MathDB

4

Part of 2005 Harvard-MIT Mathematics Tournament

Problems(3)

2005 Algebra #4: Floor Sum

Source:

4/29/2013
If a,b,c>0a,b,c>0, what is the smallest possible value of ⌊a+bc⌋+⌊b+ca⌋+⌊c+ab⌋ \left\lfloor \dfrac {a+b}{c} \right\rfloor + \left\lfloor \dfrac {b+c}{a} \right\rfloor + \left\lfloor \dfrac {c+a}{b} \right\rfloor ? (Note that ⌊x⌋ \lfloor x \rfloor denotes the greatest integer less than or equal to xx.)
floor functioninequalitiesfunctionsearchalgebra
2005 Calculus #4: Smooth function - derivative at 0

Source:

4/29/2013
Let f:R→R f : \mathbf {R} \to \mathbf {R} be a smooth function such that f′(x)2=f(x)f′′(x) f'(x)^2 = f(x) f''(x) for all xx. Suppose f(0)=1f(0)=1 and f(4)(0)=9f^{(4)} (0) = 9. Find all possible values of f′(0)f'(0).
calculusfunctionderivativereal analysis
2005 Geometry #4: Crazy Circle Watermarked on Triangle

Source:

5/16/2013
Let XYZXYZ be a triangle with ∠X=60∘ \angle X = 60^\circ and ∠Y=45∘ \angle Y = 45^\circ . A circle with center PP passes through points AA and BB on side XYXY, CC and DD on side YZYZ, and EE and FF on side ZXZX. Suppose AB=CD=EFAB=CD=EF. Find ∠XPY \angle XPY in degrees.
geometryincentercongruent triangles