MathDB

5

Part of 2010 Irish Math Olympiad

Problems(2)

Find Polynomials

Source: 2010 IrMO Paper 1 Problem 5

1/30/2018
Find all polynomials f(x)=x3+bx2+cx+df(x)=x^3+bx^2+cx+d, where b,c,d,b,c,d, are real numbers, such that f(x2−2)=−f(−x)f(x)f(x^2-2)=-f(-x)f(x).
algebrapolynomial
Triangles with Equal Area

Source: 2010 IrMO Paper 2 Problem 5

1/30/2018
Suppose a,b,ca,b,c are the side lengths of a triangle ABCABC. Show that x=a(b+c−a),y=b(c+a−b),z=c(a+b−c)x=\sqrt{a(b+c-a)}, y=\sqrt{b(c+a-b)}, z=\sqrt{c(a+b-c)} are the side lengths of an acute-angled triangle XYZXYZ, with the same area as ABCABC, but with a smaller perimeter, unless ABCABC is equilateral.
geometry