MathDB

2

Part of 2012 India Regional Mathematical Olympiad

Problems(6)

a|b^3, b|c^3 , c|a^3 . Prove that abc|(a+b+c)^{13}

Source: CRMO 2012 region 1 p2

9/30/2018
Let a,b,ca,b,c be positive integers such that a∣b3,b∣c3a|b^3, b|c^3 and c∣a3c|a^3. Prove that abc∣(a+b+c)13abc|(a+b+c)^{13}
number theorydividesdivisiblepowers
a|b^4, b|c^4 , c|a^4 . Prove that abc|(a+b+c)^{21}

Source: CRMO 2012 Region 2 p2

9/30/2018
Let a,b,ca,b,c be positive integers such that a∣b4,b∣c4a|b^4, b|c^4 and c∣a4c|a^4. Prove that abc∣(a+b+c)21abc|(a+b+c)^{21}
number theorydividesdivisiblepowers
Similar to RMO 2002

Source: RMO 2012

12/2/2012
Let a,b,ca,b,c be positive integers such that a∣b5,b∣c5a|b^5, b|c^5 and c∣a5c|a^5. Prove that abc∣(a+b+c)31abc|(a+b+c)^{31}.
number theory unsolvednumber theory
a|b^2, b|c^2 , c|a^2 . Prove that abc|(a+b+c)^{7}

Source: CRMO 2012 region 4 p2

9/30/2018
Let a,b,ca,b,c be positive integers such that a∣b2,b∣c2a|b^2, b|c^2 and c∣a2c|a^2. Prove that abc∣(a+b+c)7abc|(a+b+c)^{7}
number theoryDivisibilitydivisiblepowers
169 divides 21n^2 + 89n + 44 if 13 divides n^2 + 3n + 51

Source: CRMO 2012 region 5 p2 Mumbai

9/30/2018
Prove that for all positive integers nn, 169169 divides 21n2+89n+4421n^2 + 89n + 44 if 1313 divides n2+3n+51n^2 + 3n + 51.
number theoryDivisibilitydividesdivisor
Find the polynomial P [Indian RMO 2012(b) Q2]

Source:

12/2/2012
Let P(x)=xn+an−1xn−1+⋯+a0P(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_0 be a polynomial of degree n≥3.n\geq 3. Knowing that an−1=−(n1)a_{n-1}=-\binom{n}{1} and an−2=(n2),a_{n-2}=\binom{n}{2}, and that all the roots of PP are real, find the remaining coefficients. Note that (nr)=n!(n−r)!r!.\binom{n}{r}=\frac{n!}{(n-r)!r!}.
algebrapolynomialVietainequalitiesbinomial theoremsum of roots