Olympiad problems

2019 ASDAN Math Tournament, 1

A kite is a quadrilateral with $2$ pairs of equal adjacent sides. Given a cyclic kite with side lengths $3$ and $4$, compute the distance between the intersection of its diagonals and the center of the circle circumscribing it.

2019 CMIMC, 11

Tags: 2019 , team

Let $S$ be a subset of the natural numbers such that $0\in S$, and for all $n\in\mathbb N$, if $n$ is in $S$, then both $2n+1$ and $3n+2$ are in $S$. What is the smallest number of elements $S$ can have in the range $\{0,1,\ldots, 2019\}$?

2019 CMIMC, 10

Tags: 2019 , team

Let $\triangle ABC$ be a triangle with side lengths $a$, $b$, and $c$. Circle $\omega_A$ is the $A$-excircle of $\triangle ABC$, defined as the circle tangent to $BC$ and to the extensions of $AB$ and $AC$ past $B$ and $C$ respectively. Let $\mathcal{T}_A$ denote the triangle whose vertices are these three tangency points; denote $\mathcal{T}_B$ and $\mathcal{T}_C$ similarly. Suppose the areas of $\mathcal{T}_A$, $\mathcal{T}_B$, and $\mathcal{T}_C$ are $4$, $5$, and $6$ respectively. Find the ratio $a:b:c$.

2019 CMIMC, 1

Tags: 2019 , algebra , geometric sequence

Let $a_1$, $a_2$, $\ldots$, $a_n$ be a geometric progression with $a_1 = \sqrt{2}$ and $a_2 = \sqrt[3]{3}$. What is \[\displaystyle{\frac{a_1+a_{2013}}{a_7+a_{2019}}}?\]

2019 CMIMC, 4

Tags: geometry , 2019 , team

Let $\triangle A_1B_1C_1$ be an equilateral triangle of area $60$. Chloe constructs a new triangle $\triangle A_2B_2C_2$ as follows. First, she flips a coin. If it comes up heads, she constructs point $A_2$ such that $B_1$ is the midpoint of $\overline{A_2C_1}$. If it comes up tails, she instead constructs $A_2$ such that $C_1$ is the midpoint of $\overline{A_2B_1}$. She performs analogous operations on $B_2$ and $C_2$. What is the expected value of the area of $\triangle A_2B_2C_2$?

2019 CMIMC, 4

Tags: greatest common divisor , 2019 , number theory

Determine the sum of all positive integers $n$ between $1$ and $100$ inclusive such that \[\gcd(n,2^n - 1) = 3.\]