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Showing posts from December, 2020

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 positiv...

TOP 10 problems of Week#2

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...

TOP 10 problems of Week#1

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 fo...