Subcontests
(4)A classic extremal combi on sets and memberships
Let {E1,E2,…,Em} be a collection of sets such that Ei⊆X={1,2,…,100}, Ei=X, i=1,2,…,m. It is known that every two elements of X is contained together in exactly one Ei for some i. Determine the minimum value of m. Two tangent circles, prove complementary
Circles Ω and ω are tangent at a point P (ω lies inside Ω). A chord AB of Ω is tangent to ω at C; the line PC meets Ω again at Q. Chords QR and QS of Ω are tangent to ω. Let I,X, and Y be the incenters of the triangles APB, ARB, and ASB, respectively. Prove that ∠PXI+∠PYI=90∘. Loyal subsets and functions F and G
We call a subset B of natural numbers loyal if there exists natural numbers i≤j such that B={i,i+1,…,j}. Let Q be the set of all loyal sets. For every subset A={a1<a2<…<ak} of {1,2,…,n} we set
f(A)=1≤i≤k−1maxai+1−aiandg(A)=B⊆A,B∈Qmax∣B∣. Furthermore, we define F(n)=A⊆{1,2,…,n}∑f(A)andG(n)=A⊆{1,2,…,n}∑g(A). Prove that there exists m∈N such that for each natural number n>m we have F(n)>G(n). (By ∣A∣ we mean the number of elements of A, and if ∣A∣≤1, we define f(A) to be zero).Proposed by Javad Abedi a configuration with gergonne point and its cevians
In a non-isosceles triangle ABC, let I be its incenter. The incircle of ABC touches BC, CA, and AB at D, E, and F, respectively. A line passing through D and perpendicular to AD intersects IB and IC at Ab and Ac, respectively. Define the points Bc, Ba, Ca, and Cb similarly. Let G be the intersection of the cevians AD, BE, and CF. The points O1 and O2 are the circumcenter of the triangles AbBcCa and AcBaCb, respectively. Prove that IG is the perpendicular bisector of O1O2. already well-known, but yet strangely difficult
Let a,b be two positive integers, such that ab=1. Find all the integer values that f(a,b) can take, where f(a,b)=ab−1a2+ab+b2.