MathDB

9

Part of 2010 Harvard-MIT Mathematics Tournament

Problems(3)

2010 Algebra #9: Real Recursive Roots

Source:

7/15/2012
Let f(x)=cx(x−1)f(x)=cx(x-1), where cc is a positive real number. We use fn(x)f^n(x) to denote the polynomial obtained by composing ff with itself nn times. For every positive integer nn, all the roots of fn(x)f^n(x) are real. What is the smallest possible value of cc?
algebrapolynomial
2010 Calculus #9: Trigonometric Differential Equation

Source:

7/15/2012
Let x(t)x(t) be a solution to the differential equation (x+x′)2+x⋅x′′=cos⁡t\left(x+x^\prime\right)^2+x\cdot x^{\prime\prime}=\cos t with x(0)=x′(0)=25x(0)=x^\prime(0)=\sqrt{\frac{2}{5}}. Compute x(π4)x\left(\dfrac{\pi}{4}\right).
calculustrigonometry
2010 Geometry #9

Source:

1/2/2024
Let ABCDABCD be a quadrilateral with an inscribed circle centered at II. Let CICI intersect ABAB at EE. If ∠IDE=35∘\angle IDE=35^\circ, ∠ABC=70∘\angle ABC=70^\circ, and ∠BCD=60∘\angle BCD=60^\circ, then what are all possible measures of ∠CDA\angle CDA?
geometry