6

Part of 2005 Ukraine National Mathematical Olympiad

Problems(4)

equality of angles

Source: Ukraine 2005 grade 8

A convex quadrilateral

ABCD

with BC\equal{}CD and \angle CBA\plus{}\angle DAB>180^{\circ} is given. Suppose that

W

Q

are points on the sides

BC

DC

respectively (distinct from the vertices) such that AD\equal{}QD and

WQ \parallel AD.

Also suppose that

AQ

BD

intersect at a point

M

that is equidistant from the lines

AD

BC

. Prove that angle \angle BWD\equal{}\angle ADW.

geometry proposedgeometry

inequality in positive numbers

Source: Ukraine 2005 grade 10

a,b,c

are positive real numbers, prove the inequality: \frac {a^2}{b} \plus{} \frac {b^3}{c^2} \plus{} \frac {c^4}{a^3} \ge \minus{} a \plus{} 2b \plus{} 2c. P.S. Thank you for the observation. I've already corrected it.

inequalitiesinequalities unsolved

Source: Ukraine 2005 grade 9

For every positive integer

n

, prove the inequality: \frac{3}{1!\plus{}2!\plus{}3!}\plus{}\frac{4}{2!\plus{}3!\plus{}4!}\plus{}...\plus{}\frac{n\plus{}2}{n!\plus{}(n\plus{}1)!\plus{}(n\plus{}2)!}<\frac{1}{2}.

inequalitiesinequalities proposed

Source: Ukraine 2005 grade 11

A polygon on a coordinate grid is built of

2005

1 \times 2

. What is the smallest number of sides of an even length such a polygon can have?

analytic geometryinductionmodular arithmeticgeometryperimetercalculusintegration