MathDB

3

Part of 2020 OMpD

Problems(2)

incenter of tangential quad collinear with midpoints of diagonals

Source: 2020 1st OMpD L3 P4 - Brazil - Olimpíada Matemáticos por Diversão

9/23/2022
Let ABCDABCD be a quadrilateral and let Γ\Gamma be a circle of center OO that is internally tangent to its four sides. If MM is the midpoint of ACAC and NN is the midpoint of BDBD, prove that M,O,NM,O, N are collinear.
geometryincentertangential quadrilateraltangential
Polynomials are prime numbers

Source: 2020 1st OMpD L2 P3 - Brazil - Olimpíada Matemáticos por Diversão

7/8/2023
Determine all integers nn such that both of the numbers: ∣n3−4n2+3n−35∣ and ∣n2+4n+8∣|n^3 - 4n^2 + 3n - 35| \text{ and } |n^2 + 4n + 8| are both prime numbers.
algebrapolynomialnumber theoryprime numbers