Putnam Competition

Putnam competition is annual math test across the schools in United States. It is organized on the first Saturday in December in two three hour sittings with a lunch break in between.

The test consists of 12 questions that requires basic college mathematics and creative thinking. One can only participate at most four times. The top five scorers are named Putnam Fellows.

References
William Lowell Putnam Competition at wikipedia
Additional information about Putnam competition
Archive of putnam problems and solutions
Good collection of putnam problems and solutions cataloged by year
Putnam and beyond problem book and study guide

Great circle distance

Earth is sphere and more accurately a spheroid. Calculating distance between two points on earth is different to calculating the distance between the two points on flat surface. The Circle that connects the two points with center of the sphere as the center is called great circle. The distance between the points is the length of the arc between the points on great circle.

Approximate method:
Circumference of earth at equator is 40,076km and we have 360 degrees in circle and so each degree of longitude at equator is about to 111.32km or 69 miles. The distance between two longitude lines decrease from the equator to poles.

Around 30 degrees north or south from the equator, one degree of longitude is about 96.41km. Around 45 degrees north or south of equator, one degree of longitude is 78.71km. Around 60 degrees, one degree of longitude is 55.66km. Around 75 degrees north or south of equator, one degree of longitude is 28.82km. Around 90 degrees from the equator, that will be poles and one degree of longitude is zero distance.  

However the distance between two lines of latitude stay almost same anywhere from equator to poles. Small difference is because of the shape of the earth. One degree of latitude is 110.57km at equator and 111.69km at poles.

Once we have longitudinal distance and latitude distance between two points, we can compute the distance between two points. The Pythagoras theorem can be used here.

Using cosine law:
Suppose the geographical latitude and longitude of two points on the earth are a1, a2 and b1. We can compute the central angle between them is given by the spherical law of cosines
c = arccos(sin(a1).sin(a2) + cos(a1).cos(a2).cos(|b2-b1|))
The distance d, or the arc length on a sphere of radius r formed by spherical angle c in radians  = r. c; 

Using haversine law:
Alternatively if we know the linear distance between the points as w and using sine law, length of arc on great circle can be calculated as 2*r*arcsin(w/(2.r)).

References
Lat Lang Story
Great circle distance using cosine law
Great circle distance using sine law
Longititude and latitude measuring

Indiana Pi

Once upon a time, one man thought he found a way to squaring the circle(that also means representing the exact value of pi on paper) and had copyrighted it and proposed it as bill in a state assembly. It was almost passed by the assembly. See Indiana Pi history and Indiana state Bill 246 for details.

It starts with the claim that it as "A Bill for an act introducing a new mathematical truth ... to education to be used only by the State of Indiana free of cost ..." and supposed to be used on royalty by the rest of the world.