Problems

A target consists of a triangle divided by three families of parallel lines into 100 equilateral unit triangles. A sniper shoots at the target. He aims at a particular equilateral triangle and either hits it or hits one of the adjacent triangles that share a side with the one he was aiming for. He can see the results of his shots and can choose when to stop shooting. What is the largest number of triangles that the sniper can guarantee he can hit exactly 5 times?

Every day, James bakes a square cake size $3\times 3$. Jack immediately cuts out for himself four square pieces of size $1\times 1$ with sides parallel to the sides of the cake (not necessarily along the $3\times 3$ grid lines). After that, Sarah cuts out from the rest of the cake a square piece with sides, also parallel to the sides of the cake. What is the largest piece of cake that Sarah can count on, regardless of Jack’s actions?

In a $10\times 10$ square, all of the cells of the upper left $5\times 5$ square are painted black and the rest of the cells are painted white. What is the largest number of polygons that can be cut from this square (on the boundaries of the cells) so that in every polygon there would be three times as many white cells than black cells? (Polygons do not have to be equal in shape or size.)

Is it possible to cut out such a hole in a sheet of paper through which a person could climb through?