Olympiad problems

2015 ASDAN Math Tournament, 6

Let $f(x)=x^4-4x^3-3x^2-4x+1$. Compute the sum of the real roots of $f(x)$.

2015 ASDAN Math Tournament, 10

Tags: 2015 , Guts Test

Alice, Bob, and Conway are playing rock-paper-scissors. Each player plays against each of the other $2$ players and each pair plays until a winner is decided (i.e. in the event of a tie, they play again). What is the probability that each player wins exactly once?

2015 ASDAN Math Tournament, 20

Tags: 2015 , Guts Test

The sequence $a_1,a_2,\dots,a_{13}$ is a geometric sequence with $a_1=a$ and common ratio $r$, where $a$ and $r$ are positive integers. Given that $$\log_{2015}a_1+\log_{2015}a_2+\dots+\log_{2015}a_{13}=2015,$$ find the number of possible ordered pairs $(a,r)$.

2015 ASDAN Math Tournament, 31

Tags: 2015 , Guts Test

Compute the sum of the irrational solutions of the equation $$\frac{x^2+16x+54}{x^2+11x+35}=\frac{x^2+13x+35}{x^2+14x+54}.$$

2015 ASDAN Math Tournament, 1

Tags: 2015 , Algebra Tiebreaker

Let $a_n$ be a sequence defined as $a_1=1$, $a_2=2$, and $a_n=a_{n-1}-a_{n-2}$. Compute $a_{2015}$.