MathDB

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Part of 2014 Czech-Polish-Slovak Junior Match

Problems(2)

equal segments wanted, starting with intersecting circles and a parallel

Source: Czech-Polish-Slovak Junior Match 2014, Individual p1 CPSJ

3/13/2020
On the plane circles kk and ℓ\ell are intersected at points CC and DD, where circle kk passes through the center LL of circle ℓ\ell. The straight line passing through point DD intersects circles kk and ℓ\ell for the second time at points AA and BB respectively in such a way that DD is the interior point of segment ABAB. Show that AB=ACAB = AC.
equal segmentsgeometrycirclesparallelogram
max of gcd of 3 products of elements of 3 disjoint partitions of {1,2,...,63}

Source: Czech-Polish-Slovak Junior Match 2014, Team p1 CPSJ

3/23/2020
The set of {1,2,3,...,63}\{1,2,3,...,63\} was divided into three non-empty disjoint sets A,BA,B. Let a,b,ca,b,c be the product of all numbers in each set A,B,CA,B,C respectively and finally we have determined the greatest common divisor of these three products. What was the biggest result we could get?
number theorygreatest common divisorSubsetsProduct