MathDB

5

Part of 2016 Postal Coaching

Problems(6)

Angle AIO is a right angle

Source: India Postal Set1 P5 2016

1/18/2017
Let II and OO be respectively the incentre and circumcentre of a triangle ABCABC. If AB=2AB = 2, AC=3AC = 3 and ∠AIO=90∘\angle AIO = 90^{\circ}, find the area of △ABC\triangle ABC.
geometryCircumcenterincenter
Far less interesting than Poncelet's Porism

Source: India Postal Set 2 P 5 2016

1/18/2017
Two triangles ABCABC and DEFDEF have the same incircle. If a circle passes through A,B,C,D,EA,B,C,D,E prove that it also passes through FF.
geometryincircle3D geometryprism
Simple diophantine equation

Source: India Postal Set 3 P 5 2016

1/18/2017
Find all nonnegative integers k,nk, n which satisfy 22k+1+9⋅2k+5=n2.2^{2k+1} + 9\cdot 2^k + 5 = n^2.
number theoryDiophantine equation
Group on Z with every element self-inverse

Source: India Postal Set 4 P 5

1/18/2017
Is it possible to define an operation ⋆\star on Z\mathbb Z such that[*] for any a,b,ca, b, c in Z,(a⋆b)⋆c=a⋆(b⋆c)\mathbb Z, (a \star b) \star c = a \star (b \star c) holds; [*] for any x,yx, y in Z,x⋆x⋆y=y⋆x⋆x=y\mathbb Z, x \star x \star y = y \star x \star x=y?
set theorygroup theoryBinary operationnumber theory
Putting numbers on a chessboard

Source: India Postal Set 5 P 5 2016

1/18/2017
For even positive integer nn we put all numbers 1,2,⋯ ,n21, 2, \cdots , n^2 into the squares of an n \times n chessboard (each number appears once and only once). Let S1S_1 be the sum of the numbers put in the black squares and S2S_2 be the sum of the numbers put in the white squares. Find all nn such that it is possible to have S1S2=3964\frac{S_1}{S_2}=\frac{39}{64}.
combinatoricsnumber theory
Permutation of coefficients of polynomial

Source: India postal set 6 P 5 2016

1/18/2017
A real polynomial of odd degree has all positive coefficients. Prove that there is a (possibly trivial) permutation of the coefficients such that the resulting polynomial has exactly one real zero.
algebrapolynomial