MathDB

4

Part of 2010 Princeton University Math Competition

Problems(5)

2010 PUMaC Algebra A4/B6: nested radical

Source:

8/20/2011
Define f(x)=x+x+x+x+x+…\displaystyle{f(x) = x + \sqrt{x + \sqrt{x + \sqrt{x + \sqrt{x + \ldots}}}}}. Find the smallest integer xx such that f(x)≥50xf(x)\ge50\sqrt{x}.
(Edit: The official question asked for the "smallest integer"; the intended question was the "smallest positive integer".)
quadraticscalculusintegrationalgebraquadratic formula
2010 PUMaC Combinatorics A4: expected value from grid walk

Source:

8/21/2011
Erick stands in the square in the 2nd row and 2nd column of a 5 by 5 chessboard. There are \1billsinthetopleftandbottomrightsquares,andthereare$5billsinthetoprightandbottomleftsquares,asshownbelow.1 bills in the top left and bottom right squares, and there are \$5 bills in the top right and bottom left squares, as shown below. \begin{tabular}{|p{1em}|p{1em}|p{1em}|p{1em}|p{1em}|} \hline \$1 & & & & \$5 \\ \hline & E & & &\\ \hline & & & &\\ \hline & & & &\\ \hline \$5 & & & & \$1 \\ \hline \end{tabular}Everysecond,Erickrandomlychoosesasquareadjacenttotheonehecurrentlystandsin(thatis,asquaresharinganedgewiththeonehecurrentlystandsin)andmovestothatsquare.WhenErickreachesasquarewithmoneyonit,hetakesitandquits.TheexpectedvalueofErick′swinningsindollarsis Every second, Erick randomly chooses a square adjacent to the one he currently stands in (that is, a square sharing an edge with the one he currently stands in) and moves to that square. When Erick reaches a square with money on it, he takes it and quits. The expected value of Erick's winnings in dollars is m/n,where, where mand and narerelativelyprimepositiveintegers.Find are relatively prime positive integers. Find m+n$.
symmetryprobabilityexpected valuenumber theoryrelatively prime
2010 PUMaC Geometry A4/B6: segments in hexagon

Source:

8/21/2011
In regular hexagon ABCDEFABCDEF, ACAC, CECE are two diagonals. Points MM, NN are on ACAC, CECE respectively and satisfy AC:AM=CE:CN=rAC: AM = CE: CN = r. Suppose B,M,NB, M, N are collinear, find 100r2100r^2. [asy] size(120); defaultpen(linewidth(0.7)+fontsize(10)); pair D2(pair P) { dot(P,linewidth(3)); return P; } pair A=dir(0), B=dir(60), C=dir(120), D=dir(180), E=dir(240), F=dir(300), N=(4*E+C)/5,M=intersectionpoints(A--C,B--N)[0]; draw(A--B--C--D--E--F--cycle); draw(A--C--E); draw(B--N); label("AA",D2(A),plain.E); label("BB",D2(B),NE); label("CC",D2(C),NW); label("DD",D2(D),W); label("EE",D2(E),SW); label("FF",D2(F),SE); label("MM",D2(M),(0,-1.5)); label("NN",D2(N),SE); [/asy]
geometrytrigonometryratio
2010 PUMaC NT A4: n*phi(n) is perfect square

Source:

8/22/2011
Find the largest positive integer nn such that nφ(n)n\varphi(n) is a perfect square. (φ(n)\varphi(n) is the number of integers kk, 1≤k≤n1 \leq k \leq n that are relatively prime to nn)
number theoryrelatively prime
2010 PUMaC Geometry B4: BF+DH=FH

Source:

8/21/2011
Unit square ABCDABCD is divided into four rectangles by EFEF and GHGH, with BF=14BF = \frac14. EFEF is parallel to ABAB and GHGH parallel to BCBC. EFEF and GHGH meet at point PP. Suppose BF+DH=FHBF + DH = FH, calculate the nearest integer to the degree of ∠FAH\angle FAH. [asy] size(100); defaultpen(linewidth(0.7)+fontsize(10)); pair D2(pair P) { dot(P,linewidth(3)); return P; } // NOTE: I've tampered with the angles to make the diagram not-to-scale. The correct numbers should be 72 instead of 76, and 45 instead of 55. pair A=(0,1), B=(0,0), C=(1,0), D=(1,1), F=intersectionpoints(A--A+2*dir(-76),B--C)[0], H=intersectionpoints(A--A+2*dir(-76+55),D--C)[0], E=F+(0,1), G=H-(1,0), P=intersectionpoints(E--F,G--H)[0]; draw(A--B--C--D--cycle); draw(F--A--H); draw(E--F); draw(G--H); label("AA",D2(A),NW); label("BB",D2(B),SW); label("CC",D2(C),SE); label("DD",D2(D),NE); label("EE",D2(E),plain.N); label("FF",D2(F),S); label("GG",D2(G),W); label("HH",D2(H),plain.E); label("PP",D2(P),SE); [/asy]
geometryrectangletrigonometry