MathDB

4

Part of 1997 Ukraine National Mathematical Olympiad

Problems(4)

two players

Source: Ukraine 1997

7/18/2009
Two players alternately write the numbers 1,2,...,7 1,2,...,7 in the cells of the figure on the picture. After the figure is filled up, the sum of numbers along each line on the picture is calculated. If three of these sums are equal, the first player wins; otherwise the second player wins. Which of the players has a winning strategy?
combinatorics proposedcombinatorics
positive integer a

Source: Ukraine 1997 grade 9

7/18/2009
Does there exist a positvie integer a a such that all the numbers a,2a,3a,...,1997a a,2a,3a,...,1997a are perfect powers (i.e. numbers of the form mk m^k, where m,k∈N m,k \in \mathbb{N}, k≥2 k \ge 2)?
number theory unsolvednumber theory
perpendicular planes

Source: Ukraine 1997 grade 10

7/21/2009
Two regular pentagons ABCDE ABCDE and AEKPL AEKPL are placed in space so that \angle DAK\equal{}60^{\circ}. Prove that the planes ACK ACK and BAL BAL are perpendicular.
geometry proposedgeometry
equations

Source: Ukraine 1997 grade 11

7/22/2009
The equation ax^3\plus{}bx^2\plus{}cx\plus{}d\equal{}0 has three distinct solutions. How many distinct solutions does the following equation have: 4(ax^3\plus{}bx^2\plus{}cx\plus{}d)(3ax\plus{}b)\equal{}(3ax^2\plus{}2bx\plus{}c)^2?
algebra proposedalgebra