MathDB

5

Part of 2010 Postal Coaching

Problems(6)

Angle bisectors all the way through...

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10/21/2010
A point PP lies on the internal angle bisector of ∠BAC\angle BAC of a triangle △ABC\triangle ABC. Point DD is the midpoint of BCBC and PDPD meets the external angle bisector of ∠BAC\angle BAC at point EE. If FF is the point such that PAEFPAEF is a rectangle then prove that PFPF bisects ∠BFC\angle BFC internally or externally.
geometryrectanglegeometric transformationreflectioncircumcircleangle bisectorgeometry unsolved
A nice Inequality

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10/22/2010
For any positive real numbers a,b,ca, b, c, prove that
∑cyclic(b+c)(a4−b2c2)ab+2bc+ca≥0\sum_{cyclic} \frac{(b + c)(a^4 - b^2 c^2 )}{ab + 2bc + ca} \ge 0
inequalitiesrearrangement inequalityinequalities unsolved
Prove that product is a perfect cube...

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12/9/2010
Let a,b,ca, b, c be integers such that ab+bc+ca=3\frac ab+\frac bc+\frac ca= 3 Prove that abcabc is a cube of an integer.
searchnumber theory unsolvednumber theory
set of integers

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10/1/2010
Prove that there exist a set of 20102010 natural numbers such that product of any 10061006 numbers is divisible by product of remaining 10041004 numbers.
inequalitiesnumber theory unsolvednumber theory
Polynomial And Degrees

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12/9/2010
Let pp be a prime and Q(x)Q(x) be a polynomial with integer coefficients such that Q(0)=0, Q(1)=1Q(0) = 0, \ Q(1) = 1 and the remainder of Q(n)Q(n) is either 00 or 11 when divided by pp, for every n∈Nn \in \mathbb{N}. Prove that Q(x)Q(x) is of degree at least p−1p - 1.
algebrapolynomialnumber theory unsolvednumber theory
RMS of First n Natural Nos.

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12/9/2010
Find the first integer n>1n > 1 such that the average of 12,22,⋯ ,n21^2 , 2^2 ,\cdots, n^2 is itself a perfect square.
searchnumber theory unsolvednumber theory