Olympiad problems

2015 ASDAN Math Tournament, 21

Parallelogram $ABCD$ has $AB=CD=6$ and $BC=AD=10$, where $\angle ABC$ is obtuse. The circumcircle of $\triangle ABD$ intersects $BC$ at $E$ such that $CE=4$. Compute $BD$.

2015 ASDAN Math Tournament, 30

Tags: 2015 , Guts Test

Suppose that $10$ mathematics teachers gather at a circular table with $25$ seats to discuss the upcoming mathematics competition. Each teacher is assigned a unique integer ID number between $1$ and $10$, and the teachers arrange themselves in such a way that teachers with consecutive ID numbers are not separated by any other teacher (IDs $1$ and $10$ are considered consecutive). In addition, each pair of teachers is separated by at least one empty seat. Given that seating arrangements obtained by rotation are considered identical, how many ways are there for the teachers to sit at the table?

2015 ASDAN Math Tournament, 28

Tags: 2015 , Guts Test

Consider $13$ marbles that are labeled with positive integers such that the product of all $13$ integers is $360$. Moor randomly picks up $5$ marbles and multiplies the integers on top of them together, obtaining a single number. What is the maximum number of different products that Moor can obtain?

2015 ASDAN Math Tournament, 3

Tags: 2015 , Guts Test

Consider a unit circle with center $O$. Let $P$ be a point outside the circle such that the two line segments passing through $P$ and tangent to the circle form an angle of $60^\circ$. Compute the length of $OP$.

2015 ASDAN Math Tournament, 26

Tags: 2015 , Guts Test

Lennart and Eddy are playing a betting game. Lennart starts with $7$ dollars and Eddy starts with $3$ dollars. Each round, both Lennart and Eddy bet an amount equal to the amount of the player with the least money. For example, on the first round, both players bet $3$ dollars. A fair coin is then tossed. If it lands heads, Lennart wins all the money bet; if it lands tails, Eddy wins all the money bet. They continue playing this game until one person has no money. What is the probability that Eddy ends with $10$ dollars?

2015 ASDAN Math Tournament, 2

Tags: 2015 , Guts Test

Meena owns a bottle cap collection. While on a vacation, she finds a large number of bottle caps, increasing her collection size by $40\%$. Later on her same vacation, she decides that she does not like some of the bottle caps, so she gives away $20\%$ of her current collection. Suppose that Meena owns $21$ more bottle caps after her vacation than before her vacation. How many bottle caps did Meena have before her vacation?

2015 ASDAN Math Tournament, 12

Tags: 2015 , Guts Test

The rectangular faces of rectangular prism $A$ have perimeters $12$, $16$, and $24$. The rectangular faces of rectangular prism $B$ have perimeters $12$, $16$, and $20$. Let $V_A$ denote the volume of $A$ and $V_B$ denote the volume of $B$. Find $V_A-V_B$.

2015 ASDAN Math Tournament, 6

Tags: 2015 , Guts Test

A circle $A$ is circumscribed about a unit square and a circle $B$ is inscribed inside the same unit square. Compute the ratio of the area of $A$ to the area of $B$.

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 $$\frac{x^2+16x+54}{x^2+11x+35}=\frac{x^2+13x+35}{x^2+14x+54}.$$