MathDB

6

Part of 2009 Harvard-MIT Mathematics Tournament

Problems(5)

2009 Algebra #6 - Sum of Ratios Given Trig Relationships

Source:

1/2/2012
Let xx and yy be positive real numbers and θ\theta an angle such that θ≠π2n\theta \neq \frac{\pi}{2}n for any integer nn. Suppose sin⁡θx=cos⁡θy\frac{\sin\theta}{x}=\frac{\cos\theta}{y} and cos⁡4θx4+sin⁡4θy4=97sin⁡2θx3y+y3x. \frac{\cos^4 \theta}{x^4}+\frac{\sin^4\theta}{y^4}=\frac{97\sin2\theta}{x^3y+y^3x}. Compute xy+yx.\frac xy+\frac yx.
ratiotrigonometryLaTeX
2009 Calculus #6: Class of Polynomials

Source:

6/23/2012
Let p0(x),p1(x),p2(x),…p_0(x),p_1(x),p_2(x),\ldots be polynomials such that p0(x)=xp_0(x)=x and for all positive integers nn, ddxpn(x)=pn−1(x)\dfrac{d}{dx}p_n(x)=p_{n-1}(x). Define the function p(x):[0,∞)→Rp(x):[0,\infty)\to\mathbb{R} by p(x)=pn(x)p(x)=p_n(x) for all x∈[n,n+1)x\in [n,n+1). Given that p(x)p(x) is continuous on [0,∞)[0,\infty), compute ∑n=0∞pn(2009).\sum_{n=0}^\infty p_n(2009).
calculusalgebrapolynomialfunction
2009 Combinatorics #6 - Sequence of 5 Positive Integers

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1/7/2012
How many sequences of 55 positive integers (a,b,c,d,e)(a,b,c,d,e) satisfy abcde≤a+b+c+d+e≤10abcde\leq a+b+c+d+e\leq10?
2009 Geometry #2: Corners of a Unit Cube

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6/23/2012
The corner of a unit cube is chopped off such that the cut runs through the three vertices adjacent to the vertex of the chosen corner. What is the height of the cube when the freshly-cut face is placed on a table?
geometry3D geometry
2009 Geometry #6: Lattice Point Triangle

Source:

6/23/2012
Let ABCABC be a triangle in the coordinate plane with vertices on lattice points and with AB=1AB = 1. Suppose the perimeter of ABCABC is less than 1717. Find the largest possible value of 1/r1/r, where rr is the inradius of ABCABC.
geometryanalytic geometryperimeterinradius