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Problems Combinatorics Integer lattices Polygons and polyhedra with vertices in lattice points Calculus Real numbers Rational and irrational numbers Algebra and arithmetic Trigonometry Trigonometry (other)

14–16 3.0

A square grid on the plane and a triangle with vertices at the nodes of the grid are given. Prove that the tangent of any angle in the triangle is a rational number.

Problem #PRU-61213

Problems Calculus Derivative Increasing and decreasing. Researching functions Algebra and arithmetic Trigonometry Trigonometry (other)

14–15 3.0

Find the largest and smallest values of the functions

a) $f_{1} (x) = a \cos x + b \sin x$ ; b) $f_{2} (x) = a \cos^{2} x + b \cos x \sin x + c \sin^{2} x$ .