News: IMO Day 2 (26 July 2007)

The second IMO 2007 paper was sat in Hanoi on 26 July 2007; the problems and the countries submitting them are shown. Coordination takes place on 27–28 July and the final Jury meeting to approve scores and determine medal boundaries is scheduled for 22:00 local time (16:00 BST) on 28 July.

4. In triangle ABC the bisector of angle BCA intersects the circumcircle again at R, the perpendicular bisector of BC at P, and the perpendicular bisector of AC at Q. The midpoint of BC is K and the midpoint of AC is L. Prove that the triangles RPK and RQL have the same area.

   (Czech Republic)

5. Let a and b be positive integers. Show that if 4ab − 1 divides (4a² − 1)², then a = b.

   (United Kingdom)

6. Let n be a positive integer. Consider

   S = { (x, y, z) : x, y, z ∈ {0, 1, …, n}, x + y + z > 0 }

   as a set of (n + 1)³ − 1 points in three-dimensional space. Determine the smallest possible number of planes, the union of which contains S but does not include (0, 0, 0).

   (Netherlands)

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Next news item: IMO 2007 results (28 July 2007).