1. Let AC be a line segment in the plane and B a point between A and C. Construct isosceles triangles PAB and QBC on one side of the segment AC such that ∠APB = ∠BQC = 1200 and an isosceles triangle RAC on the otherside of AC such that ∠ARC = 1200. Show that PQR is an equilateral triangle.
2. Produce AP and CQ to meet at K. Observe that AKCR is a rhombus and BQKP is a parallelogram.(See Fig.) Put AP = x,CQ = y. Then PK = BQ = y, KQ = PB = x and AR = RC = CK = KA = x + y. Using cosine rule in triangle PKQ, we get PQ2 = x2 + y2 − 2xy cos 1200 = x2 + y2 + xy. Similarly cosine rule in triangle QCR gives QR2 = y2 +(x+y)2 −2xy cos 600 = x2 +y2 +xy and cosine rule in triangle PAR gives RP2 = x2 + (x + y)2 − 2xy cos 600 = x2 + y2 + xy. It follows that PQ = QR = RP.
3. Suppose 〈x1, x2, . . . , xn, . . .〉 is a sequence of positive real numbers such that x1 ≥ x2 ≥
x3 ≥ · · · ≥ xn · · · , and for all n
Show that for all k the following inequality is satisfied:
4. All the 7-digit numbers containing each of the digits 1, 2, 3, 4, 5, 6, 7 exactly once, and not
divisible by 5, are arranged in the increasing order. Find the 2000-th number in this list.
5. The internal bisector of angle A in a triangle ABC with AC > AB, meets the circumcircle Γ of the triangle in D. Join D to the centre O of the circle Γ and suppose DO meets AC in E, possibly when extended. Given that BE is perpendicular to AD, show that AO is parallel to BD.
6. (i) Consider two positive integers a and b which are such that aabb is divisible by 2000. What is the least possible value of the product ab?
(ii) Consider two positive integers a and b which are such that aabb is divisible by 2000. What is the least possible value of the product ab?
7. Find all real values of a for which the equation x4 −2ax2 +x+a2 −a = 0 has all its roots real.