Author Topic: Recreational Problems (SM level) (Read 76317 times) Tweet Share

Ahmad · « **on:** October 20, 2007, 11:43:27 am »

In the hope of getting this place a little more active, I shall post a few interesting (hopefully) problems. Many can be solved with methods knowledge alone.

Ask for hints if you want!

$\mbox{1. Simplify }\frac{(1+i)^{2004}}{(1-i)^{2000}}$

$\mbox{2. Using the fact that for complex numbers, }a\mbox{ and }b\mbox{, }|a||b| = |ab|\mbox{, express }(2^2 + 3^2)(5^2 + 7^2)\mbox{ as the sum of two squares.}$

$\mbox{3. Given that }\log_{ab}a = 4\mbox{. Find }\log_{ab}\frac{a^{\frac{1}{3}}}{b^{\frac{1}{2}}}}$

$\mbox{4. Simplify }\log(\tan 1\textdegree) + \log(\tan 2\textdegree) + ... + \log(\tan 89\textdegree)$

$\mbox{5. Solve }x^4+x^3-4x^2+x+1=0$

$\mbox{6. Describe a method to solve }6x^4+25x^3+x^2+25x+6=0\mbox{ (This one is out to enwiabe)}$

$\mbox{7. If }f\mbox{ is continuous and }f(x) + f(-x) = x^2\mbox{, what is }\displaystyle\int^1_{-1} f(x)\,dx\mbox{?}$

$\mbox{8. Let }f(x) = \frac{3x^4+16}{2x^4+11}\mbox{. Find the whole number nearest to }f(2^{32})$

$\mbox{9. Find the smallest possible constant }A\mbox{ such that }\log_ex\leq Ax^2\mbox{ for all }x > 0$

$\mbox{10. Evaluate }\displaystyle\int^{\frac{\pi}{2}}_{-\frac{\pi}{2}} \sin (x)\cdot\sin (2x)\,dx$

$\mbox{11. Evaluate }\displaystyle\int^{\frac{\pi}{4}}_{-\frac{\pi}{4}} \frac{\sin^2x}{\cos^2x}\,dx$

$\mbox{12. Let }f(x) = \sin^4x + \cos^4x\mbox{. Find the 100th derivative of }f\mbox{. Hint: }(\sin^2x + \cos^2x)^2 = 1$

13. If $\sqrt{x+\sqrt{x+\sqrt{x+\sqrt{\cdots}}}} = 2$ made sense, what would $x$ equal?

14. Find the real roots of $\sqrt{x+3-4\sqrt{x-1}} + \sqrt{x+8-6\sqrt{x-1}} = 1$ .

15. Solve $(x - 7)(x - 3)(x + 5)(x + 1) = 1680$ .

16. Solve the equation $6x^4 - 25x^3 + 12x^2 + 25x + 6 = 0$ .

17. Solve $6x^4 - 24x^3 + 12x^2 + 24x + 6 = 0$ .

18. $\sqrt{x + 2\sqrt{x + 2\sqrt{x + \cdots 2\sqrt{x+2\sqrt{3x}}}}} = x$ (n-radicals).

Some solutions by Coblin
1-4
http://vcenotes.com/forum/index.php/topic,11.msg15876.html#msg15876

7-9
http://vcenotes.com/forum/index.php/topic,11.msg15885.html#msg15885

10-12
http://vcenotes.com/forum/index.php/topic,11.msg15893.html#msg15893

rustic_metal · « **Reply #1 on:** October 20, 2007, 11:48:02 am »

should i know how to do any of this?

enwiabe · « **Reply #2 on:** October 20, 2007, 11:52:14 am »

3. 22/3

EDIT: I forgot to put b to the 4th power in my working. Correct answer is -1/6.

enwiabe · « **Reply #3 on:** October 20, 2007, 11:54:16 am »

enwiabe · « **Reply #4 on:** October 20, 2007, 11:56:44 am »

1. -4

Ahmad · « **Reply #5 on:** October 20, 2007, 11:57:23 am »

Quote from: "rustic_metal"

should i know how to do any of this?

No, you could get 50 in methods and spesh and still not know. Although, you might like to have a crack at problems 3 and 4.

enwiabe, how'd you get that for question 3? Question 4 is correct.

Edit: 1 is correct also.

enwiabe · « **Reply #6 on:** October 20, 2007, 11:58:31 am »

2. 11^2 + 29^2

rustic_metal · « **Reply #7 on:** October 20, 2007, 12:01:40 pm »

Quote from: "Ahmad"

Quote from: "rustic_metal"
should i know how to do any of this?

No, you could get 50 in methods and spesh and still not know. Although, you might like to have a crack at problems 3 and 4.

enwiabe, how'd you get that for question 3? Question 4 is correct.

Edit: 1 is correct also.

we dont even start logs till this week :oops:

Ahmad · « **Reply #8 on:** October 20, 2007, 12:07:06 pm »

That's fine, maybe someone will be willing to start a methods equivalent of this.

enwiabe · « **Reply #9 on:** October 20, 2007, 12:10:21 pm »

x = 1, (-3 + sqrt(5))/2, (-3 - sqrt(5))/2

Khangfu · « **Reply #10 on:** October 20, 2007, 02:36:00 pm »

wait haha dw mistake

3. 17/6 ?

enwiabe · « **Reply #11 on:** October 20, 2007, 02:54:20 pm »

nup. It's -1/6.

Khangfu · « **Reply #12 on:** October 20, 2007, 02:59:40 pm »

Wait Enwiabe i think i see your mistake. You didn't see the divide sign

Ahmad · « **Reply #13 on:** October 20, 2007, 03:10:13 pm »

It's a bit hard to write stuff here, the most I can do is avoid ambiguity and hope that you didn't miss anything

enwiabe · « **Reply #14 on:** October 20, 2007, 03:21:47 pm »

Quote from: "Khangfu"

Wait Enwiabe i think i see your mistake. You didn't see the divide sign

Or... not. Check your working.

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