MathDB

2

Part of 2008 Postal Coaching

Problems(6)

p(A)p(B)p(C) /s^3 <= 1/8

Source: Indian Postal Coaching 2008 set 1 p2

5/25/2020
Let ABCABC be a triangle, ADAD be the altitude from AA on to BCBC. Draw perpendiculars DD1DD_1 and DD2DD_2 from DD on to ABAB and ACAC respectively and let p(A)p(A) be the length of the segment D1D2D_1D_2. Similarly define p(B)p(B) and p(C)p(C). Prove that p(A)p(B)p(C)s3≤18\frac{p(A)p(B)p(C)}{s^3}\le \frac18 , where s is the semi-perimeter of the triangle ABCABC.
geometryperpendiculardistanceGeometric Inequalities
circle with diameter BC covers triangle bounded by AK,BL,CM

Source: Indian Postal Coaching 2008 set 2 p2

5/25/2020
Let ABCABC be an equilateral triangle, and let K,L,MK, L,M be points respectively on BC,CA,ABBC, CA, AB such that BK/KC=CL/LA=AM/MB=λBK/KC = CL/LA = AM/MB =\lambda . Find all values of λ\lambda such that the circle with BCBC as a diameter completely covers the triangle bounded by the lines AK,BL,CMAK,BL,CM.
geometrycircleratio
sidelengths of triangle are rational, BC equals to the altitude from A,

Source: Indian Postal Coaching 2008 set 3 p2

5/25/2020
Does there exist a triangle ABCABC whose sides are rational numbers and BCBC equals to the altitude from AA?
altitudesidelenghtsgeometryrational
prime cirterion, \phi (n) divides (n - 1) and (n + 1) divides \sigma (n)

Source: Indian Postal Coaching 2008 set 4 p2

5/25/2020
Prove that an integer n≥2n \ge 2 is a prime if and only if ϕ(n)\phi (n) divides (n−1)(n - 1) and (n+1)(n + 1) divides σ(n)\sigma (n).
[Here ϕ\phi is the Totient function and σ\sigma is the divisor - sum function.]
nn is squarefree
number theoryprimesprimeprime numbersdivides
a, b \in N and a+b is a square , then P(a) + P(b) is also a square

Source: Indian Postal Coaching 2008 set 5 p2

5/25/2020
Find all polynomials PP with integer coefficients such that wherever a,b∈Na, b \in N and a+ba+b is a square we have P(a)+P(b)P(a) + P(b) is also a square.
polynomialInteger PolynomialPerfect Squarealgebra
[2^n / n] is a power of 2, then n is a power of 2

Source: Indian Postal Coaching 2008 set 6 p2

5/25/2020
Show that if n≥4,n∈Nn \ge 4, n \in N and [2nn]\big [ \frac{2^n}{n} ] is a power of 22, then nn is a power of 22.
floor functionpower of 2number theory