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Part of 1999 Putnam

Problems(2)

Putnam 1999 A1

Source:

12/22/2012
Find polynomials f(x)f(x), g(x)g(x), and h(x)h(x), if they exist, such that for all xx, ∣f(x)∣−∣g(x)∣+h(x)={−1if x<−13x+2if −1≤x≤0−2x+2if x>0.|f(x)|-|g(x)|+h(x)=\begin{cases}-1 & \text{if }x<-1\\3x+2 &\text{if }-1\leq x\leq 0\\-2x+2 & \text{if }x>0.\end{cases}
Putnamalgebrapolynomialcalculusderivativeanalytic geometrygraphing lines
Putnam 1999 B1

Source:

12/22/2012
Right triangle ABCABC has right angle at CC and ∠BAC=θ\angle BAC=\theta; the point DD is chosen on ABAB so that ∣AC∣=∣AD∣=1|AC|=|AD|=1; the point EE is chosen on BCBC so that ∠CDE=θ\angle CDE=\theta. The perpendicular to BCBC at EE meets ABAB at FF. Evaluate lim⁡θ→0∣EF∣\lim_{\theta\to 0}|EF|.
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