MathDB

7

Part of 2005 Ukraine National Mathematical Olympiad

Problems(3)

game

Source: Ukraine 2005 grade 9

7/26/2009
Under the rules of the "Sea battle" game (no two ships may have a common point), can the following sets of rectangular ships be arranged on a 10×10 10 \times 10 square board: (a) (a) two ships 1×4 1 \times 4, 4 4 ships 1×3 1 \times 3, 6 6 ships 1×2 1 \times 2, and 8 8 ships 1×1 1 \times 1; (b) (b) two ships 1×4 1 \times 4, 4 4 ships 1×3 1 \times 3, 6 6 ships 1×2 1 \times 2, 6 6 ships 1×1 1 \times 1, and 1 1 ship 2×2 2 \times 2; (c) (c) two ships 1×4 1 \times 4, 4 4 ships 1×3 1 \times 3, 6 6 ships 1×2 1 \times 2, 4 4 ships 1×1 1 \times 1, and 2 2 ships 2×2 2 \times 2?
combinatorics unsolvedcombinatorics
find the measure of an angle

Source: Ukraine 2005 grade 10

7/28/2009
A point M M is taken on the perpendicular bisector of the side AC AC of an acute-angled triangle ABC ABC so that M M and B B are on the same side of AC AC. If \angle BAC\equal{}\angle MCB and \angle ABC\plus{}\angle MBC\equal{}180^{\circ}, find ∠BAC. \angle BAC.
geometryperpendicular bisectorgeometry proposed
divisibility

Source: Ukraine 2005 grade 11

7/28/2009
Prove that for any integers n≥2 n \ge 2 there is a set An A_n of n n distinct positive integers such that for any two distinct elements i,j \in A_n, |i\minus{}j| divides i^2\plus{}j^2.
inductionnumber theory unsolvednumber theory