5 problems

These problems were selected from the book "In Polya's Footsteps, Miscellaneous problems and Essays" by Ross Honsberger. Each problem requires different problem solving skills. It will be good to note down what was the thinking process behind each solution.

1) 4 consecutive even numbers are removed from the set A={1,2,...n} If the average of the remaining numbers is 51.5625, which 4 numbers are removed?

2) Suppose u and v are real numbers such that

u+u^2+...+u^8+10u+^9=v+v^2+...+v^(10)+10v^(11)=8

Which is bigger? u or v?

3) Suppose a quadilateral ABCD has a circumcircle with centre O and incircle with centre I. Let E be the intersection of the diagonals AC and BD. Show that O, I and E are collinear.

4) Consider an acute triangle ABC. Suppose the median, angle bisector and altitude from A to BC cuts the angle BAC into 4 equal angles. What is the angle BAC?

5) Show that for any positive integer, there exists n consecutive integers such that none of which is an integral power of a prime.

Logical thinking in mathematical problem solving

This example requires some logical thinking:

4 boys 


4 boys A,B,C,D sit in a straight line, facing 1 direction. They cannot turn their heads or leave their chairs. A wall seperates A and B. They are told that 2 black and 2 white hats are placed on their heads.

A and C are wearing black hats, while B and D are wearing white hats.

D can see B and C's hats, C can see B's hat. B is facing the wall, and A cannot see any hat, since he is on the other side of the wall.
They are only given 1 chance to get their own hats right by shouting out the answer. After 1 minute, someone calls out?

1) Who called out? How can he be 100% sure?
2) Explain how each boy managed to figure out their hat's color.

( Problem courtesy of Cassandra)

Spot the difference

AMTUVWY
BCDEK
NSZ
HIOX
FGJLPQR

The alphabet is grouped into 5 rows as above. How are these rows classified? What is the difference between each row?

Evil numbers

In arithmetic, there are common terms such as rational numbers, irrational numbers, whole numbers, natural numbers etc. Here are some less known numbers:

1) Evil numbers: Numbers with a binary representation with even number of 1s. An example of an evil number is pi
2)Happy numbers
3) Unhappy numbers, ie numbers which are not happy.
4) Wasteful numbers: If x is a wasteful number, its prime factorisation has more digits (including the powers)than the number of digits of x. An example is 4,6,8,9,12,18 etc.
5) Frugal numbers: A non wasteful number and 1.
6) Economic numbers: Frugal numbers excluding 1.
7) I like this one the most: Lucky numbers

There are many other types of numbers. Here's a nice article on number gossip.

Constructing square root 2 for lower sec

Squares 


In the secondary 1/2 syllabus for math, we talk about numbers. For instance, rational numbers and irrational numbers. Now how can we show the students a line segment of length square root 2? Moreover, these students need not have prior knowledge of pythagoras' theorem. So showing them an isoceleles triangle with 2 sides 1 and hypothenus sqrt2 may not be the best way to go.

Let us consider 2 squares each of length 1. So each has area 1 (unit, can be cm, m or anything else to make things concrete, I'll just discuss in abstract) Question: From these 2 squares, can you construct a square of area 2?

The above diagram represents the solution. Cut the 2 small squares into half along the diagonal and piece them together to form the square on the right. That square will be of area 2. So the length of the square is sqrt2. And hence we have demonstrated the construction of a line segment of sqrt2 without using any fancy maths.

Math Olympiad Question

From a math olympiad question targeted at lower secondary students:

Suppose a number x can be expressed as a sum of squares of the 4 of its smallest divisors, what is the highest prime number that can divide x?

For example,
130=(1^2)+(2^2)+(5^2)+(10^2)