MathDB

Problem 6

Part of 2007 Mongolian Mathematical Olympiad

Problems(2)

bicentric quadrilateral

Source: Mongolian MO 2007 Grade 11 P6

4/8/2021
Given a quadrilateral ABCDABCD simultaneously inscribed and circumscribed, assume that none of its diagonals or sides is a diameter of the circumscribed circle. Let PP be the intersection point of the external bisectors of the angles near AA and BB. Similarly, let QQ be the intersection point of the external bisectors of the angles CC and DD. If JJ and OO respectively are the incenter and circumcenter of ABCDABCD prove that OJ⊥PQOJ\perp PQ.
geometry
prove number divides sum of a^a

Source: Mongolian MO 2007 Teachers P6

4/8/2021
Let n=p1α1⋯psαs≥2n=p_1^{\alpha_1}\cdots p_s^{\alpha_s}\ge2. If for any α∈N\alpha\in\mathbb N, pi−1∤αp_i-1\nmid\alpha, where i=1,2,…,si=1,2,\ldots,s, prove that n∣∑α∈Zn∗ααn\mid\sum_{\alpha\in\mathbb Z^*_n}\alpha^{\alpha} where Zn∗={a∈Zn:gcd⁡(a,n)=1}\mathbb Z^*_n=\{a\in\mathbb Z_n:\gcd(a,n)=1\}.
number theory