Problems(6)
Double Summation
Source:
3/2/2008
Compute \sum_{n \equal{} 1}^\infty\sum_{k \equal{} 1}^{n \minus{} 1}\frac {k}{2^{n \plus{} k}}.
floor functionratiofunctionquadraticscalculusintegrationgeometric series
Limit of Cube Roots
Source: HMMT 2008 Calculus Problem 7
3/2/2008
(5) Find so that \lim_{x\rightarrow\infty}x^p\left(\sqrt[3]{x\plus{}1}\plus{}\sqrt[3]{x\minus{}1}\minus{}2\sqrt[3]{x}\right) is some non-zero real number.
geometry3D geometrylimitfunctioncalculusderivativecalculus computations
Connecting Points on a Circle
Source:
3/3/2008
Let be distinct points on a circle. Determine the number of possible configurations made by drawing a set of line segments connecting pairs of these points, such that: each is the endpoint of at most one segment and no two segments intersect. (The configuration with no edges drawn is allowed. An example of a valid configuration is shown below.)
[asy]unitsize(1cm);
pair[] P = new pair[8];
align[] A = {E, NE, N, NW, W, SW, S, SE};
for (int i = 0; i < 8; ++i) {
P = dir(45*i);
dot(P);
label("", P, A,fontsize(8pt));
}
draw(unitcircle);
draw(P[0]--P[1]);
draw(P[2]--P[4]);
draw(P[5]--P[6]);[/asy]
Root Transformation
Source:
3/17/2008
The equation x^3 \minus{} 9x^2 \plus{} 8x \plus{} 2 \equal{} 0 has three real roots , , . Find \frac {1}{p^2} \plus{} \frac {1}{q^2} \plus{} \frac {1}{r^2}.
algebrapolynomialVieta
Tangent Circles
Source:
3/9/2008
Let and be externally tangent circles with radius 2 and 3, respectively. Let be a circle internally tangent to both and at points and , respectively. The tangents to at and meet at , and TA \equal{} 4. Determine the radius of .
trigonometryHMMTHarvardcollegeMITtrig identitiesgeometry
x+sin(y)
Source:
3/23/2008
Given that x \plus{} \sin y \equal{} 2008 and x \plus{} 2008 \cos y \equal{} 2007, where , find the value of x \plus{} y.
trigonometry