Thinking Skills Troubles ( Yours and Mine)

[abcde]

AoA!
Those Paper 1's and Paper 3's that sometimes knock you down don't have to do that anymore. (I'm sorry; good punchlines are harder to come up with than you think.)
I'm making this thread so that anyone can post questions that fall in the category of "problem solving" aka "the maths portion of TS". There aren't many books that cover it sufficiently well, the mark schemes have no explanations, and students often have no body to turn to when they need help. So the idea is to post as many problems and solutions as you can. Even if thinking skills is not your subject and you like a good puzzle/short mental challenge, by all means, do contribute!

 

[VelaneDeBeaute]

How many pairs of integers (x,y) are there such that 2x2−2xy+y2=225?
Clarification: x and y can be negative, and need not be distinct.

 

[abcde]

2x^2 - 2xy + y^2 = 225
=> x^2 + x^2 - 2xy + y^2 = 225
=> x^2 + (x-y)^2 = 15^2
Now find a Pythagorean Triple with 15 as the hypotenuse. Here's one: 9^2 + 12^2 = 15^2.
1) 9^2 + [9-(-3)]^2 = 15^2 ----> (9, -3)
2) 12^2 + (12-3)^2 = 15^2 ------> (12,3)
3) 15^2 + (15-15)^2 = 15^2 -----> (15,15)

So, 3 integer pairs? Let me know if that's wrong.

 

[VelaneDeBeaute]

2x^2 - 2xy + y^2 = 225
=> x^2 + x^2 - 2xy + y^2 = 225
=> x^2 + (x-y)^2 = 15^2
Now find a Pythagorean Triple with 15 as the hypotenuse. Here's one: 9^2 + 12^2 = 15^2.
1) 9^2 + [9-(-3)]^2 = 15^2 ----> (9, -3)
2) 12^2 + (12-3)^2 = 15^2 ------> (12,3)
3) 15^2 + (15-15)^2 = 15^2 -----> (15,15)

So, 3 integer pairs? Let me know if that's wrong.

Ingenious! I did not consider the first one!

Here's another.

A cube is inscribed within the sphere such that all the vertices of the cube touch the sphere. The volume of the cube is equal to 2×96√3, what is the radius of the sphere? ​

I was getting something close to 492, but that's supposed to be incorrect. ​

 

[abcde]

The volume is = 192 sqr 3, right? If so:
length of cube, l = cube root (192 sqr 3) = 6.92820
Use Pythagoras Theorem to get: r^2 + r^2 = l^2
=> r^2 = 48/2 = 24.
=> r = sqr 48 = 4.90. Is that the answer?

 

[VelaneDeBeaute]

The volume is = 192 sqr 3, right? If so:
length of cube, l = cube root (192 sqr 3) = 6.92820
Use Pythagoras Theorem to get: r^2 + r^2 = l^2
=> r^2 = 48/2 = 24.
=> r = sqr 48 = 4.90. Is that the answer?

OH MY GOD! Just look at what I was doing. I was using the radius to find the volume of the sphere and wondering why wasn't my answer correct?
I really need a good brain-shake!
Anyway, the system says "Answer must be an integer between 0 and 999* Round off to 5, doesn't work either.

 

[Sanzaid Sian]

If I start preparing from now, can I give AS thinking skills in May/June through self study. I want to take it as my 7th subject. What do u guys think? Can u recommend sum gud books? Is it easy? Thanks in advance.

 

[Sanzaid Sian]

?

 

[abcde]

If I start preparing from now, can I give AS thinking skills in May/June through self study. I want to take it as my 7th subject. What do u guys think? Can u recommend sum gud books? Is it easy? Thanks in advance.

Yes, you can. It isn't intrinsically easy, but gets so with practice. If you can push yourself to be regular, go for it. Not because "a 7th subject would look cool", but because the subject is good, especially if you're into problem solving (P1 + P3) and debating (P4).

 

[Sanzaid Sian]

Yes, you can. It isn't intrinsically easy, but gets so with practice. If you can push yourself to be regular, go for it. Not because "a 7th subject would look cool", but because the subject is good, especially if you're into problem solving (P1 + P3) and debating (P4).

can u recommend sum gud books for me?

 

[VelaneDeBeaute]

Another question
ABC is a scalene (non-degenerate) triangle with integer side lengths. If AB=2 and BC=3, what is the perimeter of triangle ABC?
Note: A scalene triangle has no equal sides. A degenerate triangle has 0 area.

 

[abcde]

can u recommend sum gud books for me?

The Cambridge endorsed one by Geoff Butler is pretty good. Apart from that, search logical fallacies on the internet and solve maximum questions.

Another question
ABC is a scalene (non-degenerate) triangle with integer side lengths. If AB=2 and BC=3, what is the perimeter of triangle ABC?
Note: A scalene triangle has no equal sides. A degenerate triangle has 0 area.

The perimieter is 9. The third side must be between 1 (3-2) and 5 (3+2). Since its scalene, it can't be 2 or 3. That leaves the 3rd side with length 4. Hence, perimeter = 4+2+3 = 9 units.