MathDB

3

Part of 2015 India Regional MathematicaI Olympiad

Problems(5)

Find all fractions which can be written simultan(7k-5)/(5k-3) and (6l-1)/(4l-3)

Source: CRMO 2015 region 1 p3

9/30/2018
Find all fractions which can be written simultaneously in the forms 7k−55k−3\frac{7k- 5}{5k - 3} and 6l−14l−3\frac{6l - 1}{4l - 3} , for some integers k,lk, l.
number theoryIntegerFractions
Question 3

Source:

12/6/2015
Find all integers a,b,ca,b,c such that a2=bc+1a^2 = bc + 1 and b2=ac+1b^2 = ac + 1
RMO Delhi Q.3

Source:

12/20/2015
3. Show that there are infinitely many triples (x,y,z) of integers such that x3+y4=z31x^3 + y^4 = z^{31}.
solve in integers a^{2}=bc+4 and b^{2}=ca+4

Source: CRMO 2015 region 4 (Karnataka) p3

9/30/2018
Find all integers a,b,ca,b,c such that a2=bc+4a^{2}=bc+4 and b2=ca+4b^{2}=ca+4.
number theoryDiophantine EquationsDiophantine equationsystem of equations
Polynomial divides another one for all $n$ in $N$

Source: RMO (Mumbai Region) 2015 P3

12/6/2015
Let P(x)P(x) be a polynomial whose coefficients are positive integers. If P(n)P(n) divides P(P(n)−2015)P(P(n)-2015) for every natural number nn, prove that P(−2015)=0P(-2015)=0.
One additional condition must be given that PP is non-constant, which even though is understood.
algebrapolynomialalgebra proposedInteger PolynomialDivisibility