MathDB

9

Part of 2006 Stanford Mathematics Tournament

Problems(3)

Numerators and Denominators

Source:

3/15/2010
If to the numerator and denominator of the fraction 13 \frac{1}{3} you add its denominator 3, the fraction will double. Find a fraction which will triple when its denominator is added to its numerator and to its denominator and find one that will quadruple.
2006 SMT Algebra #9 - Figure out the Trick

Source:

8/24/2011
Principal Skinner is thinking of two integers mm and nn and bets Superintendent Chalmers that he will not be able to determine these integers with a single piece of information. Chalmers asks Skinner the numerical value of mn+13m+13n−m2−n2mn+13m+13n-m^2-n^2. From the value of this expression alone, he miraculously determines both mm and nn. What is the value of the above expression.
2006 SMT Team Round #9 - Geometry

Source:

8/22/2011
△ABC\triangle ABC has AB=ACAB=AC. Points MM and NN are midpoints of AB‾\overline{AB} and AC‾\overline{AC}, respectively. The medians MC‾\overline{MC} and NB‾\overline{NB} intersect at a right angle. Find (ABBC)2(\tfrac{AB}{BC})^2.
geometryPythagorean Theorem