MathDB

3

Part of 2006 Taiwan National Olympiad

Problems(4)

Ellipse

Source: Taiwan NMO 2006

3/21/2006
Let the major axis of an ellipse be ABAB, let OO be its center, and let FF be one of its foci. PP is a point on the ellipse, and CDCD a chord through OO, such that CDCD is parallel to the tangent of the ellipse at PP. PFPF and CDCD intersect at QQ. Compare the lengths of PQPQ and OAOA.
conicsellipseanalytic geometrygraphing linesslopegeometrygeometric transformation
Sigma and pi

Source: Taiwan NMO 2006

3/21/2006
a1,a2,...,a95a_1, a_2, ..., a_{95} are positive reals. Show that ∑k=195ak≤94+∏k=195max⁡{1,ak}\displaystyle \sum_{k=1}^{95}{a_k} \le 94+ \prod_{k=1}^{95}{\max{\{1,a_k\}}}
inequalitiesinductionfunctioninequalities proposed
Polynomial

Source: Taiwan NMO 2006

3/21/2006
f(x)=x3−6x2+17xf(x)=x^3-6x^2+17x. If f(a)=16,f(b)=20f(a)=16, f(b)=20, find a+ba+b.
algebrapolynomialfunctionalgebra proposed
Quadratic equation

Source: Taiwan NMO 2006

3/21/2006
If positive integers p,q,rp,q,r are such that the quadratic equation px2−qx+r=0px^2-qx+r=0 has two distinct real roots in the open interval (0,1)(0,1), find the minimum value of pp.
quadraticscalculusintegrationalgebraalgebra proposed