MathDB

2

Part of 2006 Harvard-MIT Mathematics Tournament

Problems(4)

2006 Algebra #2: Symmetric-like System

Source:

7/30/2012
Find all real solutions (x,y)(x,y) of the system x2+y=12=y2+xx^2+y=12=y^2+x.
2006 Calculus #2: Trigonometric Limit in e

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7/30/2012
Compute lim⁡x→0excos⁡x−1−xsin⁡(x2).\displaystyle\lim_{x\to 0}\dfrac{e^{x\cos x}-1-x}{\sin(x^2)}.
calculuslimittrigonometry
2006 Combinatorics #2: Compound Sum of Ones

Source:

7/31/2012
Compute ∑n60=02∑n59=0n60⋯∑n2=0n3∑n1=0n2∑n0=0n11.\sum_{n_{60}=0}^2\sum_{n_{59}=0}^{n_{60}}\cdots\sum_{n_2=0}^{n_3}\sum_{n_1=0}^{n_2}\sum_{n_0=0}^{n_1}1.
2006 Geometry #2: 45 Degree Angle from Hypotenuse

Source:

7/31/2012
Suppose ABCABC is a scalene right triangle, and PP is the point on hypotenuse AC‾\overline{AC} such that ∠ABP=45∘\angle ABP=45^\circ. Given that AP=1AP=1 and CP=2CP=2, compute the area of ABCABC.
geometry