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3

Part of 2006 Federal Competition For Advanced Students, Part 2

Problems(2)

37th Austrian Mathematical Olympiad 2006

Source: round 3, day1, problem 3

2/10/2009
The triangle ABC ABC is given. On the extension of the side AB AB we construct the point R R with BR \equal{} BC, where AR>BR AR > BR and on the extension of the side AC AC we construct the point S S with CS \equal{} CB, where AS>CS AS > CS. Let A1 A_1 be the point of intersection of the diagonals of the quadrilateral BRSC BRSC. Analogous we construct the point T T on the extension of the side BC BC, where CT \equal{} CA and BT>CT BT > CT and on the extension of the side BA BA we construct the point U U with AU \equal{} AC, where BU>AU BU > AU. Let B1 B_1 be the point of intersection of the diagonals of the quadrilateral CTUA CTUA. Likewise we construct the point V V on the extension of the side CA CA, where AV \equal{} AB and CV>AV CV > AV and on the extension of the side CB CB we construct the point W W with BW \equal{} BA and CW>BW CW > BW. Let C1 C_1 be the point of intersection of the diagonals of the quadrilateral AVWB AVWB. Show that the area of the hexagon AC1BA1CB1 AC_1BA_1CB_1 is equal to the sum of the areas of the triangles ABC ABC and A1B1C1 A_1B_1C_1.
geometryincenterparallelogramgeometry unsolved
37th Austrian Mathematical Olympiad 2006

Source: round 3, day2, problem 3

2/10/2009
Let A A be an integer not equal to 0 0. Solve the following system of equations in Z3 \mathbb{Z}^3. x \plus{} y^2 \plus{} z^3 \equal{} A \frac {1}{x} \plus{} \frac {1}{y^2} \plus{} \frac {1}{z^3} \equal{} \frac {1}{A} xy^2z^3 \equal{} A^2
algebrasystem of equationsalgebra unsolved