MathDB

8

Part of 2007 Harvard-MIT Mathematics Tournament

Problems(4)

2007 Algebra #8: Functional Equation

Source:

6/21/2012
Let A :=Q∖{0,1}A \text{ :}= \mathbb{Q}\setminus \{0,1\} denote the set of all rationals other than 00 and 11. A function f:A→Rf:A\to \mathbb{R} has the property that for all x∈Ax\in A, f(x)+f(1−1x)=log⁡∣x∣.f(x)+f\left(1-\dfrac{1}{x}\right)=\log |x|. Compute the value of f(2007)f(2007).
functionlogarithms
2007 Calculus #8: Sum of Primitive Roots of Unity

Source:

6/21/2012
Suppose that ω\omega is a primitive 2007th2007^{\text{th}} root of unity. Find (22007−1)∑j=1200612−ωj\left(2^{2007}-1\right)\displaystyle\sum_{j=1}^{2006}\dfrac{1}{2-\omega^j}.
calculusalgebrapolynomialVietabinomial theorem
2007 Geometry #8: Determine Angle of Quadrilateral

Source:

6/22/2012
ABCDABCD is a convex quadrilateral such that AB<ADAB<AD. The diagonal AC‾\overline{AC} bisects ∠BAD\angle BAD, and m∠ABD=130∘m\angle ABD=130^\circ. Let EE be a point on the interior of AD‾\overline{AD}, and m∠BAD=40∘m\angle BAD=40^\circ. Given that BC=CD=DEBC=CD=DE, determine m∠ACEm\angle ACE in degrees.
geometry
2007 Guts #8: Circle in a Square

Source:

6/22/2012
A circle inscribed in a square, Has two chords as shown in a pair. It has radius 22, And PP bisects TUTU. The chords' intersection is where?
Answer the question by giving the distance of the point of intersection from the center of the circle.
[asy] size(100); defaultpen(linewidth(0.8)); draw(unitcircle); draw((-1,-1)--(1,-1)--(1,1)--(-1,1)--cycle); label("AA",(-1,1),SE); label("BB",(1,1),SE); label("CC",(1,-1),SE); label("DD",(-1,-1),SE); pair M=(1,0),N=(0,-1),T=(-1,0),U=(0,1),P=dir(135); draw(P--M^^(-1,-1)--(1,1)); label("MM",M,SE); label("NN",N,SE); label("TT",T,SE); label("UU",U,SE); label("PP",P,dir(270)); dot(origin^^(-1,1)^^(-1,-1)^^(1,-1)^^(1,1)^^M^^N^^T^^U^^P); [/asy]