MathDB

4

Part of 2013 Princeton University Math Competition

Problems(7)

2013 PuMAC Algebra A4/B6

Source:

11/22/2013
Suppose a,ba,b are nonzero integers such that two roots of x3+ax2+bx+9ax^3+ax^2+bx+9a coincide, and all three roots are integers. Find ∣ab∣|ab|.
LaTeX
2013 PUMaC Algebra B4

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11/22/2013
Let f(x)=1−∣x∣f(x)=1-|x|. Let \begin{align*}f_n(x)&=(\overbrace{f\circ \cdots\circ f}^{n\text{ copies}})(x)\\g_n(x)&=|n-|x| |\end{align*} Determine the area of the region bounded by the xx-axis and the graph of the function ∑n=110f(x)+∑n=110g(x).\textstyle\sum_{n=1}^{10}f(x)+\textstyle\sum_{n=1}^{10}g(x).
geometryfunction
2013 PUMaC Combinatorics A4/B8

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11/23/2013
You roll three fair six-sided dice. Given that the highest number you rolled is a 55, the expected value of the sum of the three dice can be written as ab\tfrac ab in simplest form. Find a+ba+b.
probabilityexpected value
2013 PUMaC Geometry A4/B6

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11/22/2013
Draw an equilateral triangle with center OO. Rotate the equilateral triangle 30∘,60∘,90∘30^\circ, 60^\circ, 90^\circ with respect to OO so there would be four congruent equilateral triangles on each other. Look at the diagram. If the smallest triangle has area 11, the area of the original equilateral triangle could be expressed as p+qrp+q\sqrt r where p,q,rp,q,r are positive integers and rr is not divisible by a square greater than 11. Find p+q+rp+q+r.
geometryrotation
2013 PUMaC Number Theory A4/B6

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11/24/2013
Let dd be the greatest common divisor of 23010−22^{30^{10}}-2 and 23045−22^{30^{45}}-2. Find the remainder when dd is divided by 20132013.
modular arithmeticgreatest common divisor
2013 PUMaC Number Theory B4

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11/24/2013
Compute the smallest integer n≥4n\geq 4 such that (n4)\textstyle\binom n4 ends in 44 or more zeroes (i.e. the rightmost four digits of (n4)\textstyle\binom n4 are 00000000).
modular arithmetic
2013 PUMaC Team 4

Source:

11/24/2013
Find the sum of all positive integers mm such that 2m2^m can be expressed as a sum of four factorials (of positive integers).
Note: The factorials do not have to be distinct. For example, 24=162^4=16 counts, because it equals 3!+3!+2!+2!3!+3!+2!+2!.
factorial