31 May 2014

IrMO 1999 P1 Q5: Upper Secondary Mathematics Competition Question


Three real numbers a, b, c with a < b < c, are said to be in arithmetic progression if c - b = b - a.

Define a sequence u(n), n = 0, 1, 2, 3, ... as follows: u(0) = 0, u(1) = 1 and, for each n > 0, u(n+1) is the smallest positive integer such that u(n+1) > u(n) and {u(0), u(1),... u(n), u(n+1)} contains no three elements that are in arithmetic progression.

Find u(100).


[Irish MO, Paper 1, Question 5, 1999]


Sums of Naturals: Middle Secondary Mathematics Competition Question

Let N be a natural number with the property that it is the sum of 3 consecutive natural numbers, of 4 consecutive naturals and also of 5 consecutive naturals.

Find the smallest value of N such that the 3 sequences above are disjoint, that is, there is no number that is in more than one sequence.

25 May 2014

Pairs of Unit Fractions: Upper Secondary Mathematics Competition Question

For each positive integer n, let S(n) be the set of ordered pairs (x,y) of positive integers such that

1/n = 1/x + 1/y

and let T(n) be the number of ordered pairs in S(n).

For example, for n=2, S(2)={(3,6), (4,4), (6,3)} and hence T(2)=3.

a) Determine T(n) for all n.

b) Hence, calculate T(2014).


24 May 2014

Four Perimeters: Upper Primary Mathematics Competition Question

A rectangle is divided into four smaller rectangles using two perpendicular lines, as shown in the diagram.

The number inside each small rectangle indicates the length of its perimeter. The diagram is not drawn to scale.

What value should go into the empty rectangle?






Six Eggs in a Box: Lower Secondary Mathematics Competition Question

Edward likes eggs. Not necessarily to eat them; he likes playing with them too. Today, he has taken eggs from different boxes so that he has 3 white eggs and 3 brown eggs sitting in a 6-egg box. He is thinking about how many patterns he can make with his 6 eggs.

How many different arrangements can Edward make using all 6 eggs in the one box?

As the box has a lid, any similar patterns under rotation count as two distinct arrangements.


Feel free to comment, ask questions and even check your answer in the comments box below powered by  Disqus Google+.

}

This space is here to avoid seeing the answers before trying the problem!

}

If you enjoy using this website then please consider making a donation - every little helps :-)

You can receive these questions directly to your email box or read them in an RSS reader. Subscribe using the links on the right.

Don’t forget to follow Gifted Mathematics on Google+Facebook or Twitter. You may add your own interesting questions on our Google+ Community and Facebook..

You can also subscribe to our Bookmarks on StumbleUpon and Pinterest. Many resources never make it onto the pages of Gifted Mathematics but are stored in these bookmarking websites to share with you.

Divisibility by 99: Middle Secondary Mathematics Competition Question

You have nine cards numbered from 1 to 9. If you pick each card randomly and lay them out in order, find the probability that the resulting 9-digit number is divisible by 99.

Express this probability as m/n, where m and n are relatively prime. What is the sum (m + n)?



Feel free to comment, ask questions and even check your answer in the comments box below powered by Google+.

}

This space is here to avoid seeing the answers before trying the problem!

}

If you enjoy using this website then please consider making a donation - every little helps :-)

You can receive these questions directly to your email box or read them in an RSS reader. Subscribe using the links on the right.

Don’t forget to follow Gifted Mathematics on Google+Facebook or Twitter. You may add your own interesting questions on our Google+ Community and Facebook..

You can also subscribe to our Bookmarks on StumbleUpon and Pinterest. Many resources never make it onto the pages of Gifted Mathematics but are stored in these bookmarking websites to share with you.
Related Posts Plugin for WordPress, Blogger...