Subcontests
(4)Number Theory
a) Find the minimum positive integer k so that for every positive integers (x,y), for which x/y2 and y/x2, then xy/(x+y)k
b) Find the minimum positive integer l so that for every positive integers (x,y,z), for which x/y2, y/z2 and z/x2, then xyz/(x+y+z)l Number Theory with sequences
Let a0,a1,... be a sequence of non-negative integers and b0,b1,... be a sequence of non-negative integers defined by the following rule:
bi=gcd(ai,ai+1) for every i=>0
Is it possible every positive integer to occur exactly once in the sequence b0,b1,... Algebra functions
Let n be a nonzero natural number. We say about a function f ∶ R ⟶ R that is n-positive
if, for any real numbers x1,x2,...,xn
with the property that x1+x2+...+xn=0,
the inequality f(x1)+f(x2)+...+f(xn)=>0 is true
a) Is it true that any 2020-positive function is also 1010-positive?
b) Is it true that any 1010-positive function is 2020-positive? Geometry
Given are four distinct points A,B,E,P so that P is the middle of AE and B is on the segment AP. Let k1 and k2 be two circles passing through A and B. Let t1 and t2 be the tangents of k1 and k2, respectively, to A.Let C be the intersection point of t2 and k1 and Q be the intersection point of t2 and the circumscribed circle of the triangle ECB. Let D be the intersection posit of t1 and k2 and R be the intersection point of t1 and the circumscribed circle of triangle BDE. Prove that P,Q,R are collinear.