2020 Greece Junior Math Olympiad

Part of Greece Junior Math Olympiad

Subcontests

(4)

1

(x+2)^4/x^3 - (x+2)^2/2x >= - x/16 (Greece Junior 2020 p1)

Solve in real numbers

\frac{(x+2)^4}{x^3}-\frac{(x+2)^2}{2x}\ge - \frac{x}{16}

1

Combinatorics

We are having 99 equal circles in a row and in the interior, we write inside them all the numbers from 1 up to 99 (one number in each circle).We color each of the circles with one of the two colors available: red and green. A coloring is good if it has the ability: Red circles lying in the interval of the numbers from 1 up to 50 are more than the red circles lying in the interval of the numbers from 51 up to 99 . a) Find how many different colorings can be constructed. b) Find how many different good colorings can be constructed. (Note: Two colorings are different, if they have different color in at least one of their circles.)

1

Number Theory

Find all positive integers

x

, for which the equation

a+b+c=xabc

has solution in positive integers. Solve the equation for these values of

x

1

angle chasing, in terms of angles (Greece junior 2019 p2)

ABC

be an acute-angled triangle with

AB<AC

D

be the midpoint of side

BC

BE,CZ

be the altitudes of the triangle

ABC

ZE

intersects line

BC

O

. (i) Find all the angles of the triangle

ZDE

in terms of angle

\angle A

of the triangle

ABC

(ii) Find the angle

\angle BOZ

in terms of angles

\angle B

\angle C

of the triangle

ABC