Subcontests
(16)hat in 1 of 100 colors, 1 girl is known to 2015 boys, 1 boy knows 2015 girls
Is it true that for all natural n, it is always possible to give each of the n children a hat painted in one of 100 colors so that if a girl is known to more than 2015 boys, then not all of these boys have hats of the same color, and, if a boy is acquainted with more than 2015 girls, don't all these girls have hats of the same color?[hide=original wording]Vai tiesa, ka visiem naturāliem n vienmēr iespējams katram no n bērniem iedot pa cepurei, kas nokrāsota vienā no 100 krāsām tā, ka, ja kāda meitene ir pazīstama ar vairāk nekā 2015 zēniem, tad ne visiem šiem zēniem cepures ir vienā krāsā, un, ja kāds zēns ir pazīštams ar vairāk nekā 2015 meitenēm, tad ne visām šīm meitenēm cepures ir vienā krāsā? charismatic numbers x= prod (q+i)^{a_i}
Given a fixed rational number q. Let's call a number x charismatic if we can find a natural number n and integers a1,a2,..,an such that
x=(q+1)a1⋅(q+2)a2⋅...⋅(q+n)an.
i) Prove that one can find a q such that all positive rational numbers are charismatic.
ii) Is it true that for all q, if the number x is charismatic, then x+1 is also charismatic? AB=CD wanted, 3 circles of diameters XZ, XY , YZ related where X-Y-Z (collinear)
Points X , Y, Z lie on a line k in this order. Let ω1, ω2, ω3 be three circles of diameters XZ, XY , YZ , respectively. Line ℓ passing through point Y intersects ω1 at points A and D, ω2 at B and ω3 at C in such manner that points A,B,Y,X,D lie on ℓ in this order. Prove that AB=CD.