MathDB

4

Part of 2016 India Regional Mathematical Olympiad

Problems(7)

Maximum value of non-homogenous expression

Source: RMO Delhi 2016, P4

10/11/2016
Let a,b,ca,b,c be positive real numbers such that a+b+c=3a+b+c=3. Determine, with certainty, the largest possible value of the expression aa3+b2+c+bb3+c2+a+cc3+a2+b \frac{a}{a^3+b^2+c}+\frac{b}{b^3+c^2+a}+\frac{c}{c^3+a^2+b}
Inequalityalgebrathree variable inequalitymaximum valueinequalities
Points cyclic iff angle is right

Source: RMO Maharashtra and Goa 2016, P4

10/11/2016
Let △ABC\triangle ABC be scalene, with BCBC as the largest side. Let DD be the foot of the perpendicular from AA on side BCBC. Let points K,LK,L be chosen on the lines ABAB and ACAC respectively, such that DD is the midpoint of segment KLKL. Prove that the points B,K,C,LB,K,C,L are concyclic if and only if ∠BAC=90∘\angle BAC=90^{\circ}.
geometry
No. of six digits numbers

Source: RMO Mumbai 2016, P4

10/11/2016
Find the number of all 6-digits numbers having exactly three odd and three even digits.
countingcombinatorics
Combinatorics

Source: RMO 2016 Karnataka Region P4

10/16/2016
There are 100100 countries participating in an olympiad. Suppose nn is a positive integers such that each of the 100100 countries is willing to communicate in exactly nn languages. If each set of 2020 countries can communicate in exactly one common language, and no language is common to all 100100 countries, what is the minimum possible value of nn?
Counting 6 digit numbers

Source: RMO Hyderabad 2016 , P4

10/12/2016
Find all 66 digit natural numbers, which consist of only the digits 1,2,1,2, and 33, in which 33 occurs exactly twice and the number is divisible by 99.
combinatoricsDigits
box contains answer 4032 scripts with exactly half have odd number of marks

Source: RMO 2016 Odisha Region p4

9/30/2018
A box contains answer 40324032 scripts out of which exactly half have odd number of marks. We choose 2 scripts randomly and, if the scores on both of them are odd number, we add one mark to one of them, put the script back in the box and keep the other script outside. If both scripts have even scores, we put back one of the scripts and keep the other outside. If there is one script with even score and the other with odd score, we put back the script with the odd score and keep the other script outside. After following this procedure a number of times, there are 3 scripts left among which there is at least one script each with odd and even scores. Find, with proof, the number of scripts with odd scores among the three left.
number theoryoddgame
2016 Chandigarh RMO (4cos^2 9^o - 3) (4 cos^2 27^o -3) = tan 9^o

Source:

8/9/2019
Prove that (4cos⁡29o–3)(4cos⁡227o–3)=tan⁡9o(4\cos^29^o – 3) (4 \cos^227^o– 3) = \tan 9^o.
trigonometry