MathDB

2

Part of 2010 JBMO Shortlist

Problems(3)

JBMO 2010 Shortlist A2

Source: JBMO 2010 Shortlist

6/28/2017
Determine all four digit numbers aˉbˉcˉdˉ\bar{a}\bar{b}\bar{c}\bar{d} such that
a(a+b+c+d)(a2+b2+c2+d2)(a6+2b6+3c6+4d6)=aˉbˉcˉdˉa(a+b+c+d)(a^{2}+b^{2}+c^{2}+d^{2})(a^{6}+2b^{6}+3c^{6}+4d^{6})=\bar{a}\bar{b}\bar{c}\bar{d}
algebranumber theoryequationJBMO Shortlist
Geometry

Source: JBMO 2010 Shortlist

10/18/2012
Let ABCABC be acute-angled triangle . A circle ω1(O1,R1)\omega_1(O_1,R_1) passes through points BB and CC and meets the sides ABAB and ACAC at points DD and EE ,respectively . Let ω2(O2,R2)\omega_2(O_2,R_2) be the circumcircle of triangle ADEADE . Prove that O1O2O_1O_2 is equal to the circumradius of triangle ABCABC .
geometrycircumcircleparallelogramperpendicular bisectorgeometry unsolved
Number theory

Source: JBMO shortlist 2010

4/25/2016
Find n such that 36n−636^n-6 is the product of three consecutive natural numbers
number theory