MathDB

3

Part of 2016 Postal Coaching

Problems(6)

For triangles with equal inradii

Source: India Postal Set 1 P3 2016

1/18/2017
Four points lie on a plane such that no three of them are collinear. Consider the four triangles formed by taking any three points at a time. If the inradii of these four triangles are all equal, prove that the four triangles are congruent.
geometrycongruent triangles
Circle containing a polygon

Source: India Postal Set 2 P3 2016

1/18/2017
Given a convex polygon, show that it has three consecutive vertices such that the circle through them contains the polygon.
geometrycombinatorial geometry
Properly Connected Triples

Source: India Postal Set 3 P3 2016

1/18/2017
Five airlines operate in a country consisting of 3636 cities. Between any pair of cities exactly one airline operates two way flights. If some airlines operates between cities A,BA,B and B,CB,C we say that the ordered triple A,B,CA,B,C is properly-connected. Determine the largest possible value of kk such that no matter how these flights are arranged there are at least kk properly-connected triples.
combinatoricsoptimization
Equal areas in a hexagon

Source: India Postal Set 4 P3 2016

1/18/2017
The diagonals AD,BEAD, BE and CFCF of a convex hexagon concur at a point MM. Suppose the six triangles ABM,BCM,CDM,DEM,EFMABM, BCM, CDM, DEM, EFM and FAMFAM are all acute-angled and the circumcentre of all these triangles lie on a circle. Prove that the quadrilaterals ABDE,BCEFABDE, BCEF and CDFACDFA have equal areas.
geometryhexagon
Composition is real implies polynomials are real

Source: India Postal Set 5 P 3 2016

1/18/2017
Call a non-constant polynomial real if all its coecients are real. Let PP and QQ be polynomials with complex coefficients such that the composition P \circ Q is real. Show that if the leading coefficient of QQ and its constant term are both real, then PP and QQ are real.
algebrapolynomial
Find all reals a

Source: India Postal Set 6 P 3 2016

1/18/2017
Find all real numbers aa such that there exists a function f:R→Rf:\mathbb R\to \mathbb R such that the following conditions are simultaneously satisfied: (a) f(f(x))=xf(x)−ax,  ∀x∈R;f(f(x))=xf(x)-ax,\;\forall x\in\mathbb{R}; (b) ff is not a constant function; (c) ff takes the value aa.
functionfunctional equationalgebra