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TOP 10 problems of Week#3

 This week was full algebra biased!! 😄 

This is my first time trying inequalities, so this pure beginners level.  Do try all the problems first!! And if you guys get any nice solutions , do post in the comments section! 

These problem uses only Power mean Inequality  and Titu's lemma.

The First few problems happen to be  not problems, but tricks(?) which are extensively used.. 

Here are the walkthroughs of this week's top 10 Inequality problems!

10th position: Prove that for any real $a>0$ , $a+\frac{1}{a}\ge 2$

Walkthrough: a.  only AM-GM

9th position:Prove that for any real $a>0$ , $\frac{a}{1+a^2}\le \frac {a}{2a}$.

Walkthrough: a. Only AM-GM 

b. Use AM-GM to show that  $1+a^2\ge 2a $ 

8th position: Prove that for any real $x,y>0$ ,$\frac{1}{x+y}\le \frac{1}{4x}+\frac{1}{4y}$

Walkthrough: a. AM-HM inequality (cute 💖) 

7th position : Prove that for any real positive $p,q >0$ and $p+q=1$, then $\left(p+\frac{1}{p} \right)^2+ \left(q+\frac{1}{q} \right)^2 \ge \frac{25}{2}$

Walkthrough: a. Open the brackets , what do we get?

b. $p^2+\frac{1}{p^2}+2+q^2+\frac{1}{q^2}+2$  $ \ge \frac{25}{2}$

c. Note that by AM-GM, $p^2+q^2\le \left(\frac{p+q}{2}\right )^2 =\frac{1}{2}$

d. Again by AM-GM $\frac{1}{p^2}+\frac{1}{q^2}\ge \frac{2}{p\cdot q}$

e. hmm..still not achievable, so use this $(p+a)^2\ge 4pq$ which is true by AM-GM. 

f. conclude !

6th position: Prove that $\dfrac{ab}{c^3}+\dfrac{bc}{a^3}+\dfrac{ca}{b^3}> \dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}$, where $a,b,c$  are different positive real numbers

Walkthrough: a. It's pure AM-GM

b. Note that by AM-GM $$\frac{ab}{c^3}+\frac{bc}{a^3}\ge \frac{2b}{ac}$$ 

c. Again by AM-GM $$\frac {b}{ac}+\frac{c}{ab}\ge \frac{2}{a}$$. 

5th position : Prove that for any real $a,b,c>0$ , $a^2+b^2+c^2\ge ab+bc+ca$.

Walkthrough: This is only AM-GM

a.  Note that $a^2+b^2\ge 2ab$ , sum stuffs and conclude!

4th Position:  Prove that if $a,b,c >0$ and $a^2+b^2+c^2=3 $ , then $\sum_{cyc}\frac{1}{1+ab}\ge \frac{3}{2}$

Walkthrough: Thankyou Dapper :) Do check out his blog in AOPS !!

a. So by Titu, $ \sum_{cyc} \frac {1^2}{1+ab}\ge \frac {3^2}{1+ab+1+bc+1+ca}= \frac{9}{3+ab+bc+ca}$

b. Use problem 5th and get that $\frac{9}{3+ab+bc+ca} \ge \frac{9}{6}=\frac{3}{2}$ .

3rd position : Prove that for any real $a+b\ge 1$ , then $a^4+b^4\ge \frac{1}{8}$

Walkthrough: a. Apply Quadratic Mean $\ge$ Arithmetic Mean 

I added it in the 3rd position , because I didn't know what Quadratic Mean was :P

2nd position : Let $x_0>x_1>x_2>\ldots>x_n$ be real numbers.

Prove $ x_{0}+\frac{1}{x_{0}-x_{1}}+\frac{1}{x_{1}-x_{2}}+\ldots+\frac{1}{x_{n-1}-x_{n}}\geq x_{n}+2n$.

Walkthrough: I love this guy's solution and couldn't resist to post walkthrough here, so full credits to him!

a.  Take $a_k=x_{k-1}-x_k$, simplify LHS.

b. Okie this is why problem 1 is so important , use the fact that $a+\frac{1}{a}\ge 2$ and you are done!! cute right?

1st Position:  Solve the system of equations in $\Bbb{R}^{+}$  

$$ a+b+c+d=12 $$ and $$abcd=27+ab+ac+ad+bc+bd+cd$$ 

Walkthrough: Oooo okie 4 variables 2 equations, seems unsolvable right? :P

a. Apply AM-GM in first equation and get $81\le abcd$.

b. Again apply AM-GM in the second equation (on $9+9+9+ab+ac+ad+bc+bd+cd$ ) , and get $abcd\ge 81$ .

c. Note that equality happens in AM-GM only when $a=b=c=d$.

d. Hence solution is $\boxed{a=b=c=d=3}$.

PS. : This YT video and series has so nice inequalities video, one can try it!!

---

So these were my top 10 !

What are your top 10s ? Do write in the comments section (at least write something ! I will be happy to hear your comments ). Follow this blog if you want to see more contest math problems! To follow ( if you want to ) click the 3 bars thing on the right.  See you all soon 😊.

Sunaina 💜

Comments

  1. Oh no! Inequalities😔. Luckily, it's only the basics!

    ReplyDelete
  2. Hey ! Can you please recommend me some books for contest math? I'm in grade 10 and I know geometry I learnt that from Evan Chen's book and I know little about combinatorics like algorithms etc. and a little about inequality, AM-GM-HM, Muirhead, Jensen, Karamata. So is there a book that you can suggest such that it helps me to brush up my skills? Thanks in advance!

    ReplyDelete
    Replies
    1. Hey ! Firstly I can't recommend because I am not in level to recommend someone, right? And then high five I am in grade 10 too.
      TBH I haven't done a single book completely. I do handouts a bit. But I think OTIS excerpts with Evan's handouts is really nice pack . And if you get any doubts then Evan clears it too. Again for geo , I think A beautiful journey through Olympiad geo V4, is really good and it covers a lot of part that were uncovered in EGMO. Rest is problem solving :)

      Delete
    2. This comment has been removed by the author.

      Delete
    3. ay man thanks for ur reply! And high five on being grade 10. I'm studying NT now I will go through A beautiful journey through Olympiad geo soon! Thanks!

      Delete
    4. oh..also I saw the prev comment before u deleted it :P, thanks will go through the book (ofc it's a pr0 book ). BTW how did u come to know about this blog ?

      Delete
    5. I was asking for some books recommendation at MSE and someone linked your post. I saw your profile and there was the link. Make sure to try that book too its a good read!!!

      Delete
    6. certified MO person moment, ik many people who just study from handouts and im also one of them (i dont classify myself as a nice MO person yet tho :|)

      Delete
  3. I am bad at inequalities 🥲
    Btw Lenin must have banned inequalities in math contests in soviet Russia 🥲

    ReplyDelete
    Replies
    1. arey chill prabh, I will make a post on Jensen ineq's prerequisites, rest is practice. I am bad too, so same party!

      Delete
    2. Lol I am chill. I just like saying how bad I am at math

      Delete

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