Olympiad problems

2017 ASDAN Math Tournament, 2

An equilateral triangle $ABC$ shares a side with a square $BCDE$. If the resulting pentagon has a perimeter of $20$, what is the area of the pentagon? (The triangle and square do not overlap).

2017 CMIMC Individual Finals, 1

Tags: 2017 , Tiebreaker , number theory

Let $\tau(n)$ denote the number of positive integer divisors of $n$. For example, $\tau(4) = 3$. Find the sum of all positive integers $n$ such that $2 \tau(n) = n$.

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 $\alpha\beta$?

2017 ASDAN Math Tournament, 5

Tags: 2017 , Algebra Test

Compute $$\sum_{i=0}^\infty(-1)^i\sum_{j=i}^\infty(-1)^j\frac{2}{j^2+4j+3}.$$

2017 ASDAN Math Tournament, 3

Tags: 2017 , Discrete Math Tiebreaker

Alex and Zev are two members of a group of $2017$ friends who all know each other. Alex is trying to send a package to Zev. The delivery process goes as follows: Alex sends the package randomly to one of the people in the group. If this person is Zev, the delivery is done. Otherwise, the person who received the package also randomly sends it to someone in the group who hasn't held the package before and this process repeats until Zev gets the package. What is the expected number of deliveries made?

2017 ASDAN Math Tournament, 2

Tags: 2017 , Calculus Tiebreaker

Let $f$ be a linear function. Compute the slope of $f$ if $$\int_3^5f(x)dx=0\text{ and }\int_5^7f(x)dx=12.$$

2017 ASDAN Math Tournament, 6

Tags: 2017 , Geometry Test

Let $\triangle ABC$ be a right triangle with right angle $\angle B$. Suppose the angle bisector $l$ of $B$ divides the hypotenuse $AC$ into two segments of length $\sqrt{3}-1$ and $\sqrt{3}+1$. What is the measure of the smaller angle between $l$ and $AC$, in radians?

2017 CMIMC Computer Science, 2

Tags: algorithm , 2017 , computer science

We are given the following function $f$, which takes a list of integers and outputs another list of integers. (Note that here the list is zero-indexed.) \begin{tabular}{l} 1: \textbf{FUNCTION} $f(A)$ \\ 2: $\quad$ \textbf{FOR} $i=1,\ldots, \operatorname{length}(A)-1$: \\ 3: $\quad\quad$ $A[i]\leftarrow A[A[i]]$ \\ 4: $\quad\quad$ $A[0]\leftarrow A[0]-1$ \\ 5: $\quad$ \textbf{RETURN} $A$ \end{tabular} Suppose the list $B$ is equal to $[0,1,2,8,2,0,1,7,0]$. In how many entries do $B$ and $f(B)$ differ?