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Suppose that the polynomial \(f\left(x\right)=2x^5-9x^3+2x^2+9x-3\) has 5 solutions x₁ ; x₂; x_3; x₄; x₅. The other polynomial \(k\left(x\right)=x^2-4\)_. Find the value of \(P=k\left(x_1\right)\times k\left(x_2\right)\times k\left(x_3\right)\times k\left(x_4\right)\times k\left(x_5\right)\).

Let N be the largest number of region that can be formed by drawing 2016 straight lines on a plane. Find the sum of all digits of N.

There are 2017 points on the plane. The area of any triangle with verticles does not exceed 1. Assume that in any case, all these points can be placed in a triangle whose area is a positive integer K. Find the least value of K.

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Five pirates have obtained 100 gold coins and have to divide up the loot. The pirates are all extremely intelligent, treatrous and selfish (especially the captain). The captain always proposes a distribution of the loot. All pirates vote on the proposal and if at least half of the crew (which includes himself) approve, the loot will be divided as proposal, as no pirates would be willing to take on the captain without superior force on their side. Otherwise he will face a mutiny: all pirates will turn against him and make him walk the plank. The pirates will start over again with the next senior pirate as captain. What's the maximum number of coins the captain can keep without risking his life?

In trapezoid ABCD with AB parallel to CD, the diagonals AC and BD
intersect at E. If the area of triangle ABE is 50 square units, and the
area of triangle ADE is 20 square units, what is the area of trapezoid ABCD?
 

A trapezium ABCD has two paralell sides AB and CD. The diagonals AC and BD intesect at E. If the areas of triangles CDE and CDB are 1 and 4 respectively, what is the area of trapezium ABCD? 

Five pirates have obtained 100 gold coins and have to divide up the loot. The pirates are all extremely intelligent, treatrous and selfish (especially the captain). The captain always proposes a distribution of the loot. All pirates vote on the proposal and if at least half of the crew (which includes himself) approve, the loot will be divided as proposal, as no pirates would be willing to take on the captain without superior force on their side. Otherwise he will face a mutiny: all pirates will turn against him and make him walk the plank. The pirates will start over again with the next senior pirate as captain. What's the maximum number of coins the captain can keep without risking his life?