MathDB

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Part of 2010 Princeton University Math Competition

Problems(10)

2010 PUMaC Algebra A1: Sum of coefficients

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8/20/2011
Find the sum of the coefficients of the polynomial (63x−61)4(63x-61)^4.
algebrapolynomial
2010 PUMaC Combinatorics A1/B2: new customer order

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8/21/2011
PUMaCDonalds, a newly-opened fast food restaurant, has 5 menu items. If the first 4 customers each choose one menu item at random, the probability that the 4th customer orders a previously unordered item is m/nm/n, where mm and nn are relatively prime positive integers. Find m+nm+n.
probabilitynumber theoryrelatively prime
2010 PUMaC Geometry A1/B3: minimal triangle area

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8/21/2011
As in the following diagram, square ABCDABCD and square CEFGCEFG are placed side by side (i.e. CC is between BB and EE and GG is between CC and DD). If CE=14CE = 14, AB>14AB > 14, compute the minimal area of △AEG\triangle AEG. [asy] size(120); defaultpen(linewidth(0.7)+fontsize(10)); pair D2(real x, real y) { pair P = (x,y); dot(P,linewidth(3)); return P; } int big = 30, small = 14; filldraw((0,big)--(big+small,0)--(big,small)--cycle, rgb(0.9,0.5,0.5)); draw(scale(big)*unitsquare); draw(shift(big,0)*scale(small)*unitsquare); label("AA",D2(0,big),NW); label("BB",D2(0,0),SW); label("CC",D2(big,0),SW); label("DD",D2(big,big),N); label("EE",D2(big+small,0),SE); label("FF",D2(big+small,small),NE); label("GG",D2(big,small),NE); [/asy]
geometry
2010 PUMaC NT A1/B3: n^4+(n+1)^4 composite

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8/22/2011
Find the smallest positive integer nn such that n4+(n+1)4n^4 + (n+1)^4 is composite.
2010 PUMaC Individual Finals A1: combinatorial identity

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8/31/2011
Show that ∑i=1n(−1)n+i(ni)(inn)=nn\displaystyle{\sum_{i=1}^{n}(-1)^{n+i}\binom{n}{i}\binom{in}{n} = n^{n}}.
2010 PUMaC Algebra B1: Star operation

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8/21/2011
Let the operation ★\bigstar be defined by x★y=yx−xyx\bigstar y=y^x-xy. Calculate (3★4)−(4★3)(3\bigstar4)-(4\bigstar3).
2010 PUMaC Combinatorics B1: band song list

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8/21/2011
The Princeton University Band plays a setlist of 8 distinct songs, 3 of which are tiring to play. If the Band can't play any two tiring songs in a row, how many ways can the band play its 8 songs?
Princetoncollege
2010 PUMaC Geometry B1: external angle = internal/6

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8/21/2011
In a polygon, every external angle is one sixth of its corresponding internal angle. How many sides does the polygon have?
geometry
2010 PUMaC NT B1: small integer with many divisors

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8/22/2011
Find the positive integer less than 18 with the most positive divisors.
number theoryprime factorization
2010 PUMaC Ind. Finals B1: GCD of consecutive triangular #s

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8/31/2011
Show that the GCD of three consecutive triangular numbers is 1.
number theorygreatest common divisor