there be two cats. The rat is on the interior field. Is it possible to put the cats on such a fields on the border that they will be able to catch the rat? b) Let there be three cats, but the rat moves twice during the first turn. Prove that the rat can escape.

Each of the side of the convex hexagon (6-angle) is longer than 1. Does it necessary have a diagonal longer than 2? Each of the main diagonals of the convex hexagon is longer than 2. Does it necessary have a side longer than 1?

Among all the numbers representable as 36k - 5m (k and m are natural numbers) find the smallest. Prove that it is really the smallest.

but how do you 'know' these can't be further optimized?

just equal areas

elaborate please

Let us define a line segment as a segment that joins two vertices either vertically or horizontally. So in the image below, drawing a line from top to bottom vertically can have a maximum of 5 line segments. The tasks: Divide the shape into two figures with identical areas using.... the least amount of line segments the most amount of line segments **It should be noted that the shape is "missing" one square in the upper left and two in the lower right intentionally. I realize that the image may make the missing square(s) appear present.**

yes, the machine reports an incorrect false score.

Create a statement that is true that the machine will report as false