Virginia Tech Regional Mathematics Contest

- An isosceles triangle with an inscribed circle is labeled as shown in
the figure. Find an expression, in terms of the angle
α and
the length
*a*, for the area of the curvilinear triangle bounded by sides*AB*and*AC*and the arc*BC*.

- Find all differentiable functions
*f*which satisfy*f*(*x*)^{3}= ∫_{0}^{x}*f*(*t*)^{2}*dt*for all real*x*. - Prove that if
α is a real root of
(1 -
*x*^{2})(1 +*x*+*x*^{2}+ ... +*x*^{n}) -*x*= 0 which lies in (0, 1), with*n*= 1, 2,..., then α is also a root of (1 -*x*^{2})(1 +*x*+*x*^{2}+ ... +*x*^{n+1}) - 1 = 0. - Prove that if
*x*> 0 and*n*> 0, where*x*is real and*n*is an integer, then*x*^{n}/((*x*+ 1)^{n+1})≤*n*^{n}/((*n*+ 1)^{n+1}). - Let
*f*(*x*) =*x*^{5}-5*x*^{3}+ 4*x*. In each part (i)-(iv), prove or disprove that there exists a real number*c*for which*f*(*x*) -*c*= 0 has a root of multiplicity(i) one, (ii) two, (iii) three, (iv) four.

- Let
*a*_{0}= 1 and for*n*> 0, let*a*_{n}be defined by*a*_{n}= - ∑_{k=1}^{n}*a*_{n-k}/*k*!.*a*_{n}= (- 1)^{n}/*n*!, for*n*= 0, 1, 2,.... - A and B play the following money game, where
*a*_{n}and*b*_{n}denote the amount of holdings of A and B, respectively, after the*n*th round. At each round a player pays one-half his holdings to the bank, then receives one dollar from the bank if the*other*player had less than*c*dollars at the end of the*previous*round. If*a*_{0}= .5 and*b*_{0}= 0, describe the behavior of*a*_{n}and*b*_{n}when*n*is large, for(i)

*c*= 1.24 and (ii)*c*= 1.26. - Mathematical National Park has a collection of trails. There are
designated campsites along the trails, including a campsite at each
intersection of trails. The rangers call each stretch of trail
between adjacent campsites a ``segment". The trails have been laid
out so that it is possible to take a hike that starts at any
campsite, covers each segment exactly once, and ends at the beginning
campsite. Prove that it is possible to plan a collection
C of hikes with all of the following properties:
- (i)
- Each segment is covered exactly once in one hike
*h*∈C and never in any of the other hikes of C. - (ii)
- Each
*h*∈C has a base campsite that is its beginning and end, but which is never passed in the middle of the hike. (Different hikes of C may have different base campsites.) - (iii)
- Except for its base campsite at beginning and end, no hike in C passes any campsite more than once.

Peter Linnell 2009-06-24