2022 Azerbaijan National Mathematical Olympiad

Part of Azerbaijan Senior National Olympiad

Subcontests

(4)

1

well known property of centroid with parallel line from A

\omega

be the circumcircle of an acute angled tirangle

ABC.

The line tangent to

\omega

A

intersects the line

BC

T.

Let the midpoint of segment

AT

N,

and the centroid of

\triangle ABC

G.

The other tangent line drawn from

N

\omega

\omega

L.

LG

\omega

S\neq L.

AS\parallel BC.

1

Grasshopper jumps to adjacent cells

Each cell of the 4x4 board has a grasshopper. When a grasshopper jumps, it moves to one of the adjacent cells (down, up, right, or left). The grasshopper cannot move diagonally or go off the board. At most how many cells can remain empty after each grasshopper jumps once?

1

system 6x4 , x_1 + x_2 = x^2_3 + x^2_4 + 6x_3x_4

Find all quadruplets

(x_1, x_2, x_3, x_4)

of real numbers such that the next six equalities apply:

\begin{cases} x_1 + x_2 = x^2_3 + x^2_4 + 6x_3x_4\\ x_1 + x_3 = x^2_2 + x^2_4 + 6x_2x_4\\ x_1 + x_4 = x^2_2 + x^2_3 + 6x_2x_3\\ x_2 + x_3 = x^2_1 + x^2_4 + 6x_1x_4\\ x_2 + x_4 = x^2_1 + x^2_3 + 6x_1x_3 \\ x_3 + x_4 = x^2_1 + x^2_2 + 6x_1x_2 \end{cases}

1

set of positive integers such that 2x^2 + 3y^3 = 4z^4

A

be the set of all triples

(x, y, z)

of positive integers satisfying

2x^2 + 3y^3 = 4z^4

. a) Show that if

(x, y, z) \in A

6

x, y, z

A

is an infinite set.