MathDB

7

Part of 1997 Ukraine National Mathematical Olympiad

Problems(3)

inequality

Source: Ukraine 1997 grade 9

7/21/2009
Triangles ABC ABC and A1B1C1 A_1 B_1 C_1 are non-congruent, but AC\equal{}A_1 C_1\equal{}b, BC\equal{}B_1 C_1\equal{}a, and BH\equal{}B_1 H_1, where BH BH and B1H1 B_1 H_1 are the altitudes. Prove the inequality: a \cdot AB\plus{}b \cdot A_1 B_1 \le \sqrt{2}(a^2\plus{}b^2).
inequalitiesgeometry proposedgeometry
equal areas

Source: Ukraine 1997 grade 10

7/21/2009
In a parallelogram ABCD ABCD, M M is the midpoint of BC BC and N N an arbitrary point on the side AD AD. Let P P be the intersection of MN MN and AC AC, and Q Q the intersection of AM AM and BN BN. Prove that the triangles BDQ BDQ and DMP DMP have equal areas.
geometryparallelogramgeometry unsolved
find the smallest natural number

Source: Ukraine 1997 grade 11

7/22/2009
Determine the smallest natural number n n such that among any n n integers one can choose 18 18 integers whose sum is divisible by 18 18.
number theory proposednumber theory