2

Part of 2003 Tuymaada Olympiad

Problems(2)

A quadrilateral with two equal sides. One angle to find.

Source: Tuymaada 2003, day 1, problem 2.

In a quadrilateral

ABCD

AB

CD

\angle A=150^\circ,

\angle B=44^\circ,

\angle C=72^\circ.

Perpendicular bisector of the segment

AD

BC

P.

\angle APD.

Proposed by F. Bakharev

trigonometrygeometryperpendicular bisectorgeometry proposed

Positive integers of the form 2x² - 3y² and 10xy - x² - y².

Source: Tuymaada 2003, day 2, problem 2.

Which number is bigger : the number of positive integers not exceeding 1000000 that can be represented by the form

2x^{2}-3y^{2}

x

y

or that of positive integers not exceeding 1000000 that can be represented by the form

10xy-x^{2}-y^{2}

x

y?

Proposed by A. Golovanov

modular arithmeticnumber theory proposednumber theory