The roots of the equation $x^2+4x-5 = 0$ are also the roots of the equation $2x^3+9x^2-6x-5 = 0$. What is the third root of the second equation?

The numbers $a,b,c$ are the digits of a three digit number which satisfy $49a+7b+c = 286$. What is the three digit number $(100a+10b+c)$?

The vertices of a right-angled triangle are on a circle of radius $R$ and the sides of the triangle are tangent to another circle of radius $r$ (this is the circle that is inside triangle). If the lengths of the sides about the right angles are 16 and 30, determine the value of $R+r$.

Determine the smallest positive integer, $n$, which satisfies the equation $n^3+2n^2 = b$, where $b$ is the square of an odd integer.

Edward starts in his house, which is at (0,0) and needs to go point (6,4), which is coordinate for his school. However, there is a park that shaped as a square and has coordinates (2,1),(2,3),(4,1), and (4,3). There is no road for him to walk inside the park but there is a road for him to walk around the perimeter of the square. How many different shortest road routes are there from Edward's house to his school?

In a 14 team baseball league, each team played each of the other teams 10 times. At the end of the season, the number of games won by each team differed from those won by the team that immediately followed it by the same amount. Determine the greatest number of games the last place team could have won, assuming that no ties were allowed.

Triangle $ABC$ is right angled at $A$. The circle with center $A$ and radius $AB$ cuts $BC$ and $AC$ internally at $D$ and $E$ respectively. If $BD = 20$ and $DC = 16$, determine $AC^2$.

Determine all pairs of integers $(x,y)$ which satisfy the equation $$6x^2-3xy-13x+5y = -11$$

If $\log_{2n} 1994 = \log_n \left(486 \sqrt{2}\right)$, compute $n^6$.

Determine the sum of angles $A,B,$ where $0^\circ \leq A,B, \leq 180^\circ$ and $$\sin A + \sin B = \sqrt{\frac{3}{2}}, \cos A + \cos B = \sqrt{\frac{1}{2}}$$