Soal Matematika : Olympiad Corner - Mathematical Excalibur

The following were the problems of the first day of the 2008 Chinese Girls’ Math Olympiad.

Problem 1. (a) Determine if the set {1,2,⋯,96} can be partitioned into 32 sets of equal size and equal sum.

(b) Determine if the set {1,2,⋯,99} can be partitioned into 33 sets of equal size and equal sum.

Problem 2. Let φ(x) = ax³+bx²+cx+d be a polynomial with real coefficients. Given that φ(x) has three positive real roots and that φ(x) < 0, prove that 2b³ + 9a2d − 7abc ≤ 0.

Problem 3. Determine the least real number a greater than 1 such that for any point P in the interior of square ABCD, the area ratio between some two of the triangles PAB, PBC, PCD, PDA lies in the interval [1/a, a].

Problem 4. Equilateral triangles ABQ, BCR, CDS, DAP are erected outside the (convex) quadrilateral ABCD. Let X, Y, Z, W be the midpoints of the segments PQ, QR, RS, SP respectively. Determine the maximum value of

Diambil dari Mathematical Exalibur Vol.14 No.2