30th Spanish 1994


A1. Show that if an (infinite) arithmetic progression includes a square, then it must include infinitely many squares.
A2. Take three-dimensional coordinates with origin O. C is the point (0,0,c). P is a point on the x-axis, and Q is a point on the y-axis such that OP + OQ = k, where k is fixed. Let W be the center of the sphere through O, C, P, Q. Let W' be the projection of W on the xy-plane. Find the locus of W' as P and Q vary. Find also the locus of W as P and Q vary.
A3. The tourism office is collecting figures on the number of sunny days and the number of rainy days in the regions A, B, C, D, E, F. sunny/rainy unclassifiable A 336 29 B 321 44 C 335 30 D 343 22 E 329 36 F 330 35 If one region is excluded then the total number of rainy days in the other regions is one-third of the total number of sunny days in those regions. Which region is excluded?
B1. The triangle ABC has ∠A = 36^o, ∠B = 72^o, ∠C = 72^o. The bisector of ∠C meets AB at D. Find the angles of BCD. Express the length BC in terms of AC, without using any trigonometric functions.
B2. 21 counters are arranged in a 3 x 7 grid. Some of the counters are black and some white. Show that one can always find 4 counters of the same color at the vertices of a rectangle.
B3. A convex n-gon is divided into m triangles, so that no two triangles have interior points in common, and each side of a triangle is either a side of the polygon or a side of another triangle. Show that m + n must be even. Given m, n, find the number of triangle sides in the interior of the polygon and the number of vertices in the interior of the polygon.

To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.