MathDB

5

Part of 2013 Princeton University Math Competition

Problems(6)

2013 PUMaC Algebra A5

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11/22/2013
Suppose w,x,y,zw,x,y,z satisfy \begin{align*}w+x+y+z&=25,\\wx+wy+wz+xy+xz+yz&=2y+2z+193\end{align*} The largest possible value of ww can be expressed in lowest terms as w1/w2w_1/w_2 for some integers w1,w2>0w_1,w_2>0. Find w1+w2w_1+w_2.
AMCUSA(J)MOUSAMOconicsparabola
2013 PUMaC Combinatorics A5/B4

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11/23/2013
Mereduth has many red boxes and many blue boxes. Coloon has placed five green boxes in a row on the ground, and Mereduth wants to arrange some number of her boxes on top of his row. Assume that each box must be placed so that it straddles two lower boxes. Including the one with no boxes, how many arrangements can Mereduth make?
2013 PUMaC Geometry A5

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11/22/2013
Suppose you have a sphere tangent to the xyxy-plane with its center having positive zz-coordinate. If it is projected from a point P=(0,b,a)P=(0,b,a) to the xyxy-plane, it gives the conic section y=x2y=x^2. If we write a=pqa=\tfrac pq where p,qp,q are integers, find p+qp+q.
geometry3D geometrysphereconics
2013 PUMaC Geometry B5

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11/22/2013
Circle ww with center OO meets circle Γ\Gamma at X,Y,X,Y, and OO is on Γ\Gamma. Point Z∈ΓZ\in\Gamma lies outside ww such that XZ=11XZ=11, OZ=15OZ=15, and YZ=13YZ=13. If the radius of circle ww is rr, find r2r^2.
geometrytrigonometry
2013 PUMaC Number Theory A5/B7

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11/24/2013
Define a "digitized number" as a ten-digit number a0a1…a9a_0a_1\ldots a_9 such that for k=0,1,…,9k=0,1,\ldots, 9, aka_k is equal to the number of times the digit kk occurs in the number. Find the sum of all digitized numbers.
2013 PUMaC Team 5

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11/24/2013
A palindrome number is a positive integer that reads the same forward and backward. For example, 12211221 and 88 are palindrome numbers whereas 6969 and 157157 are not. AA and BB are 44-digit palindrome numbers. CC is a 33-digit palindrome number. Given that A−B=CA-B=C, what is the value of CC?