MathDB

2

Part of 2014 India IMO Training Camp

Problems(4)

Eighth power inequality

Source: IMOTC 2014 Practice Test 1 Problem 2

7/11/2014
Let a,ba,b be positive real numbers.Prove that (1+a)8+(1+b)8≥128ab(a+b)2(1+a)^{8}+(1+b)^{8}\geq 128ab(a+b)^{2}.
inequalitiesinequalities unsolved
Triangulation

Source: IMOTC 2014 Practice Test 2 Problem 2

7/11/2014
Let nn be a natural number.A triangulation of a convex n-gon is a division of the polygon into n−2n-2 triangles by drawing n−3n-3 diagonals no two of which intersect at an interior point of the polygon.Let f(n)f(n) denote the number of triangulations of a regular n-gon such that each of the triangles formed is isosceles.Determine f(n)f(n) in terms of nn.
AMCUSA(J)MOUSAMOcombinatorics unsolvedcombinatorics
Exactly one negative among six reals

Source: Indian TST Day 2 Problem 2

7/11/2014
For j=1,2,3j=1,2,3 let xj,yjx_{j},y_{j} be non-zero real numbers, and let vj=xj+yjv_{j}=x_{j}+y_{j}.Suppose that the following statements hold:
x1x2x3=−y1y2y3x_{1}x_{2}x_{3}=-y_{1}y_{2}y_{3}
x12+x22+x32=y12+y22+y32x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=y_{1}^{2}+y_{2}^{2}+y_{3}^2
v1,v2,v3v_{1},v_{2},v_{3} satisfy triangle inequality
v12,v22,v32v_{1}^{2},v_{2}^{2},v_{3}^{2} also satisfy triangle inequality.
Prove that exactly one of x1,x2,x3,y1,y2,y3x_{1},x_{2},x_{3},y_{1},y_{2},y_{3} is negative.
inequalitiestriangle inequalityalgebra unsolvedalgebra
Diophantine equation involving powers

Source: Indian TST Day 4 Problem 2

7/11/2014
Find all positive integers xx and yy such that xx+y=y3xx^{x+y}=y^{3x}.
number theory unsolvednumber theory