MathDB

12.3

Part of VMEO IV 2015

Problems(2)

VMEO IV P3/12 - Junior Division

Source:

5/14/2016
Find all integes a,b,c,da,b,c,d that form an arithmetic progression satisfying dāˆ’c+1d-c+1 is prime number and a+b2+c3=d2ba+b^2+c^3=d^2b
number theoryarithmetic sequenceprime numbers
PA=PL wanted, circumcircle, symmetrics, projection,

Source: 2015 VMEO IV Seniors 12.3 Vietnamese Mathematics e - Olympiad https://artofproblemsolving.com/community/c2463156_2015_vmeo_iv

9/19/2021
Triangle ABCABC is inscribed in circle (O)(O). P P is a point on arc BCBC that does not contain A A such that APAP is the symmedian of triangle ABCABC. E,FE ,F are symmetric of PP wrt CA,ABCA, AB respectively . KK is symmetric of AA wrt EFEF. LL is the projection of KK on the line passing through AA and parallel to BCBC. Prove that PA=PLPA=PL.
geometrycircumcircleequal segmentssymmedian