Found problems: 105
2015 ASDAN Math Tournament, 30
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, 6
Find all triples of integers $(x,y,z)$ which satisfy the equations
\begin{align*}
x^2-y-2z&=4\\
y^2-2z-3x&=-2\\
2z^2-3x-5y&=-22.\\
\end{align*}
2015 ASDAN Math Tournament, 2
Nick is taking a $10$ question test where each answer is either true or false with equal probability. Nick forgot to study, so he guesses randomly on each of the $10$ problems. What is the probability that Nick answers exactly half of the questions correctly?
2015 ASDAN Math Tournament, 8
Let $f(x)=\tfrac{x+a}{x+b}$ for real numbers $x$ such that $x\neq -b$. Compute all pairs of real numbers $(a,b)$ such that $f(f(x))=-\tfrac{1}{x}$ for $x\neq0$.
2015 ASDAN Math Tournament, 28
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
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, 3
You have a circular necklace with $10$ beads on it, all of which are initially unpainted. You randomly select $5$ of these beads. For each selected bead, you paint that selected bead and the two beads immediately next to it (this means we may paint a bead multiple times). Once you have finished painting, what is the probability that every bead is painted?
2015 Turkey Team Selection Test, 2
There are $2015$ points on a plane and no two distances between them are equal. We call the closest $22$ points to a point its $neighbours$. If $k$ points share the same neighbour, what is the maximum value of $k$?
2015 ASDAN Math Tournament, 26
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
Heesu plays a game where he starts with $1$ piece of candy. Every turn, he flips a fair coin. On heads, he gains another piece of candy, unless he already has $5$ pieces of candy, in which case he loses $4$ pieces of candy and goes back to having $1$ piece of candy. On tails, the game ends. What is the expected number of pieces of candy that Heesu will have when the game ends?
2015 ASDAN Math Tournament, 1
Rachel has $3$ children, all of which are at least $2$ years old. The ages of the children are all pairwise relatively prime, but Rachel’s age is a multiple of each of her children’s ages. What is Rachel’s minimum possible age?
2015 ASDAN Math Tournament, 1
A rectangle $ABCD$ is split into four smaller non-overlapping rectangles by two perpendicular line segments, whose endpoints are on the sides of $ABCD$. If the smallest three rectangles have areas of $48$, $18$, and $12$, what is the area of $ABCD$?
2015 ASDAN Math Tournament, 8
Let $\{x\}$ denote the fractional part of $x$, which means the unique real $0\leq\{x\}<1$ such that $x-\{x\}$ is an integer. Let $f_{a,b}(x)=\{x+a\}+2\{x+b\}$ and let its range be $[m_{a,b},M_{a,b})$. Find the minimum value of $M_{a,b}$ as $a$ and $b$ range along all real numbers.
2015 ASDAN Math Tournament, 4
In trapezoid $ABCD$ with $AD\parallel BC$, $AB=6$, $AD=9$, and $BD=12$. If $\angle ABD=\angle DCB$, find the perimeter of the trapezoid.
2015 China Team Selection Test, 6
There are some players in a Ping Pong tournament, where every $2$ players play with each other at most once. Given:
\\(1) Each player wins at least $a$ players, and loses to at least $b$ players. ($a,b\geq 1$)
\\(2) For any two players $A,B$, there exist some players $P_1,...,P_k$ ($k\geq 2$) (where $P_1=A$,$P_k=B$), such that $P_i$ wins $P_{i+1}$ ($i=1,2...,k-1$).
\\Prove that there exist $a+b+1$ distinct players $Q_1,...Q_{a+b+1}$, such that $Q_i$ wins $Q_{i+1}$ ($i=1,...,a+b$)
2015 ASDAN Math Tournament, 2
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, 4
Let $f(x)=(x-a)^3$. If the sum of all $x$ satisfying $f(x)=f(x-a)$ is $42$, find $a$.
2015 ASDAN Math Tournament, 12
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, 4
Given a positive integer $x>1$ with $n$ divisors, define $f(x)$ to be the product of the smallest $\lceil\tfrac{n}{2}\rceil$ divisors of $x$. Let $a$ be the least value of $x$ such that $f(x)$ is a multiple of $X$, and $b$ be the least value of $n$ such that $f(y)$ is a multiple of $y$ for some $y$ that has exactly $n$ factors. Compute $a+b$.
2015 ASDAN Math Tournament, 10
Let $\sigma(n)$ be the sum of all the positive divisors of $n$. Let $a$ be the smallest positive integer greater than or equal to $2015$ for which there exists some positive integer $n$ satisfying $\sigma(n)=a$. Finally, let $b$ be the largest such value of $n$. Compute $a+b$.
2015 ASDAN Math Tournament, 3
Let $a_1,a_2,a_3,\dots,a_6$ be an arithmetic sequence with common difference $3$. Suppose that $a_1$, $a_3$, and $a_6$ also form a geometric sequence. Compute $a_1$.
2015 Turkey Team Selection Test, 8
Let $ABC$ be a triangle with incenter $I$ and circumcenter $O$ such that $|AC|>|BC|>|AB|$ and the incircle touches the sides $BC, CA, AB$ at $D, E, F$ respectively. Let the reflection of $A$ with respect to $F$ and $E$ be $F_1$ and $E_1$ respectively. The circle tangent to $BC$ at $D$ and passing through $F_1$ intersects $AB$ a second time at $F_2$ and the circle tangent to $BC$ at $D$ and passing through $E_1$ intersects $AC$ a second time at $E_2$. The midpoints of the segments $|OE|$ and $|IF|$ are $P$ and $Q$ respectively. Prove that \[|AB| + |AC| = 2|BC| \iff PQ\perp E_2F_2 \].
2015 ASDAN Math Tournament, 6
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, 14
For a given positive integer $m$, the series
$$\sum_{k=1,k\neq m}^{\infty}\frac{1}{(k+m)(k-m)}$$
evaluates to $\frac{a}{bm^2}$, where $a$ and $b$ are positive integers. Compute $a+b$.
2015 ASDAN Math Tournament, 5
The eight corners of a cube are cut off, yielding a polyhedron with $6$ octagonal faces and $8$ triangular faces. Given that all polyhedron's edges have length $2$, compute the volume of the polyhedron.