1988 Greece Junior Math Olympiad

Part of Greece Junior Math Olympiad

Subcontests

(4)

1

1,2 or 4 axes of symmetry - Greece Juniors 1988 p2

Draw the smaller number of line segments connecting points of the figure such that the new figure obtained to have exactly: https://cdn.artofproblemsolving.com/attachments/d/1/098e03714904573a1eacd2d3dc28b4e8c42c7c.png i) one axis of symmetry ii) two axes of symmetry iii) four axes of symmetry Draw a new figure, at each case.

1

algebraic exrpessions - Greece Juniors 1988 p4

b^2+c^2=a^2, \,\,\,\, b\ne \pm c

, calculate the expression

\frac{b^3+c^3}{b+c}+\frac{b^3-c^3}{b-c}

a+\frac{1}{a}=k, a\ne 0

, find the expression

a^4+\frac{1}{a^4}

k

1

factor polynomials - Greece Juniors 1988 p3

Consider the polynomials

P(x)=x^4-3x^3+x-3,\,\,\,\,Q(x)=x^2-2x-3 \,\,\,\, R(x)=-x^2-5x+a

a \in

R such that polynomial

R(x)

x-2

ii) Factor polynomials

P(x),Q(x)

iii) Prove that exrpession

-x^2+x+\frac{P(x)}{Q(x)}+15

is a perfect square.

1

simplify fractions - Greece Juniors 1988 p1

\left(a-\frac{4ab}{a+b}+b\right): \left(\frac{a}{a+b}-\frac{b}{b-a}-\frac{2ab}{a^2-b^2}\right)

\frac{2x^2-(3a+b)x+a^2+ab}{2x^2-(a+3b)x+ab+b^2}