## imqwerty one year ago emergency!!!

1. imqwerty

2. ganeshie8

which q

3. ParthKohli

Last one x = y = z = 4

4. imqwerty

any question that u like...these r the question which i wsn't able to solve

5. Empty

First one probably some fermat's little theorem nonsense, the second one looks like part of the Riemann zeta function so you could probably just multiply two geometric series or something together for it, and 3rd one looks like it's probably a telescoping series, just my first guesses

6. imqwerty

@ParthKohli how did u get x=y=z=4

7. ParthKohli

It's a symmetric situation. Apply all inequalities you please, but you'll finally get to this.

8. Empty

Simplest way to think of it (not a proof) is that $n^2 > (n-a)(n+a)$ when a is not equal to zero, so that's the max when they're all the same.

9. ganeshie8

first q doesnt look easy

10. Empty

I think we can split that 3rd one up using partial fractions, then it becomes a telescoping series that way.

11. Empty

Good questions by the way!

12. imqwerty

thanks @Empty @ParthKohli i can't get to the result x=y=z. can u tell how to start ..

13. ParthKohli

I've only given a qualitative argument here so far - the whole situation you have here is symmetric in nature. x can be interchanged with y, y with z or z with x, and the situation won't change. So if the triplet $$(a, b , c)$$ gives you a maximum value then so should $$(b, a, c)$$ or $$(b, c, a)$$ or $$(c, a, b)$$, which means that $$x = y = z$$ when the given expression is maximum. Let me think of an inequality that can be applied here. You'll see that we'll find the maximum when the equality of two expressions is considered, which is almost always when all variables are equal.

14. Empty

Ok so for the second one I'm getting: $$\frac{1}{1-9}\frac{1}{1-4}\frac{5}{6}$$ Does that match your answer?

15. imqwerty

wait m doing the 1st one

16. ganeshie8

for first one you may want to try factoring it http://www.wolframalpha.com/input/?i=factor+5%5E5%5E%28n%2B1%29%2B5%5E5%5En%2B1

17. Empty

Yeah I think it is more clear how to solve the first one if you do that, here to make factoring easier substitute: $$5^{5^n}=a$$ then you end up with the expression: $$a^5+a+1$$ which somehow wolfram is able to factor out into: $$(a^2+a+1)(a^3-a^2+1)$$

18. imqwerty

@ParthKohli if we are given a question like - x+y=4 and x belongs to positive integers. and we are supposed to find out maximum of xy + x/y +y/x then by symmetric approach x=y=2 but in this question x=3 and y=1 gives the maximum so the symmetric approach is violated.

19. imqwerty

@Empty we can't write 5^5^(n+1) as a^5 if a=5^5^n

20. ParthKohli

OK, I didn't say that x = y gives you the maximum. I said it gives you the maximum if it exists. Here, x = y is most probably giving you the minimum.

21. imqwerty

yes so how do we know that in which case x=y gives the max

22. Empty

$$5^{5^{n+1}}=5^{5*5^n} = (5^{5^n})^5$$ @imqwerty

23. ParthKohli

You have to check if x = y is giving you the maximum/minimum by entering another pair of values. If that pair gives you a higher value, then x = y gives you the minimum.

24. ganeshie8

i feel #1 is still a stupid q, you're not gona learn anything frm it

25. imqwerty

wait sry yes we can put a^5...

26. imqwerty

yes it is @ganeshie8 :D

27. ParthKohli

Y'know, qwerty, it's just like that with single variable functions. You can differentiate a function and equate it to zero, but you still have to check if that gives you the maxima or minima by other methods.

28. Empty

Ok so I also found a good way of solving the second one: We only want odd total powers, so the way we can do that is only combine even powers with odd and odd with evens. So in order to do that, we multiply the geometric series' together: $$\left(\sum_{i=0}^{\infty} \frac{1}{2^{2i}} \right) \left(\sum_{j=0}^{\infty} \frac{1}{3^{2j+1}} \right) +\left(\sum_{i=0}^{\infty} \frac{1}{3^{2i}} \right) \left(\sum_{j=0}^{\infty} \frac{1}{2^{2j+1}} \right)$$ factor out a 1/3 and 1/2 so you get only geometric series in terms of 1/4 and 1/9, and you're done!

29. ParthKohli

Isn't that the standard way to do it though? How did you originally do it?

30. Empty

I factored it

31. imqwerty

the 1st one is solved thanks guys :)

32. ganeshie8

\begin{align} x^5 + x + 1 & = (x^5-x^2)+(x^2+x+1)\\ \\ & = x^2(x^3 - 1) + (x^2 + x + 1) \\ \\ & = x^2(x-1)\color{blue}{\bf (x^2 + x + 1)} + \color{blue}{\bf (x^2 + x + 1)}\end{align} http://math.stackexchange.com/questions/477295/factor-x5-x-1

33. ParthKohli

To be fair, that solution was most likely obtained by working backwards. :P

34. ganeshie8

that is okay for a proof

35. ParthKohli

Yes of course, but you can't really think of that on your own.

36. ganeshie8

that all goes as "scratch work" :P

37. Empty

ahhh thanks @ganeshie8 that's a clever trick I think I can even remember that as adding fancy form of zero and then abusing the geometric series yet again lol

38. ParthKohli

I'm ashamed that I have no other ideas other than telescoping for #3.

39. Empty

It wants to telescope, why force it to be anything it's probably not? :P

40. Empty

When you separate it out, the coefficients are both 1, it's truly a beautiful result

41. ganeshie8

I'm either dumb or too much drunk to notice it but why do you think telescoping is a bad idea ?

42. Empty

Yeah you're not drunk enough ganeshie, cause this baby wants to see the moon

43. ParthKohli

I didn't say telescoping is a bad idea. I just said that this series doesn't seem to be telescoping nicely and I have no other ideas.

44. imqwerty

m really confused nd annoyed nd kinda drunk ..cheers @ganeshie8 @Empty ..can u please tell me what u did with the 2nd question

45. imqwerty

my mind is jst nt wrkin normal today :(

46. ParthKohli

OK, it's actually telescoping really nicely.

47. Empty

Here I made it smaller, part of the total sum, just write out all these terms (it's not tooo many, and it's good for you) $\left(\sum_{i=0}^{3} \frac{1}{2^{2i}} \right) \left(\sum_{j=0}^{3} \frac{1}{3^{2j+1}} \right) +\left(\sum_{i=0}^{3} \frac{1}{3^{2i}} \right) \left(\sum_{j=0}^{3} \frac{1}{2^{2j+1}} \right)$

48. imqwerty

no i mean hw did u split that

49. Empty

No, I _created_ this

50. Empty

so by multiplying it out you can kinda see how I came up with this too... I'll draw a picture of how I think about it

51. ParthKohli

$\frac{6^k}{(3^k - 2^k)(3^{k+1 } - 2^{k+1})}$$= \frac{3^{k+1}}{2^{k+1} - 3^{k+1}} - \frac{3^k}{3^k - 2^k}$

52. ParthKohli

There's a typo.

53. Empty

|dw:1438453754914:dw| So this is the first half of the terms! @imqwerty

54. ParthKohli

Supposed to be$= \frac{3^{k+1}}{2^{k+1} - 3^{k+1}} - \frac{3^k}{2^k - 3^k}$

55. Empty

Specifically I knew to write that because the only way for i+j=odd is when one is even and the other is odd since odd+odd=even and even+even=even. So combine all the 2s with even powers with all the 3s with odd powers, and add it to all the 2s with odd powers with all the 3s with even powers.

56. Empty

so that's where the 2i and 2i+1 is coming from, keeping the powers even and odd: $\left(\sum_{i=0}^{\infty} \frac{1}{2^{2i}} \right) \left(\sum_{j=0}^{\infty} \frac{1}{3^{2j+1}} \right) +\left(\sum_{i=0}^{\infty} \frac{1}{3^{2i}} \right) \left(\sum_{j=0}^{\infty} \frac{1}{2^{2j+1}} \right)$

57. imqwerty

thank u so much @Empty :)

58. ParthKohli

OK, so we're sorta done with all of the questions.

59. ganeshie8

whats the ansur for q2

60. Empty

61. imqwerty

62. ParthKohli

@Empty Did you check out the solution to the problem I gave you yesterday?

63. Empty

Oh no I didn't, also I ended up getting 2 to that, I'll go check real fast

64. ParthKohli

2 is correct.

65. imqwerty

:) a big THANKS TO ALL OF YOU @Empty @ParthKohli @ganeshie8 :)

66. ganeshie8

#2 im getting 5/144

67. ParthKohli

Ganeshie and Qwerty haven't done that problem so I'll ask them to do it too.$\sum_{n=1}^{\infty}\frac{F_n}{2^n}$where $$F_n$$ is the $$n$$-th Fibonacci number. $$F_1 = F_2 = 1$$.

68. imqwerty

my answer the #2 is 121/144

69. ganeshie8

I am familiar with that problem $\sum_{n=1}^{\infty}F_nx^n= \frac{x}{1-x-x^2}$ plugin $$x=1/2$$

70. Empty

Here's the solution: $\left(\sum_{i=0}^{\infty} \frac{1}{2^{2i}} \right) \left(\sum_{j=0}^{\infty} \frac{1}{3^{2j+1}} \right) +\left(\sum_{i=0}^{\infty} \frac{1}{3^{2i}} \right) \left(\sum_{j=0}^{\infty} \frac{1}{2^{2j+1}} \right)$ $\left(\sum_{i=0}^{\infty} \frac{1}{2^{2i}} \right) \frac{1}{3}\left(\sum_{j=0}^{\infty} \frac{1}{3^{2j}} \right) +\left(\sum_{i=0}^{\infty} \frac{1}{3^{2i}} \right) \frac{1}{2} \left(\sum_{j=0}^{\infty} \frac{1}{2^{2j}} \right)$ $\left( \frac{1}{3} + \frac{1}{2} \right) \left(\sum_{i=0}^{\infty} \frac{1}{3^{2i}} \right) \left(\sum_{j=0}^{\infty} \frac{1}{2^{2j}} \right)$ $\left( \frac{1}{3} + \frac{1}{2} \right) \left( \frac{1}{1-3^2} \right) \left(\frac{1}{1-2^2} \right)$ $\frac{5}{144}$

71. imqwerty

damn my mind is nt ok today ....instead of starting with i=0 i started with i=1. :( feeling bad at the silly mistakes m doin :(

72. Empty

Haha nah it's fine that's why I wrote it all out for you. And hey, if you want you can always come back tomorrow and look, no reason to feel bad XD

73. Empty

I'm pretty sure "feeling bad" is not supposed to be a step in solving problems haha, even though sometimes it does tend to be when those problems are proofs in real analysis for me lolol

74. ganeshie8

sometimes i hate problems from indian competitive exams/tests mostly because they are meaning less, how do they expect a 10/12th grader to know about generating functions and stuff ?

75. ganeshie8

they just want the exams to be tough, they dont bother to look at the syllabus of 12th grader

76. imqwerty

the reason y m feeling bad and annoyed - today the results of the last test held at of coaching institute came- the 1st ranker got 360/360 nd i ws like O_o nd i ws 12th with score 329. i could've done way better if i wuld have rechecked my paper after completing it instead of looking at other ppl faces :(

77. imqwerty

at my home m nicknamed - 'silly mistake'

78. ganeshie8

trust me, you and PK are way better than me when i was in my 12th

79. Empty

Yeah I don't understand the obsession with rote memorizing all these useless one-time-use tricks for problems mankind invented calculators and wolfram alpha for. Time to change the curriculum to a modern age I think.

80. imqwerty

@Empty so true^ .

81. Empty

I always see these questions like "find the number of roots of this polynomial" or "show this ridiculous trig identity is true without any geometric picture that it came from" ...why?

82. Empty

It's like teachers want people to believe math is useless AND ugly.

83. ParthKohli

Whut, qwerty? The top-ranker got 360?! Surely the questions must have been really simple. The topper at my coaching (me) got 307/360 which was an exception to the rule that the previous years' toppers got no more than 250-260.

84. ParthKohli

humblebrag

85. Empty

lol duuuude gotta go for that $$2 \pi$$ score man!

86. imqwerty

the paper wasn't enough easy to get 360/360. the guys are too crazy...they are like 24 x 7 studying.

87. Empty

Their parents must be awful people

88. Empty

Ok ok maybe I'm being pessimistic here haha

89. imqwerty

The guy who got 1st rank is a vry good frnd of mine. He says his parents think that he can do whatever he wants but he must remain healthy eating proper food.

90. imqwerty

his parents don't interfere his personal life

91. Empty

Woah he must be a genius :O

92. imqwerty

this guy got 1st rank in 2/4 test held till date

93. ParthKohli

Kaunsa institute hai ye?

94. ganeshie8

see you cant study more than 14 hours each day, you need at least 6 hours of sleep and 4 hours for maintenance

95. imqwerty

allen

96. ParthKohli

Kota wala?

97. imqwerty

han

98. ParthKohli

Tumhe internet mil jaata hai?

99. imqwerty

mai to kota ka hi hu

100. ParthKohli

101. imqwerty

:) badiya bhi hai aur thodi prblems bhi hai....roz ghar par koi na koi guest aa jate hai...roz tv etc etc ka shore distrb karta hai...nd many more prblems

102. ParthKohli

Guest kyun aa jaate hain?

103. imqwerty

pata nai... btw mere relatives paas me hi rahte hai to wo dadi dada se milne aa jate hai.

104. imqwerty

btw @ganeshie8 did u got the answer to the question - dIzI/dz z=complex number

105. ganeshie8

common sense : common lannguage guys, when others are there in the thread

106. imqwerty

ok :) sry

107. ganeshie8

thats okay, just saying..

108. ParthKohli

あなたは何について話していますか？理解できません。

109. ganeshie8

dIzI/dz derivative doesnt exist

110. ParthKohli

What kind of derivative is this?

111. ganeshie8

z is a complex number f(z) = |z| is a real valued function that takes complex number as input

112. ParthKohli

Yeah, I can see that. But it's a complex number. How do you compute derivatives of complex-valued functions? Is it that instead of taking a small interval, you take a small circular area?

113. ganeshie8

Exactly, the "interval on real number line" becomes a "ball in argand plane"

114. ParthKohli

@imqwerty knows all this?! o.O

115. imqwerty

not too much..

116. ParthKohli

Well, there goes my self-esteem.

117. ganeshie8

if you're in 12th grade and if you say you know all this, then you're just lying

118. ParthKohli

119. ParthKohli

So am I.

120. EclipsedStar

Me too...lol

121. imqwerty

i asked my teacher to explain this but he refused :( he said - its of no use

122. ParthKohli

That's stupid. "It's not there in the JEE examination" should not be restated as "it's of no use".

123. imqwerty

^true that

124. ParthKohli

I think complex derivatives are a hell lot more useful that the four questions you gave us.

125. imqwerty

yes!! the teaching methodology nd the mindset of ppl must change.

126. ganeshie8

hmm your teacher knows more than you, he might be thinking that teaching you about complex derivatives is useless because you ppl are not ready yet

127. ParthKohli

What if he straight-up doesn't know what they are? That's a possibility too. I don't know much about Allen teachers but I don't expect mine to give a satisfactory explanation.

128. imqwerty

es this can be the reason @ganeshie8 nd we nt taught complex numbers till nw ..but still he said that its far away frm JEE syllabus. but i'll ask him when the chapter gets over. I asked Anna sir [HOD math allen] nd m sure he knws it

129. ParthKohli

Haha, of course it's beyond the reaches of JEE syllabi. It's not just the complex numbers we're taught in 11th and 12th.

130. ParthKohli

Wow, I just searched him up and he looks pretty young, like a passout.

131. imqwerty

m enrolled in vibrant academy too i'll go and ask Vikas gupta sir (HOD math vibrant).i think he will gimme the explanation.

132. ParthKohli

I think it's better to not ask for it...

133. ganeshie8

If you get a satisficatory explanation, please do share wid us

134. imqwerty

haahaha once i got kicked out (frnkly) fr asking such doubts.. nd yes i'll definately share it.

135. imqwerty

pk have u joined any institute?

136. ParthKohli

Yeah, Vidyamandir.

137. imqwerty

oh nice :)

138. anonymous

consider $$x^2+x+1$$ divides into both $$x^5+x+1,x^{25}+x^5+1$$. this is sufficient to prove $$5^{5^{n+1}}+5^{5^n}+1$$ is not prime for positive integer $$n$$

139. anonymous

for the next problem, you just break it into an iterated sum, though if you don't set a lower bound on $$i,j$$ individually you will run into problems

140. imqwerty

thanks @oldrin.bataku heres the new question that m solvin -

141. ParthKohli

Eh, this problem isn't that hard.

142. ParthKohli

It may be calculative, but it's not hard.

143. ganeshie8

looks that fraction evaluates to $$\left(\dfrac{n+1}{n}\right)^2$$

144. imqwerty

Yes it has lil bit calculation prt

145. Empty

You can do the gauss fliperoo method there I think.

146. anonymous

$$\sum_{k=1}^n (2k)^2=4\sum_{k=1}^n k^2=4\cdot\frac{n(n+1)(2n+1)}6\\\sum_{k=0}^n(2k+1)^2=\sum_{k=0}^n (4k^2+4k+1)=4\sum_{k=1}^n k^2+4\sum_{k=1}^n k+\sum_{k=0}^n 1$$ so the second sum gives us: $$4\cdot\frac{n(n+1)(2n+1)}6+4\cdot\frac{n(n+1)}2+(n+1)$$

147. anonymous

oops, the second should be the sum up to $$n-1$$, actually

148. Empty

@imqwerty You might like Mathematical Methods in the Physical Sciences by Mary Boas. I think you might like it, it introduces complex numbers and complex analysis in separate chapters, check it out: https://faculty.psau.edu.sa/filedownload/doc-4-pdf-0a187866618ca3049030ec5014860ae8-original.pdf It's very accessible and if you have any questions feel free to ask, that book covers so much more though than just that.

149. imqwerty

ok i am getting 151 as the answer

150. imqwerty

thank u so much @Empty ^^

151. imqwerty

wait no m getting 150 as the answer

152. anonymous

so $$\frac23\cdot n(n-1)(2n-1)+2 n(n-1)+n$$ which reduces the rational expression as $$\frac{2n+2}{2n-1}>1.01\\2n+2>1.01(2n-1)\\2+1.01>2(1.01-1)n\\3.01>0.02n\\n<50\cdot 3.01=150.5$$ so $$n\le 150$$

153. imqwerty

correct @oldrin.bataku :)

154. anonymous

for the last one $$xyz+xy+yz+zx$$ links us to $$(x+1)(y+1)(z+1)=(x+1)(yz+y+z+1)\\\quad =xyz+xy+zx+yz+x+y+z+1$$ so we ahve that $$(x+1)(y+1)(z+1)-(x+y+z)-1=xyz+xy+zx+yz$$ now, we know $$x+y+z=12$$ so: $$(x+1)(y+1)(z+1)-13=xyz+xy+zx+yz$$ now, let's use the AM-GM on $$x+1,y+1,z+1$$: $$\frac{(x+1)+(y+1)+(z+1)}3\ge \sqrt[3]{(x+1)(y+1)(z+1)}$$ so it follows: $$(x+1)(y+1)(z+1)\le \left[1+\frac{x+y+z}3\right]^3$$ which gives us: $$(x+1)(y+1)(z+1)\le \left[1+4\right]^3=125$$ so we have that $$(x+1)(y+1)(z+1)-13\le125-13=112$$

155. imqwerty

thank u so much!! @oldrin.bataku now all the questions r solved ^~^

156. imqwerty

@Empty in the second question in the last step u took the common ratios of the 2 infinite GPs wrong

157. anonymous

complex differentiation is not beyond what a high schooler can know; I used the residue theorem in calculus II which was in 11th grade

158. anonymous

anyways, $$|z|=z\bar z$$ is not differentiable on $$\mathbb{C}$$ because it depends on $$\bar z$$

159. anonymous

$$\sum_{k=0}^N\left(\frac{3^{k+1}}{2^{k+1} - 3^{k+1}} - \frac{3^k}{2^k - 3^k}\right)=\frac{3^{N+1}}{2^{N+1}-3^{N+1}}-\frac{3^0}{2^0-3^0}$$ the first term behaves like: $$\frac{1}{(2/3)^{N+1}-1}\to \frac1{-1}=-1$$as $$N\to\infty$$

160. anonymous

also it should be $$k=1$$ giving: $$-\frac{3^1}{2^1-3^1}=-\frac{3}{2-3}=-\frac{3}{-1}=3$$ so $$-1+3=2$$

161. anonymous

@imqwerty

162. imqwerty

thanks @oldrin.bataku ^~^ :)

163. imqwerty

@dan815