Subcontests
(5)Positive sequence
Let a0,a1,a2,… be a sequence of positive real numbers satisfying \, a_{i\minus{}1}a_{i\plus{}1}\leq a_{i}^{2}\, for i \equal{} 1,2,3,\ldots\; . (Such a sequence is said to be log concave.) Show that for each n>1,
\frac{a_{0}\plus{}\cdots\plus{}a_{n}}{n\plus{}1}\cdot\frac{a_{1}\plus{}\cdots\plus{}a_{n\minus{}1}}{n\minus{}1}\geq\frac{a_{0}\plus{}\cdots\plus{}a_{n\minus{}1}}{n}\cdot\frac{a_{1}\plus{}\cdots\plus{}a_{n}}{n}. Odd positive integer functions
Let a,b be odd positive integers. Define the sequence (fn) by putting f1=a, f2=b, and by letting fn for n≥3 be the greatest odd divisor of fn−1+fn−2. Show that fn is constant for n sufficiently large and determine the eventual value as a function of a and b. Convex quad
Let ABCD be a convex quadrilateral such that diagonals AC and BD intersect at right angles, and let E be their intersection. Prove that the reflections of E across AB,BC,CD,DA are concyclic. Which is larger
For each integer n≥2, determine, with proof, which of the two positive real numbers a and b satisfying an=a+1,b2n=b+3a is larger. Prove that f(x) <=cx
Consider functions f:[0,1]→R which satisfy
(i) f(x)≥0 for all x in [0,1],
(ii) f(1)=1,
(iii) f(x)+f(y)≤f(x+y) whenever x,y, and x+y are all in [0,1].Find, with proof, the smallest constant c such that
f(x)≤cx
for every function f satisfying (i)-(iii) and every x in [0,1].