MathDB

6

Part of 2010 Princeton University Math Competition

Problems(5)

2010 PUMaC Algebra A6: functional equation

Source:

8/20/2011
Assume that f(a+b)=f(a)+f(b)+abf(a+b) = f(a) + f(b) + ab, and that f(75)−f(51)=1230f(75) - f(51) = 1230. Find f(100)f(100).
2010 PUMaC Combinatorics A6: diagonals of regular decagon

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8/21/2011
All the diagonals of a regular decagon are drawn. A regular decagon satisfies the property that if three diagonals concur, then one of the three diagonals is a diameter of the circumcircle of the decagon. How many distinct intersection points of diagonals are in the interior of the decagon?
geometrycircumcircle
2010 PUMaC Geometry A6: folded semicircle

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8/21/2011
In the following diagram, a semicircle is folded along a chord ANAN and intersects its diameter MNMN at BB. Given that MB:BN=2:3MB : BN = 2 : 3 and MN=10MN = 10. If AN=xAN = x, find x2x^2. [asy] size(120); defaultpen(linewidth(0.7)+fontsize(10)); pair D2(pair P) { dot(P,linewidth(3)); return P; } real r = sqrt(80)/5; pair M=(-1,0), N=(1,0), A=intersectionpoints(arc((M+N)/2, 1, 0, 180),circle(N,r))[0], C=intersectionpoints(circle(A,1),circle(N,1))[0], B=intersectionpoints(circle(C,1),M--N)[0]; draw(arc((M+N)/2, 1, 0, 180)--cycle); draw(A--N); draw(arc(C,1,180,180+2*aSin(r/2))); label("AA",D2(A),NW); label("BB",D2(B),SW); label("MM",D2(M),S); label("NN",D2(N),SE); [/asy]
geometrygeometric transformationreflection
2010 PUMaC NT A6/B7: fraction with 0s, 1s

Source:

8/22/2011
Find the numerator of 101011…11⏞2011 ones0101110011…11⏟2011 ones0011\frac{1010\overbrace{11 \ldots 11}^{2011 \text{ ones}}0101}{1100\underbrace{11 \ldots 11}_{2011\text{ ones}}0011} when reduced.
2010 PUMaC Combinatorics B6: exploding ants in pentagon

Source:

8/21/2011
A regular pentagon is drawn in the plane, along with all its diagonals. All its sides and diagonals are extended infinitely in both directions, dividing the plane into regions, some of which are unbounded. An ant starts in the center of the pentagon, and every second, the ant randomly chooses one of the edges of the region it's in, with an equal probability of choosing each edge, and crosses that edge into another region. If the ant enters an unbounded region, it explodes. After first leaving the central region of the pentagon, let xx be the expected number of times the ant re-enters the central region before it explodes. Find the closest integer to 100x100x.
probabilityexpected value