MathDB

2

Part of 2013 India Regional Mathematical Olympiad

Problems(5)

Sum of 2013th Powers of Coefficients

Source: Indian RMO 2013 Region 1 Problem 2

2/1/2014
Let f(x)=x3+ax2+bx+cf(x)=x^3+ax^2+bx+c and g(x)=x3+bx2+cx+ag(x)=x^3+bx^2+cx+a, where a,b,ca,b,c are integers with c≠0c\not=0. Suppose that the following conditions hold:
[*]f(1)=0f(1)=0, [*]the roots of g(x)=0g(x)=0 are the squares of the roots of f(x)=0f(x)=0.
Find the value of a2013+b2013+c2013a^{2013}+b^{2013}+c^{2013}.
algebraRMO
Indian RMO- Paper 2

Source: Problem 2

12/11/2013
In a triangle ABCABC, ADAD is the altitude from AA, and HH is the orthocentre. Let KK be the centre of the circle passing through DD and tangent to BHBH at HH. Prove that the line DKDK bisects ACAC.
geometry unsolvedgeometry
Indian RMO - Paper 3

Source: RMO- Problem 2

12/11/2013
Find all 44-tuples (a,b,c,d)(a,b,c,d) of natural numbers with a≤b≤ca \le b \le c and a!+b!+c!=3da!+b!+c!=3^d
number theory unsolvednumber theory
Indian RMO - Paper -4[2]

Source:

12/12/2013
Determine the smallest prime that does not divide any five-digit number whose digits are in a strictly increasing order.
divisorprime
Diophantine Equation with Primes and Squares

Source: Indian RMO 2013 Mumbai Region Problem 2

2/1/2014
Find all triples (p,q,r)(p,q,r) of primes such that pq=r+1pq=r+1 and 2(p2+q2)=r2+12(p^2+q^2)=r^2+1.
number theory unsolvednumber theory