## A Similar Triangle and Area Question

Question & Solutions are in the images

## An Angle Bisector in a Parallelogram Question

Question & Solutions are in image below except the (a) part whose solution is as follows: EAB = 1/2(BAC) And EBA = 1/2(DBA) So EAB + EBA = 1/2(BAC + DBA) = 1/2(180) = 90 *Note: EBA = angle (EBA) So for (a), we write: AEB = 180 – (EAB + EBA) =...

## Find the Minimum Value of k

Question We are given 2014 positive numbers such that x(1) < x(2) < x(3) <… < x(2014) It is known that the number x(k) is 19 times the average of the 2014 numbers. Determine the “minimum” possible value of k A Possible Solution (Please refer to the attached image) I think the answer...

## A Basic PigeonHole Question

Question Allison has a box of coloured ribbons. There are 12 red ribbons, 11 white ribbons, 10 yellow ribbons, 7 blue ribbons and 4 black ribbons. What is the least number of coloured ribbons Allison must take to ensure that she will have 6 ribbons of the same colour? Solution The Highest number of...

## Most Number of Factors

Question Determine the three digit number with the most number of factors Solution Found in the images (with close-up shots)

## A property of Regular Octagons

Question Prove that the area of a regular octagon is the product of its longest diagonal and its shortest diagonal Solution There’s an easier solution by cutting & pasting cleverly transforming the octagon into the area of a rectangle formed by cutting and pasting (where the length & width of this rectangle are the...

## Consecutive Integers and Prime Factors

Question Find three consecutive integers greater than 2000, such that each integer is a product of exactly three distinct prime numbers. Solution A simple way for the above would be to list down the prime factors for the numbers above 2000 & note that the numbers 2013, 2014 & 2015 are a possible answer....

## A Sum of Digits Problem

Question I wrote down all integers from 1 to 711. Next, I added up all the digits that I have written. I get the number S. Prove that S is divisible by 711. Solution The idea is to separately count “the sum of digits”. For example, what is the sum of all digit “1’s”...

## Find the Maximum Value of k

Question We are given 2014 positive numbers such that x(1) < x(2) < x(3) <… < x(2014) It is known that the number x(k) is 19 times the average of the 2014 numbers. (a) Determine the maximum possible value of k (b) Give an example of 2014 numbers x(1) < x(2) < x(3)...

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