MathDB

Problem 4

Part of 2014 VJIMC

Problems(2)

parabolas and tangency, points conparabolic

Source: VJIMC 2014 1.4

5/20/2021
Let P1,P2,P3,P4P_1,P_2,P_3,P_4 be the graphs of four quadratic polynomials drawn in the coordinate plane. Suppose that P1P_1 is tangent to P2P_2 at the point q2,P2q_2,P_2 is tangent to P3P_3 at the point q3,P3q_3,P_3 is tangent to P4P_4 at the point q4q_4, and P4P_4 is tangent to P1P_1 at the point q1q_1. Assume that all the points q1,q2,q3,q4q_1,q_2,q_3,q_4 have distinct xx-coordinates. Prove that q1,q2,q3,q4q_1,q_2,q_3,q_4 lie on a graph of an at most quadratic polynomial.
geometryconic sectionsconics
double integral inequality

Source: VJIMC 2014 2.4

5/21/2021
Let 0<a<b0<a<b and let f:[a,b]→Rf:[a,b]\to\mathbb R be a continuous function with ∫abf(t)dt=0\int^b_af(t)dt=0. Show that ∫ab∫abf(x)f(y)ln⁡(x+y)dxdy≤0.\int^b_a\int^b_af(x)f(y)\ln(x+y)dxdy\le0.
inequalitiesintegrationcalculus