Prove the inequality x₁x₂ + x₂x₃ + … + x₉₉x₁₀₀ ≤ 1/4. Prove that there is no con

Prove the inequality x₁x₂ + x₂x₃ + … + x₉₉x₁₀₀ ≤ 1/4.

Prove that there is no convex polyhedron in which all faces have a different number of sides.

On the plane, 100 straight lines are drawn. No two are parallel, and no three pass through the same point. Into how many regions do the lines divide the plane?

a) Several identical coins are lying on a table without overlapping. Prove that there is a coin that touches no more than three others. b) There are 21 numbers, and the sum of any five of them is positive. Prove that the sum of all the numbers is positive.

Six numbers are arranged around a circle, and each number is equal to the absolute value of the difference of the next two numbers in clockwise order. The sum of all the numbers is equal to