Virginia Tech Regional Mathematics Contest

- In the expansion of (
*a*+*b*)^{n}, where*n*is a natural number, there are*n*+ 1 dissimilar terms. Find the number of dissimilar terms in the expansion of (*a*+*b*+*c*)^{n}. - A positive integer
*N*(in base 10) is called*special*if the operation*C*of replacing each digit*d*of*N*by its nine's-complement 9 -*d*, followed by the operation*R*of reversing the order of the digits, results in the original number. (For example, 3456 is a special number because*R*[(*C*3456)] = 3456.) Find the sum of all special positive integers less than one million which do not end in zero or nine. - Let a triangle have vertices at
*O*(0, 0),*A*(*a*, 0), and*B*(*b*,*c*) in the (*x*,*y*)-plane.- (a)
- Find the coordinates of a point
*P*(*x*,*y*) in the exterior of D*OAB*satisfying area(*OAP*) = area(*OBP*) = area(*ABP*). - (b)
- Find a point
*Q*(*x*,*y*) in the interior of D*OAQ*satisfying area(*OAQ*) = area(*OBQ*) = area(*ABQ*).

- A finite set of roads connect
*n*towns*T*_{1},*T*_{2},...,*T*_{n}where*n*2. We say that towns*T*_{i}and*T*_{j}(*i**j*) are*directly connected*if there is a road segment connecting*T*_{i}and*T*_{j}which does not pass through any other town. Let*f*(*T*_{k}) be the number of other towns directly connected to*T*_{k}. Prove that*f*is not one-to-one. - Find the function
*f*(*x*) such that for all*L*0, the area under the graph of*y*=*f*(*x*) and above the*x*-axis from*x*= 0 to*x*=*L*equals the arc length of the graph from*x*= 0 to*x*=*L*. (Hint: recall that*d*/*dx*cosh^{-1}*x*= 1/.) - Let
*f*(*x*) = 1/*x*and*g*(*x*) = 1 -*x*for*x*(0, 1). List all distinct functions that can be written in the form*f*`o`*g*`o`*f*`o`*g*`o`...`o`*f*`o`*g*`o`*f*where`o`represents composition. Write each function in the form (*ax*+*b*)/(*cx*+*d*), and prove that your list is exhaustive. - If
*a*and*b*are real, prove that*x*^{4}+*ax*+*b*= 0 cannot have only real roots. - A sequence
*f*_{n}is generated by the recurrence formula*f*_{n + 1}= (*f*_{n}*f*_{n - 1}+ 1)/*f*_{n - 2},*n*= 2, 3, 4,..., with*f*_{0}=*f*_{1}=*f*_{2}= 1. Prove that*f*_{n}is integer-valued for all integers*n*0.