MathDB

6

Part of 2012 India Regional Mathematical Olympiad

Problems(6)

not all the roots of ax^4+bx^3+x^2+x+1=0 can be real when a \ne 0

Source: CRMO 2012 Region 1 p6

9/30/2018
Let aa and bb be real numbers such that a≠0a \ne 0. Prove that not all the roots of ax4+bx3+x2+x+1=0ax^4 + bx^3 + x^2 + x + 1 = 0 can be real.
polynomialpolynomial equationReal Rootsalgebra
x^3y+ y^3z+z^3x is constant, when x + y + z = 0 and xy + yz + zx = -3

Source: CRMO 2012 Region 2 p6

9/30/2018
Show that for all real numbers x,y,zx,y,z such that x+y+z=0x + y + z = 0 and xy+yz+zx=−3xy + yz + zx = -3, the expression x3y+y3z+z3xx^3y + y^3z + z^3x is a constant.
algebraidentityalgebraic identities
Find all n

Source: RMO 2012

12/2/2012
Find all positive integers such that 32n+3n2+73^{2n}+3n^2+7 is a perfect square.
inductionalgebrapolynomialnumber theory unsolvednumber theory
positive real system \frac{1}{xy}=\frac{x}{z}+ 1,\frac{1}{yz} = \frac{y}{x} + 1

Source: CRMO 2012 Region 4 p6

9/30/2018
Solve the system of equations for positive real numbers: 1xy=xz+1,1yz=yx+1,1zx=zy+1\frac{1}{xy}=\frac{x}{z}+ 1,\frac{1}{yz} = \frac{y}{x} + 1, \frac{1}{zx} =\frac{z}{y}+ 1
system of equationsalgebraSolve system
WSUM = 3(a_1 + a_3 +..) + 2(a_2 + a_4 +...) , sum of WSUMs

Source: CRMO 2012 region 5 p6 Mumbai

9/30/2018
Let SS be the set {1,2,...,10}\{1, 2, ..., 10\}. Let AA be a subset of SS. We arrange the elements of AA in increasing order, that is, A={a1,a2,....,ak}A = \{a_1, a_2, ...., a_k\} with a1<a2<...<aka_1 < a_2 < ... < a_k. Define WSUM for this subset as 3(a1+a3+..)+2(a2+a4+...)3(a_1 + a_3 +..) + 2(a_2 + a_4 +...) where the first term contains the odd numbered terms and the second the even numbered terms. (For example, if A={2,5,7,8}A = \{2, 5, 7, 8\}, WSUM is 3(2+7)+2(5+8)3(2 + 7) + 2(5 + 8).) Find the sum of WSUMs over all the subsets of S. (Assume that WSUM for the null set is 00.)
Subsetsset theorycombinatoricsCombinatorics of set
abc=x^3; 175 positive integers [Indian RMO 2012(b) Q6]

Source:

12/2/2012
A computer program generated 175175 positive integers at random, none of which had a prime divisor grater than 10.10. Prove that there are three numbers among them whose product is the cube of an integer.
pigeonhole principle