MathDB

3

Part of 2013 Dutch IMO TST

Problems(2)

3 circles, tangents, intersections, midpoint wanted

Source: Dutch 2013 IMO TST1 p3

8/2/2019
Fix a triangle ABCABC. Let Γ1\Gamma_1 the circle through BB, tangent to edge in AA. Let Γ2\Gamma_2 the circle through C tangent to edge ABAB in AA. The second intersection of Γ1\Gamma_1 and Γ2\Gamma_2 is denoted by DD. The line ADAD has second intersection EE with the circumcircle of △ABC\vartriangle ABC. Show that DD is the midpoint of the segment AEAE.
geometrycirclesTangentsmidpoint
Sequence of integers and prime numbers

Source: Netherlands IMO Team Selection Test 2013

10/21/2014
Fix a sequence a1,a2,a3…a_1,a_2,a_3\ldots of integers satisfying the following condition:for all prime numbers pp and all positive integers kk,we have apk+1=pak−3ap+13a_{pk+1}=pa_k-3a_p+13.Determine all possible values of a2013a_{2013}.
number theoryprime numbersnumber theory unsolved