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## ARML Math Cache

A cache by tenebrus Message this owner
Hidden : 06/02/2010
Difficulty:
Terrain:

Size:  (other)

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### Geocache Description:

Every year, for quite some time, Penn State is a host for the annual American Regions Mathematics League competition. In fact, it is the flagship location, although now there are three other locations which also hold the competition. As such, I figured that a taste of the challenges that these exceptional high school students overcome would be appropriate for such a campus.

ONCE SOLVED, DO NOT USE ANY OLD OFFSET VALUES. JUST GO THERE AND FIND IT.

The cache is NOT at the posted coordinates. You will have to solve the puzzle for the real coordinates.

There are four problems. The solutions to the first three will set up the solution to the fourth (not entirely dissimilar to the relay event at ARML). Your answers must be in the simplest indicated form. Although typically, calculators would not be permitted, you may use them for this geocache (but you'll know deep down inside that you had to resort to doing so, if you do).

Question 1. In rectangle PQRS, diagonal PR is drawn, as is segment ST where T is the midpoint of side PQ. Segments PR and ST intersect at U and the following measurements are given: TU = 3 and UR = 8. Determine in simplest form, a * sqrt(b), the area of the rectangle.

Question 2. In the complex plane, z, z^2, z^3 form, in some order, three vertices of a non-degenerate square. Determine, in lowest terms, the harmonic mean, c/d, of the 3 possible areas of these squares.

Question 3. Semi-regular hexagon GHJKLM is circumscribed by circle O. If GH = 2 HJ = JK = 2 KL = LM = 2 MG and if GK = 12, determine the perimeter, e, of the hexagon.

Final Question. Using a, b, c, d, e as the consecutive sides of a hexagon all of whose angles are congruent, determine the missing side f.

Finding the cache. N 40° 48.(d*e + f) W 77° 51.(a*b - 6*c).

Just to be clear, both (a*b - 6*c) and (d*e + f) are three-digit numbers, even if that means that you need to put a placeholder 0 in the hundreds' place.

[one] fvzvyne svtherf cyhf clguntberna nytroen
[two] guvax ubj ner pbzcyrk qvfgnaprf pbzchgrq
[three] flzzrgevrf be cnggreaf
[final] cnggreaf be pbbeqvangr obbxxrrcvat
[container] pbeare, urnq uvtu
[doublecheck] fvk inevnoyr fhz vf bar uhaqerq gjragl-guerr

Decryption Key

A|B|C|D|E|F|G|H|I|J|K|L|M
-------------------------
N|O|P|Q|R|S|T|U|V|W|X|Y|Z

(letter above equals below, and vice versa)

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