MathDB

Oliforum Contest I 2008

Part of Oliforum Contest

Subcontests

(3)
1
4
2
4
3
3

Exagon and concurrency

Let C1,C2 C_1,C_2 and C3 C_3 be three pairwise disjoint circles. For each pair of disjoint circles, we define their internal tangent lines as the two common tangents which intersect in a point between the two centres. For each i,j i,j, we define (rij,sij) (r_{ij},s_{ij}) as the two internal tangent lines of (Ci,Cj) (C_i,C_j). Let r12,r23,r13,s12,s13,s23 r_{12},r_{23},r_{13},s_{12},s_{13},s_{23} be the sides of ABCA′B′C′ ABCA'B'C'. Prove that AA′,BB′ AA',BB' and CC′ CC' are concurrent. https://cdn.artofproblemsolving.com/attachments/1/2/5ef098966fc9f48dd06239bc7ee803ce4701e2.png

0<abc<4

Let a,b,c a,b,c be three pairwise distinct real numbers such that a\plus{}b\plus{}c\equal{}6\equal{}ab\plus{}bc\plus{}ca\minus{}3. Prove that 0<abc<4 0<abc<4.

Oliforum contest- final round - problem3

Let 0<a1<a2<a3<...<a10000<20000 0 < a_1 < a_2 < a_3 < ... < a_{10000} < 20000 be integers such that gcd(ai,aj)<ai,∀i<j gcd(a_i,a_j) < a_i, \forall i < j ; is 500<a1 500 < a_1 (always) true ? (own) :lol: