MathDB

10

Part of 2017 QEDMO 15th

Problems(2)

q is divisor of 2^{q-1}- 1 if q = \frac{4^p-1}{3}

Source: 15th -a QEDMO problem 10 (19. - 22. 10. 2017) https://artofproblemsolving.com/community/c1512515_qedmo_2005

5/30/2021
Let p>3p> 3 be a prime number and let q=4p−13q = \frac{4^p-1}{3}. Show that qq is a composite integer as well is a divisor of 2q−1−12^{q-1}- 1.
number theorydivisordivides
inradius of P A_j A_{j+k} independent of P, if inradii of P A_iA_{i+1} are equal

Source: 15th -b QEDMO problem 10 (19. - 22. 10. 2017) https://artofproblemsolving.com/community/c1512515_qedmo_2005

5/30/2021
Let ℓ\ell be a straight line and P∉ℓP \notin \ell be a point in the plane. On ℓ\ell are, in this arrangement, points A1,A2,...A_1, A_2,... such that the radii of the incircles of all triangles PAiAi+1P A_iA_{i + 1} are equal. Let k∈Nk \in N. Show that the radius of the incircle of the triangle PAjAj+kP A_j A_{j + k} does not depend on the choice of j∈Nj \in N .
geometryinradiusequal segments