Olympiad problems

2016 ASDAN Math Tournament, 20

Let $ABC$ be a triangle such that $AB=9$, $BC=6$, and $AC=10$. $2$ points $D_1,D_2$ are labeled on $BC$ such that $BC$ is subdivided into $3$ equal segments; $4$ points $E_1,E_2,\dots,E_4$ are labeled on $AC$ such that $AC$ is subdivided into $5$ equal segments; and $8$ points $F_1,F_2,\dots,F_8$ are labeled on $AB$ such that $AB$ is subdivided into $9$ equal segments. All possible cevians are drawn from $A$ to each $D_i$; from $B$ to each $E_j$; and from $C$ to each $F_k$. At how many points in the interior of $\triangle ABC$ do at least $2$ cevians intersect?

2017 ASDAN Math Tournament, 20

Tags: 2017 , Guts Round

Let $\alpha$ and $\beta$ be positive rational numbers so that $\alpha+\beta\sqrt{5}$ is a root of some polynomial $x^2+ax+b$ where $a$ and $b$ are integers. What is the smallest possible value of $\alpha\beta$?

2016 ASDAN Math Tournament, 4

Tags: 2016 , Guts Round

Eddy is traveling to England and needs to exchange USD to GBP (US dollars to British pounds). The current exchange rate is $1.3$ USD for $1$ GBP. He exchanges $x$ USD to GBP and while in England, uses $\tfrac{x}{2}$ GBP. When he returns, the value of the British pound has changed so that $1$ GBP equals $\alpha$ USD. After exchanging all his remaining GBP, he notes that he has $\tfrac{x}{2}$ USD left. What is $\alpha$?

2016 ASDAN Math Tournament, 27

Tags: 2016 , Guts Round

Suppose that you are standing in the middle of a $100$ meter long bridge. You take a random sequence of steps either $1$ meter forward or $1$ meter backwards each iteration. At each step, if you are currently at meter $n$, you have a $\tfrac{n}{100}$ probability of $1$ meter forward, to meter $n+1$, and a $\tfrac{100-n}{100}$ of going $1$ meter backward, to meter $n-1$. What is the expected value of the number of steps it takes for you to step off the bridge (i.e., to get to meter $0$ or $100$)? Let $C$ be the actual answer and $A$ be the answer you will submit. Your score will be given by $\max\{0,\lceil25-25|\log_6(\tfrac{A-C/2}{C/2})|^{0.8}\rceil\}$.