Advanced book on Mathematics Olympiad

4.2 Trigonometry 231

prove that the quadrilateral is a rhombus. Does the property hold ifOis some other

point in the interior of the quadrilateral?

650.Prove that the plane cannot be covered by the interiors of finitely many parabolas.

651.LetABCbe a triangle with the largest angle atA. On lineAB consider the
pointDsuch thatAlies betweenBandD, andAD=AB^3 /AC^2. Prove that
CD≤

√

3 BC^3 /AC^2.

652.Show that if all angles of an octagon are equal and all its sides have rational length,
then the octagon has a center of symmetry.

653.Show that if each of the three main diagonals of a hexagon divides the hexagon into
two parts with equal areas, then the three diagonals are concurrent.

654.Centered at every point with integer coordinates in the plane there is a disk with
radius 10001.
(a) Prove that there exists an equilateral triangle whose vertices lie in different disks.
(b) Prove that every equilateral triangle with vertices in different disks has side
length greater than 96.

655.On a cylindrical surface of radiusr, unbounded in both directions, considernpoints
and a surfaceSof area strictly less than 1. Prove that by rotating around the axis
of the cylinder and then translating in the direction of the axis by at most 4 πrn units
one can transformSinto a surface that does not contain any of thenpoints.

4.2 Trigonometry..................................................

4.2.1 Trigonometric Identities...................................

The beauty of trigonometry lies in its identities. There are two fundamental identities,

sin^2 x+cos^2 x=1 and cos(x−y)=cosxcosy−sinxsiny,

both with geometric origins, from which all the others can be derived. Our problems will
make use of addition and subtraction formulas for two, three, even four angles, double-
and triple-angle formulas, and product-to-sum formulas.

Example.Find all acute anglesxsatisfying the equation

2 sinxcos 40◦=sin(x+ 20 ◦).

Solution.Trying particular values we see thatx = 30 ◦is a solution. Are there other
solutions? Use the addition formula for sine to rewrite the equation as