MathDB

3

Part of 2013 Princeton University Math Competition

Problems(11)

2013 PUMaC Algebra A3

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11/22/2013
Let x1=10x_1=\sqrt{10} and y1=3y_1=\sqrt3. For all n≥2n\geq 2, let \begin{align*}x_n&=x_{n-1}\sqrt{77}+15y_{n-1}\\y_n&=5x_{n-1}+y_{n-1}\sqrt{77}\end{align*} Find x56+2x54−9x54y52−12x52y52+27x52y54+18y54−27y56.x_5^6+2x_5^4-9x_5^4y_5^2-12x_5^2y_5^2+27x_5^2y_5^4+18y_5^4-27y_5^6.
2013 PUMaC Algebra B3

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11/22/2013
Let x1=1/20x_1=1/20, x2=1/13x_2=1/13, and xn+2=2xnxn+1(xn+xn+1)xn2+xn+12x_{n+2}=\dfrac{2x_nx_{n+1}(x_n+x_{n+1})}{x_n^2+x_{n+1}^2} for all integers n≥1n\geq 1. Evaluate ∑n=1∞(1/(xn+xn+1))\textstyle\sum_{n=1}^\infty(1/(x_n+x_{n+1})).
2013 PUMaC Combinatorics A3

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11/23/2013
How many tuples of integers (a0,a1,a2,a3,a4)(a_0,a_1,a_2,a_3,a_4) are there, with 1≤ai≤51\leq a_i\leq 5 for each ii, so that a0<a1>a2<a3>a4a_0<a_1>a_2<a_3>a_4?
2013 PUMaC Combinatorics B3

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11/23/2013
Chris's pet tiger travels by jumping north and east. Chris wants to ride his tiger from Fine Hall to McCosh, which is 33 jumps east and 1010 jumps north. However, Chris wants to avoid the horde of PUMaC competitors eating lunch at Frist, located 22 jumps east and 44 jumps north of Fine Hall. How many ways can he get to McCosh without going through Frist?
2013 PUMaC Geometry A3

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11/22/2013
Consider the shape formed from taking equilateral triangle ABCABC with side length 66 and tracing out the arc BCBC with center AA. Set the shape down on line ll so that segment ABAB is perpendicular to ll, and BB touches ll. Beginning from arc BCBC touching ll, we roll ABCABC along ll until both points AA and CC are on the line. The area traced out by the roll can be written in the form nπn\pi, where nn is an integer. Find nn.
geometryUSAMTSrectangle
2013 PUMaC Geometry B3

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11/22/2013
Consider all planes through the center of a 2×2×22\times2\times2 cube that create cross sections that are regular polygons. The sum of the cross sections for each of these planes can be written in the form ab+ca\sqrt b+c, where bb is a square-free positive integer. Find a+b+ca+b+c.
geometry3D geometry
2013 PUMaC Number Theory A3/B5

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11/24/2013
Let AA be the greatest possible value of a product of positive integers that sums to 20142014. Compute the sum of all bases and exponents in the prime factorization of AA. For example, if A=7⋅115A=7\cdot 11^5, the answer would be 7+11+5=237+11+5=23.
number theoryprime factorization
2013 PUMaC Number Theory B3

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11/24/2013
Find the smallest positive integer xx such that
[*] xx is 11 more than a multiple of 33, [*] xx is 33 more than a multiple of 55, [*] xx is 55 more than a multiple of 77, [*] xx is 99 more than a multiple of 1111, and [*] xx is 22 more than a multiple of 1313.
modular arithmeticnumber theory
Finals 2013 A3: An edge matching

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11/17/2013
A graph consists of a set of vertices, some of which are connected by (undirected) edges. A star of a graph is a set of edges with a common endpoint. A matching of a graph is a set of edges such that no two have a common endpoint. Show that if the number of edges of a graph GG is larger than 2(k−1)22(k-1)^2, then GG contains a matching of size kk or a star of size kk.
inequalitiesevan orz1434xooksi lost the gameotis
2013 Division B Finals #3

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11/17/2013
Find the smallest positive integer nn with the following property: for every sequence of positive integers a1,a2,…,ana_1,a_2,\ldots , a_n with a1+a2+…+an=2013a_1+a_2+\ldots +a_n=2013, there exist some (possibly one) consecutive term(s) in the sequence that add up to 7070.