2002 Greece National Olympiad

Part of Greece National Olympiad

Subcontests

(4)

1

A student

A student of the National Technical University was reading advanced mathematics last summer for 37 days according to the following rules : (a) He was reading at least one hour every day. (b) He was reading an integer number of hours, but not more than 12, each day. (c) He had to read at most 60 hours in total. Prove that there were some successive days during which the student was reading exactly 13 hours in total.

1

In a triangle.

ABC

\angle C>10^0

\angle B=\angle C+10^0.

We consider point

E

AB

\angle ACE=10^0,

D

AC

\angle DBA=15^0.

Z\neq A

be a point of interection of the circumcircles of the triangles

ABD

AEC.

\angle ZBA>\angle ZCA.

1

Perfect square.

(a) Positive integers

p,q,r,a

pq=ra^2

r

p,q

are relatively prime. Prove that one of the numbers

p,q

is a perfect square. (b) Examine if there exists a prime

p

p(2^{p+1}-1)

is a perfect square.

1

Inequality

The real numbers

a,b,c

bc\neq0

\frac{1-c^2}{bc}\geq0.

10(a^2+b^2+c^2-bc^3)\geq2ab+5ac.