Problems and Solutions

Jai Ganesh · 2005-10-26 20:38:05

Problem # k + 40

How many are we?
You tell us the answer, knowing that the probability that at least two of us have birthdays on the same day is less than half, but that this would not be the case were we one more in number.

mathsyperson · 2005-10-26 23:07:28

I've seen k+40 before, so I'll leave it for someone else. It's really counter-intuitive though.

Last edited by mathsyperson (2005-10-26 23:08:24)

mathsyperson · 2005-10-27 08:57:23

That puzzle looks very similar to kind of thing you get in the UKMT maths challenges and I'm doing the senior one in *checks* 12 days, so thanks for the practice!

Jai Ganesh · 2005-10-27 15:47:43

Well done, Mathsy!

Jai Ganesh · 2005-10-27 16:48:06

Problem # k + 41

Prove that the number 512³ + 675³ + 720³ is not prime.

mathsyperson · 2005-10-27 21:34:53

Wow, that's a tough one. I tried splitting it into prime factors and looking for common ones, but that doesn't work because there aren't any. I could just actually calculate the number and then prove that it has factors, but it seems like there's an easier way that I'm missing.

Jai Ganesh · 2005-10-28 17:38:11

Problem # k + 42

On a deserted island live five people and a monkey. One day everybody gathers coconuts and puts them together in a community pile, to be divided the next day. During the night one person decides to take his share himself. He divides the coconuts into five equal piles, with one coconut left over. He gives the extra coconut to the monkey, hides his pile, and puts the other four piles back into a single pile. The other four islanders then do the same thing, one at a time, each giving one coconut to the monkey to make the piles divide equally. What is the smallest possible number of coconuts in the original pile?

mathsyperson · 2005-11-05 11:29:15

I didn't do that because I was trying to think of a more elegant way. But maybe such a way doesn't exist...

jU · 2005-11-05 11:35:06

maybe

MathsIsFun · 2005-11-05 21:42:12

For me "Elegant" usually involves simplicity, beauty and a certain effortless quality.

Imagine a movie where the master swordsman is suddenly surrounded by 100s of armed guards ... and he just pulls out a machine gun and solves his problem in 5 seconds. Simple. Effortless. Elegant ... ? But definitely effective!

BTW, where has Puzzlemaster Ganesh gone?

Jai Ganesh · 2005-11-06 16:34:19

A week Diwali break!
(Diwali is the most important festivel for Hindus, the festival of color and lights, sweets and fireworks! The festival celebrated for victory of good over evil!)

MathsIsFun · 2005-11-06 23:10:37

OK ... so where are the photos, then? Hmmm ... ??

Jai Ganesh · 2005-11-08 22:44:43

Here........

Problem # k + 43

If x² + 1/x² = 23, what is the value of x^9 + 1/x^9 ?

Last edited by Jai Ganesh (2005-11-14 18:25:40)

Jai Ganesh · 2005-11-13 17:22:44

Problem # k + 44

It is believed that James Bernoulli calculated the sum of the tenth powers of the first 1,000 natural numbers in 7½ minutes (without a calculator or computer...they were non-existent during his period!). Given the same time limit, can you tell what the last diigit of the sum of the tenth powers of the first 1,000 natural numbers would be?

mathsyperson · 2005-11-14 11:06:24

Last edited by mathsyperson (2005-11-14 11:06:35)

Jai Ganesh · 2005-11-14 18:27:59

Problem # k + 45

Find the sum of all the numbers that can be formed with the digits 1, 2, 3, 4, and 5 taken 4 at a time (without repetition).

Jai Ganesh · 2005-11-17 23:51:54

Problem # k + 46

What is the largest 2-digit prime factor of the binomial coefficient 200C100?

Jai Ganesh · 2005-11-18 19:57:51

Problem # k + 47

n digits, none of them 0, are randomly (and independently) generated, find the probability that their product is divisible by 10.

mathsyperson · 2005-11-19 02:14:18

<-- Prepare yourself, this is really ugly.

I tried k+47 but I got an answer that gave values that were >1 for larger values of n, so I've gone wrong somewhere. I'll try to fix it later.

Jai Ganesh · 2005-11-23 01:33:31

Problem # k + 48

A rectangular sheet of paper is folded so that two diagonally opposite corners come together. If the crease formed is the same length as the longer side of the sheet, what is the ratio of the longer side of the sheet to the shorter side?

Jai Ganesh · 2005-11-25 16:33:24

Problem # k + 49

Without division, tell me which of the following fractions has the highest value?
a) 13/8
b) 1 5/9
c) 23/14
d) 39/22

Wanna be math wiz · 2005-11-29 04:51:07

This is what i got for k+49:
1. 13/8= 1 5/8 (cancelling out the idea that 1 5/9 is the largest value)
2. 23/14= 1 9/14
3. 39/22= 1 17/22

Finding a common demoninator- 8x14x22= 2464

1. 1 1540/2464
2. 1 1584/2464
3. 1 1904/2464

Thus making 39/22 the biggest fraction. Please tell if i did anything wrong.

Jai Ganesh · 2005-11-29 16:37:02

You are right! Well done!
However, instead of finding a common denominator, a much simpler method exists.
If you have to tell which of the fractions a/b or c/d is greater,
multiply a and d, and b and c. ad represents the first fraction, bc the second.
If ad>bc, the first fraction is higher. If ad<bc, the second fraction is higher.
When comparing three or more fractions, rule out the options by choosing the higher or lower of the two, whichever is required.
For example,
to tell which of 3/7, 4/11, 5/13 is highest,
first compare 3/7 and 4/11. 33 represents the first fraction and 28 the second. Now, compare the first and the third. 39 represents the first fraction and 35 the third. Therefore, 3/7 is the highest of the three. Simple, isn't it?

Jai Ganesh · 2005-11-29 16:43:39

Problem # k + 50

If a chord of length 16 cm is at a distance 6cm from the centre, what would be the distance of a chord of length 12 cm from the centre in the same circle?

mathsyperson · 2005-11-30 02:25:49

Last edited by mathsyperson (2005-11-30 02:26:00)

Math Is Fun Forum

#201 2005-10-26 20:38:05

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