MathDB

4

Part of 2016 Postal Coaching

Problems(6)

The n-th smallest number in a set

Source: India Postal Set 1 P4 2016

1/18/2017
Suppose nn is a perfect square. Consider the set of all numbers which is the product of two numbers, not necessarily distinct, both of which are at least nn. Express the n−n-th smallest number in this set in terms of nn.
number theoryinequalitiesalgebra
An FE like are-you-even-serious

Source: India Postal Set 2 P 4 2016

1/18/2017
Find a real function f:[0,∞)→Rf : [0,\infty)\to \mathbb R such that f(2x+1)=3f(x)+5f(2x+1) = 3f(x)+5, for all xx in [0,∞)[0,\infty).
functionfunctional equationalgebra
Making the coefficients nonnegative

Source: India Postal Set 3 P 4

1/18/2017
Let ff be a polynomial with real coefficients and suppose ff has no nonnegative real root. Prove that there exists a polynomial hh with real coefficients such that the coefficients of fhfh are nonnegative.
polynomialalgebra
Partition of set with equal sums

Source: India Postal Set 4 P4

1/18/2017
Let n∈Nn \in \mathbb N. Prove that for each factor m \ge n of n(n+1)/2n(n + 1)/2, one can partition the set {1,2,3,⋯ ,n}\{1,2, 3,\cdots , n\} into disjoint subsets such that the sum of elements in each subset is equal to mm.
number theorycombinatorics
Diophantine equation with prime number

Source: India Postal Set 5 P 4 2016

1/18/2017
Find all triplets (x,y,p)(x, y, p) of positive integers such that pp is a prime number and xy3x+y=p.\frac{xy^3}{x+y}=p.
number theoryDiophantine equationprime numbers
Chessboard coloring

Source: India Postal Set 6 P 4 2016

1/18/2017
Consider a 2n×2n2n\times 2n chessboard with all the 4n24n^2 cells being white to start with. The following operation is allowed to be performed any number of times:
"Three consecutive cells (in a row or column) are recolored (a white cell is colored black and a black cell is colored white)."
Find all possible values of n≥2n\ge 2 for which using the above operation one can obtain the normal chess coloring of the given board.
combinatoricsChessboard