Maximum and Minimum

1. Into which a) two parts, b) n parts should a given number be divided so that the product of the parts is the greatest?
2. Prove that the sum of a) two, b) several numbers with a fixed product becomes smallest when the numbers are equal.
3. Let x₁ < x₂ < ... < xₙ. Find such a point x on the number line that the sum of the distances from x to xᵢ is minimised.
4. (Geometry problem) Find the minimum value of the expression: square root of (y - 2)² + 1 + square root of (x² + y²) + square root of (x - 2)² + 4.

5. Prove that among a) rectangles, b) rhombuses with the same perimeter, the square has the greatest area.
   c) What should be the sides of a rectangle inscribed in a circle for its area to be the greatest?
6. Among all triangles ABC with given sides AB and BC, find the triangle with the largest area.
7. A straight line l and two points A and B are given, lying a) on opposite sides, b) on the same side of the line.
   Find such a point X on line l that AX + BX is minimised.
8. A ray of light travels from point A to point B, reflecting off a flat mirror a.
   Prove that, according to the law of reflection (angle of incidence equals angle of reflection), the ray chooses the shortest path.
9. A polygon is given, symmetric with respect to point O.
   Prove that for this point, the sum of distances to the vertices of the polygon is minimised.
10. (Fermat–Torricelli problem) Find the point for which the sum of distances to the vertices of a given triangle is minimised.
11. (Fangyuan’s problem) A triangle ABC is given, which is acute-angled.
    For which points K, L, and M, lying on sides BC, AC, and AB respectively, is the perimeter of triangle KLM minimised?
12. (Dido’s problem) Prove that among all shapes with a given perimeter, the one with the largest area is the circle.

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