MathDB

Problem 1

Part of 2015 Postal Coaching

Problems(5)

Strictly increasing positive integer sequence

Source: India Postals 2015 Set 1

11/7/2015
Let n∈Nn \in \mathbb{N} be such that gcd(n,6)=1gcd(n, 6) = 1. Let a1<a2<⋯<ana_1 < a_2 < \cdots < a_n and b1<b2<⋯<bnb_1 < b_2 < \cdots < b_n be two collection of positive integers such that aj+ak+al=bj+bk+bla_j + a_k + a_l = b_j + b_k + b_l for all integers 1≤j<k<l≤n1 \le j < k < l \le n. Prove that aj=bja_j = b_j for all 1≤j≤n1 \le j \le n.
number theoryinequalities
Incentre of reflected triangle

Source: India Postals 2015 Set 2

11/7/2015
OO is the centre of the circumcircle of triangle ABCABC, and MM is its orthocentre. Point AA is reflected in the perpendicular bisector of the side BCBC,B B is reflected in the perpendicular bisector of the side CACA, and finally CC is reflected in the perpendicular bisector of the side ABAB. The images are denoted by A1,B1,C1A_1, B_1, C_1 respectively. Let KK be the centre of the inscribed circle of triangle A1B1C1A_1B_1C_1. Prove that OO bisects the line segment MKMK.
geometryincenter
Functional Equation with f(2n+1)=2f(n) and f(2n)=2f(n)+1

Source: India Postals Set 3

11/7/2015
Let f:N∪{0}→N∪{0}f:\mathbb{N} \cup \{0\} \to \mathbb{N} \cup \{0\} be defined by f(0)=0f(0)=0, f(2n+1)=2f(n)f(2n+1)=2f(n) for n≥0n \ge 0 and f(2n)=2f(n)+1f(2n)=2f(n)+1 for n≥1n \ge 1
If g(n)=f(f(n))g(n)=f(f(n)), prove that g(n−g(n))=0g(n-g(n))=0 for all n≥0n \ge 0.
algebrafunctional equation
Length of chord independent of A

Source: India Postals 2015 Set 4

11/7/2015
A circle, its chord ABAB and the midpoint WW of the minor arc ABAB are given. Take an arbitrary point CC on the major arc ABAB. The tangent to the circle at CC meets the tangents at AA and BB at points XX and YY respectively. Lines WXWX and WYWY meet ABAB at points NN and MM. Prove that the length of segment NMNM doesn’t depend on point CC.
geometry
Integers and Trigonometric functions

Source: Indian Postals 2015 Set 5

11/15/2015
Find all positive integer nn such that sin⁡nθsin⁡θ−cos⁡nθcos⁡θ=n−1\frac{\sin{n\theta}}{\sin{\theta}} - \frac{\cos{n\theta}}{\cos{\theta}} = n-1 holds for all θ\theta which are not integral multiples of π2\frac{\pi}{2}
trigonometryinequalitiesnumber theorycalculusintegrationfunction