MathDB

2

Part of 2009 Postal Coaching

Problems(6)

x^3 + 2y^3 + 4z^3 - 6xyz = 1 diophantine

Source: Indian Postal Coaching 2009 set 1 p2

5/26/2020
Determine, with proof, all the integer solutions of the equation x3+2y3+4z3−6xyz=1x^3 + 2y^3 + 4z^3 - 6xyz = 1.
diophantineDiophantine equationnumber theory
a^n \equiv -1 mod n^2

Source: Indian Postal Coaching 2009 set 2 p2

5/26/2020
Let a>2a > 2 be a natural number. Show that there are infinitely many natural numbers n such that an≡−1a^n \equiv -1 (mod n2n^2).
number theoryremainder
max area of inscribed n-gon in a circle with 2 perp. diagonals

Source: Indian Postal Coaching 2009 set 3 p2

5/26/2020
Let n≥4n \ge 4 be an integer. Find the maximum value of the area of a nn-gon which is inscribed in the circle of radius 11 and has two perpendicular diagonals.
geometrymaxareapolygonCyclicdiagonalsperpendicular
xy^2 /(x+y) is a prime

Source: Indian Postal Coaching 2009 set 4 p2

5/26/2020
Find all pairs (x,y)(x, y) of natural numbers xx and yy such that xy2x+y\frac{xy^2}{x+y} is a prime
primenumber theoryFraction
7^a = 4^b + 5^c + 6^d , diophantine

Source: Indian Postal Coaching 2009 set 5 p2

5/26/2020
Find all non-negative integers a,b,c,da, b, c, d such that 7a=4b+5c+6d7^a = 4^b + 5^c + 6^d
diophantinenumber theoryDiophantine equation
p/q - 4/ (r + 1)= 1 , diophantine

Source: Indian Postal Coaching 2009 set 6 p2

5/26/2020
Solve for prime numbers p,q,rp, q, r : pq−4r+1=1\frac{p}{q} - \frac{4}{r + 1}= 1
diophantineDiophantine equationnumber theory