MathDB

1

Part of 2009 Postal Coaching

Problems(6)

min of f(a, b, c) = (a + b)^4 + (b + c)^4 + (c + a)^4 -477 (a^4 + b^4 + c^4)

Source: Indian Postal Coaching 2009 set 1 p1

5/26/2020
Find the minimum value of the expression f(a,b,c)=(a+b)4+(b+c)4+(c+a)4−47(a4+b4+c4)f(a, b, c) = (a + b)^4 + (b + c)^4 + (c + a)^4 - \frac47 (a^4 + b^4 + c^4), as a,b,ca, b, c varies over the set of all real numbers
algebramininequalities
{P_{x\in A} x} / |A|} and P_{x\in B} x} / |B| are two relatively prime compos

Source: Indian Postal Coaching 2009 set 2 p1

5/26/2020
Let n≥1n \ge 1 be an integer. Prove that there exists a set SS of nn positive integers with the following property: if AA and BB are any two distinct non-empty subsets of SS, then the averages Px∈Ax∣A∣\frac{P_{x\in A} x}{|A|} and Px∈Bx∣B∣\frac{P_{x\in B} x}{|B|} are two relatively prime composite integers.
number theorySubsetscombinatorics
infinitely many powers of 2, a_{n+1} = a_n + b_n

Source: Indian Postal Coaching 2009 set 3 p1

5/26/2020
Let a1,a2,a3,...,an,...a_1, a_2, a_3, . . . , a_n, . . . be an infinite sequence of natural numbers in which a1a_1 is not divisible by 55. Suppose an+1=an+bna_{n+1} = a_n + b_n where bn is the last digit of ana_n, for every nn. Prove that the sequence {an}\{a_n\} contains infinitely many powers of 2.
Sequencepower of 2recurrence relationalgebra
1/gamma_a +1/ gamma_b >= p (1/a+1/b), right triangle, radii

Source: Indian Postal Coaching 2009 set 4 p1

5/26/2020
Two circles Γa\Gamma_a and Γb\Gamma_b with their centres lying on the legs BCBC and CACA of a right triangle, both touching the hypotenuse ABAB, and both passing through the vertex CC are given. Let the radii of these circles be denoted by γa\gamma_a and γb\gamma_b. Find the greatest real number pp such that the inequality 1γa+1γb≥p(1a+1b)\frac{1}{\gamma_a}+\frac{1}{\gamma_b}\ge p \left(\frac{1}{a}+\frac{1}{b}\right) (BC=a,CA=bBC = a,CA = b) holds for all right triangles ABCABC.
inequalitiesGeometric Inequalitiesright triangle
circle on AB as a diameter passes through two fixed points

Source: Indian Postal Coaching 2009 set 5 p1

5/26/2020
A circle Γ\Gamma and a line ℓ\ell which does not intersect Γ\Gamma are given. Suppose P,Q,R,SP, Q,R, S are variable points on circle Γ\Gamma such that the points A=PQ∩RSA = PQ\cap RS and B=PS∩QRB = PS \cap QR lie on ℓ\ell. Prove that the circle on ABAB as a diameter passes through two fixed points.
fixedFixed pointdiametercircle
AD/DK + BE/EL + CF/FM >= 9

Source: Indian Postal Coaching 2009 set 6 p1

5/26/2020
In a triangle ABCABC, let D,E,FD,E, F be interior points of sides BC,CA,ABBC,CA,AB respectively. Let AD,BE,CFAD,BE,CF meet the circumcircle of triangle ABCABC in K,L,MK, L,M respectively. Prove that ADDK+BEEL+CFFM≥9\frac{AD}{DK} + \frac{BE}{EL} + \frac{CF}{FM} \ge 9. When does the equality hold?
geometryGeometric Inequalitiescircumcircle