MathDB

5

Part of STEMS 2023 Math Cat A

Problems(2)

Computational polynomial

Source: STEMS 2023 Maths CAT A Part A P5

1/8/2023
Consider a polynomial P(x)∈R[x]P(x) \in \mathbb{R}[x], with degree 20232023, such that P(sin⁡2(x))+P(cos⁡2(x))=1P(\sin^2(x))+P(\cos^2(x)) =1 for all x∈Rx \in \mathbb{R}. If the sum of all roots of PP is equal to pq\dfrac{p}{q} with p,qp, q coprime, then what is the product pqpq?
algebrapolynomialSTEMSComputational
Radax on STEMS go brrr...

Source: STEMS 2023 Maths CAT A Part B P5

1/8/2023
A convex quadrilateral ABCDABCD is such that ∠B=∠D\angle B = \angle D and are both acute angles. EE is on ABAB such that CB=CECB = CE and FF is on ADAD such that CF=CDCF = CD. If the circumcenter of CEFCEF is O1O_1 and the circumcenter of ABDABD is O2O_2. Prove that C,O1,O2C,O_1,O_2 are collinear.
Proposed by Kapil Pause
geometrySTEMSradical axiscollinearity