MathDB

3

Part of 2024 Thailand October Camp

Problems(2)

Product of primitive roots is not primitive root

Source: Thailand TSTST 2024 P3

7/18/2024
Recall that for an arbitrary prime pp, we define a primitive root modulo pp as an integer rr for which the least positive integer vv such that rv≡1(modp)r^{v}\equiv 1\pmod{p} is p−1p-1.\\ Prove or disprove the following statement:
For every prime p>2023p>2023, there exists positive integers 1⩽a<b<c<p1\leqslant a<b<c<p\\ such that a,ba,b and cc are primitive roots modulo pp but abcabc is not a primitive root modulo pp.
number theoryprimitive root
square of hell :)

Source: Thailand TSTST 2024 P9

7/17/2024
Let triangle ABC ABC be an acute-angled triangle. Square AEFB AEFB and ADGC ADGC lie outside triangle ABC ABC . BD BD intersects CE CE at point H H , and BG BG intersects CF CF at point I I . The circumcircle of triangle BFI BFI intersects the circumcircle of triangle CGI CGI again at point K K . Prove that line segment HK HK bisects BC BC .
geometrySpiral Similaritycartesian coordinatescomplex bashInversion