MathDB

6

Part of 2020 Iran Team Selection Test

Problems(2)

Diameter length in triangulation

Source: Iran TST1 Day 2 P6

2/23/2020
nn positive numbers are given. Is it always possible to find a convex polygon with n+3n+3 edges and a triangulation of it so that the length of the diameters used in the triangulation are the given nn numbers?
Proposed by Morteza Saghafian
combinatoricstriangulationIranian TSTMorteza Saghafiyan
Number Theory

Source: Iranian TST 2020, second exam day 2, problem 6

3/12/2020
pp is an odd prime number. Find all p−12\frac{p-1}2-tuples (x1,x2,…,xp−12)∈Zpp−12\left(x_1,x_2,\dots,x_{\frac{p-1}2}\right)\in \mathbb{Z}_p^{\frac{p-1}2} such that ∑i=1p−12xi≡∑i=1p−12xi2≡⋯≡∑i=1p−12xip−12(modp).\sum_{i = 1}^{\frac{p-1}{2}} x_{i} \equiv \sum_{i = 1}^{\frac{p-1}{2}} x_{i}^{2} \equiv \cdots \equiv \sum_{i = 1}^{\frac{p-1}{2}} x_{i}^{\frac{p - 1}{2}} \pmod p.
Proposed by Ali Partofard
number theoryprime numbers