Mathematics Enrichment Club Problem Sheet 14, Slides of Mathematics

Download Mathematics Enrichment Club Problem Sheet 14 and more Slides Mathematics in PDF only on Docsity! MATHEMATICS ENRICHMENT CLUB. Problem Sheet 14, August 18, 20151 1. A number of cards are to be taken out of a standard 52-cards deck. If the resultant deck contains all four aces, with a probability to select all aces being 1/1001. How many cards had been taken out of the original deck? 2. Let n! denote the factorial of n; i.e n! = n× (n− 1)× (n− 2)× . . .× 2× 1. Find the largest integer n, such that 1 + 2! + 3! + . . . + (n− 1)! + n! is a perfect square. 3. Let a1, a2, . . . , a100 be a sequence of consecutive positive integers. Find the minimum value of √ a2 + a3 + . . . + a99 − √ a1 + a100. 4. A 6-digit number is increased 6 times when its last 3 digits are carried to the beginning of the number without their order being changed. Find this number. 5. A large number of brown, green and yellow frogs are wandering around on an island. Whenever two frogs of different colours meet each other, they change immediately into two frogs of the third colour. More than two frogs never meet simultaneously. If there are initially 2014 brown frogs, 2015 green frogs and 2016 yellow frogs on the island, is it possible that at some future time all the frogs will have the same colour? 6. On the sides of triangle 4ABC, three similar triangles are constructed with 4Y BA and 4ZAC in the exterior and 4XBC in the interior. Above, the vertices of the triangles are ordered so that the similarities takes vertices to corresponding vertices; for example, the similarity between 4Y BA and 4ZAC takes Y to Z, B to A and A to C. (a) Show that the triangles Y BX and ZXC are similar to ABC. (b) Use part (a) or otherwise, prove that AYXZ is a parallelogram. 1Some problems from UNSW’s publication Parabola, and the Tournament of Towns in Toronto. 1