Maths Olympiad - exercise your grey matter!

[Fat Tone]

I doubt many murcers qualify to enter the competition, but the questions are fun to think about.

http://www.math.le.ac.uk/OLYMPIAD/

Here are the problems:

1. The difference of two numbers is equal to 0.01. Can the difference of their squares be greater than 1000?


2. A box of fresh mushrooms weighs 10 kg. At first, water made up 99% of the weight of mushrooms. After a while, the mushrooms have dried up so that now water makes up only 98% of their weight. How much does the box of mushrooms weigh now?


3. Is it possible to come up with four integers such that both their product and their sum are odd?


4. Each of the following equalities is missing numerators (which are positive integers): ?/7-?/5=1/35, ?/5-?/7=1/35. Find as many possible pairs of numerators as you can. Note that irregular fractions (i.e. fractions whose numerator is larger than the denominator) are allowed. For example, 2 and 4 are NOT a solution for the first equality: 2/7-4/5=10/35-28/35=-18/35.


5. Is it possible to wrap a unit cube into a square piece of paper with side of length 3? (You may not cut the piece of paper.)


6. There are two villages on the same side of the river. How should a road be built from one village to the other if it has to be of the smallest possible length and must touch the river?


7. Given an angle and a point P inside of it draw a straight line through this point in such a way that the segment of this line inside the angle has the point P as its midpoint.


8. Among the lines passing through the point A choose the one for which the sum of the distances to the points B and C is the maximum possible.


9. Consider a unit square in the plane. Find all the points such that the sum of the shortest distances from that point to the sides of the square or their extensions is 3.


10. You have 16 coins. You know that each of them has a different weight. Find the lightest and the heaviest coins by taking at most 22 measurements using a scale (balance) with two cups without using any extra weights.

Now lets see how long it take Gurm to either answer them all, or explain why they are fundamentally flawed

[Brian Ellis]

I haven't looked at the site, but a number of the questions, as you have quoted them, appear ambiguously worded.

OK, here's no. 11. You have a perfectly spherical ball of chocolate 20 mm diameter which you wish to wrap in a piece of aluminium foil 0.05 mm thick such that it is rectangular (including square) with a minimum 2 mm overlap at all edges. What are the dimensions of the foil with the minimum surface area? Just to make things clear, I do not know the answer!!!

Now, to make things a little harder, instead of a sphere, you have a chocolate Easter Egg of perfectly circular cross section of 20 mm diameter across the middle and a perfect ellipse of 20 mm x 30 mm across the length. ???

[SitFlyer]

The answer to # 1 is yes

[Marshmallowman]

2 is 5

10 is yes

and I think 5 might be yes.

But alot of the rest are pretty vague, for instance with 6, if the river is straight and both towns are on the river its obviously a straight line. but I guess a lot of these problems involve coming up with a general math solution.eg formulae not just yes/no or simple amounts.

[Dilitante1]

#2 9.9Kg

[VJ]

#3: no (I think)
If the sum has to be odd, you can't have 4 odd numbers.
As soon as you have one even number, the product cannot be odd.


Jörg

[Marshmallowman]

#2 9.9Kg

1% of 10kg is .1 kg(mass of mushroom is constant)

2% of 5 kg is .1kg

[Umfriend]

No. 5: No. You need 4.

[Umfriend]

6.: A 2nd order function I'd say.

Edit: I'm being silly here. It most certainly is nbot a 2nd order function.

[Dilitante1]

10K x .98 (from .99 when fresh) = 9.8 kg

[Umfriend]

But Dil, if 9.8K is not equal to 98% of 9.9K...... And 9.9K is the total weight if 9.8K is water, so you're not meeting the constraint.

[Marshmallowman]

#5, rotate the cube 45 degree and place in the centre of 3x3, you will see the corners meet quite neatly on top when you wrap...actually I am not 100% sure, as I don't have a cube test it with but I am pretty sure.

#10, measure 8 pairs , putting 8 in the heavy pile and 8 in the light pile
then of those 8 in the heavies measure 4 pairs, keep the heavies, then 2 then 1
then you do the same for the lights 4,2 1 giving exactly 22 measures all up to find the lightest and heaviest

[Dilitante1]

10 kg = 1% whatever + 99% water
now = 1% whatever + 98% water so 9.9 WAS correct

[Marshmallowman]

99% water 1% dry mushroom
so 1% of 10kg is .1kg,

now we have 98% water 2% dry mushroom

eg the % of dry mushrooms has doubled so, they the total weight must have halved

5Kg * .98 = 4.9 kg
5kg * .02 = 0.1 kg
total 100% =5kg

the weight of dry mushroom must remain the same.

[Dilitante1]

MM it doesnt say the mushrooms are dry, it says they lost 1% water weight....from 99% to 98%