Sunaina thinks absurd

 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)

This week was full Number Theory and algebra biased!! 😄 Do try all the problems first!! And if you guys get any nice solutions , do post in the comments section! Here are the walkthroughs of this week's top 5 Number Theory problems! 5th position (1999 JBMO P2):  For each nonnegative integer $n$ we define $A_n = 2^{3n}+3^{6n+2}+5^{6n+2}$. Find the greatest common divisor of the numbers $A_0,A_1,\ldots, A_{1999}$. Walkthrough: It doesn't require a walkthrough, I wrote this here, cause it's a cute problem for the person who has just started Contest math :P a. What is $A_0$? b. Find out $A_1$. c. Show that $\boxed{7}$ is the required answer! 4th position (APMO, Evan Chen's orders modulo a prime handout):   Let $a,b,c$ be distinct integers. Given that $a | bc + b + c, b | ca + c + a$ and $c | ab + a + b$, prove that at least one of $a, b, c$ is not prime. Walkthrough: Fully thanks to  MSE  ! (Also one should try MSE, it has helped me a lot, ofc it's more tilted to Colle

This week was full Geo and NT 😊 . Do try all problems first!! And if you guys get any nice solutions , do post in the comments section! Here are the walkthroughs of this week's top 5 geo problems! 5th position (PUMac 2009 G8) : Consider $\Delta ABC$ and a point $M$ in its interior so that $\angle MAB = 10^{\circ}, \angle MBA = 20^{\circ}, \angle MCA =30^{\circ}$ and $\angle MAC = 40^{\circ}$. What is $\angle MBC$?  Walkthrough: a. Take $D$ as a point on $CM$ such that $\angle DAC=30^{\circ}$, and define $BD\cap AC=E$ . So $\Delta DAC$ is isosceles . b. Show M is the incentre of $\Delta ABD$ c. Show $\angle EDC=60^{\circ}$ d. Show $\Delta BAC$ is isosceles . e. So $\boxed{\angle MBC=60^{\circ}}$ 4th position (IMO SL 2000 G4):  Let $ A_1A_2 \ldots A_n$ be a convex polygon, $ n \geq 4.$ Prove that $ A_1A_2 \ldots A_n$ is cyclic if and only if to each vertex $ A_j$ one can assign a pair $ (b_j, c_j)$ of real numbers, $ j = 1, 2, \ldots, n,$ so that $ A_iA_j = b_jc_i - b_ic_j$ for all