Found problems: 84
2014 ASDAN Math Tournament, 24
It's pouring down rain, and the amount of rain hitting point $(x,y)$ is given by
$$f(x,y)=|x^3+2x^2y-5xy^2-6y^3|.$$
If you start at the origin $(0,0)$, find all the possibilities for $m$ such that $y=mx$ is a straight line along which you could walk without any rain falling on you.
2014 ASDAN Math Tournament, 8
Equilateral triangle $DEF$ is inscribed inside equilateral triangle $ABC$ such that $DE$ is perpendicular to $BC$. Let $x$ be the area of triangle $ABC$ and $y$ be the area of triangle $DEF$. Compute $\tfrac{x}{y}$.
2014 ASDAN Math Tournament, 1
Compute the remainder when $2^{30}$ is divided by $1000$.
2014 ASDAN Math Tournament, 13
Let $\alpha,\beta,\gamma$ be the three real roots of the polynomial $x^3-x^2-2x+1=0$. Find all possible values of $\tfrac{\alpha}{\beta}+\tfrac{\beta}{\gamma}+\tfrac{\gamma}{\alpha}$.
2014 ASDAN Math Tournament, 9
Find the sum of all real numbers $x$ such that $x^4-2x^3+3x^2-2x-2014=0$.
2014 ASDAN Math Tournament, 15
Antoine, Benoît, Claude, Didier, Étienne, and Françoise go to the cinéma together to see a movie. The six of them want to sit in a single row of six seats. But Antoine, Benoît, and Claude are mortal enemies and refuse to sit next to either of the other two. How many different arrangements are possible?
2014 ASDAN Math Tournament, 20
$ABCD$ is a parallelogram, and circle $S$ (with radius $2$) is inscribed insider $ABCD$ such that $S$ is tangent to all four line segments $AB$, $BC$, $CD$, and $DA$. One of the internal angles of the parallelogram is $60^\circ$. What is the maximum possible area of $ABCD$?
2014 ASDAN Math Tournament, 3
Compute the perimeter of the triangle that has area $3-\sqrt{3}$ and angles $45^\circ$, $60^\circ$, and $75^\circ$.
2014 ASDAN Math Tournament, 10
Find the area of the smallest possible square that contains the points $(2,-1)$ and $(4,4)$.
2014 ASDAN Math Tournament, 1
Points $A$, $B$, $C$, and $D$ lie in the plane with $AB=AD=7$, $CB=CD=4$, and $BD=6$. Compute the sum of all possible values of $AC$.
2014 ASDAN Math Tournament, 17
Given that the line $y=mx+k$ intersects the parabola $y=ax^2+bx+c$ at two points, compute the product of the two $x$-coordinates of these points in terms of $a$, $b$, $c$, $k$, and $m$.
2014 ASDAN Math Tournament, 9
Compute how many permutations of the numbers $1,2,\dots,8$ have no adjacent numbers that sum to $9$.
2014 ASDAN Math Tournament, 12
Find the last two digits of $\tbinom{200}{100}$. Express the answer as an integer between $0$ and $99$. (e.g. if the last two digits are $05$, just write $5$.)
2014 ASDAN Math Tournament, 4
A frog is hopping from $(0,0)$ to $(8,8)$. The frog can hop from $(x,y)$ to either $(x+1,y)$ or $(x,y+1)$. The frog is only allowed to hop to point $(x,y)$ if $|y-x|\leq1$. Compute the number of distinct valid paths the frog can take.
2014 ASDAN Math Tournament, 1
Alex gets $8$ points on an exam, while his friend gets $3$ times as many points as Alex. What is the average of their scores?
2014 ASDAN Math Tournament, 4
Let $f(x)=\sum_{i=1}^{2014}|x-i|$. Compute the length of the longest interval $[a,b]$ such that $f(x)$ is constant on that interval.
2014 ASDAN Math Tournament, 14
Consider a round table on which $2014$ people are seated. Suppose that the person at the head of the table receives a giant plate containing all the food for supper. He then serves himself and passes the plate either right or left with equal probability. Each person, upon receiving the plate, will serve himself if necessary and similarly pass the plate either left or right with equal probability. Compute the probability that you are served last if you are seated $2$ seats away from the person at the head of the table.
2014 ASDAN Math Tournament, 6
Consider a circle of radius $4$ with center $O_1$, a circle of radius $2$ with center $O_2$ that lies on the circumference of circle $O_1$, and a circle of radius $1$ with center $O_3$ that lies on the circumference of circle $O_2$. The centers of the circle are collinear in the order $O_1$, $O_2$, $O_3$. Let $A$ be a point of intersection of circles $O_1$ and $O_2$ and $B$ be a point of intersection of circles $O_2$ and $O_3$ such that $A$ and $B$ lie on the same semicircle of $O_2$. Compute the length of $AB$.
2014 ASDAN Math Tournament, 4
Consider a square $ABCD$ with side length $4$ and label the midpoint of side $BC$ as $M$. Let $X$ be the point along $AM$ obtained by dropping a perpendicular from $D$ onto $AM$. Compute the product of the lengths $XC$ and $MD$.
2014 ASDAN Math Tournament, 9
We have squares $ABCD$ and $EFGH$. Square $ABCD$ has points with coordinates $A=(1,1,-1)$, $B=(1,-1,-1)$, $C=(-1,-1,-1)$ and $D=(-1,1,-1)$. Square $EFGH$ has points with coordinates $E=(\sqrt{2},0,1)$, $F=(0,-\sqrt{2},1)$, $G=(-\sqrt{2},0,1)$, and $H=(0,\sqrt{2},1)$. Consider the solid formed by joining point $A$ to $H$ and $E$, point $B$ to $E$ and $F$, point $C$ to $F$ and $G$, and point $D$ to $G$ and $H$. Compute the volume of this solid.
2014 ASDAN Math Tournament, 3
Compute
$$\sin\left(\frac{\pi}{9}\right)\sin\left(\frac{2\pi}{9}\right)\sin\left(\frac{4\pi}{9}\right).$$
2014 ASDAN Math Tournament, 15
A point is "bouncing" inside a unit equilateral triangle with vertices $(0,0)$, $(1,0)$, and $(1/2,\sqrt{3}/2)$. The point moves in straight lines inside the triangle and bounces elastically off an edge at an angle equal to the angle of incidence. Suppose that the point starts at the origin and begins motion in the direction of $(1,1)$. After the ball has traveled a cumulative distance of $30\sqrt{2}$, compute its distance from the origin.
2014 ASDAN Math Tournament, 5
Compute the smallest $9$-digit number containing all the digits $1$ to $9$ that is divisible by $99$.
2014 ASDAN Math Tournament, 2
Compute the number of integers between $1$ and $100$, inclusive, that have an odd number of factors. Note that $1$ and $4$ are the first two such numbers.
2014 ASDAN Math Tournament, 9
A sequence $\{a_n\}_{n\geq0}$ obeys the recurrence $a_n=1+a_{n-1}+\alpha a_{n-2}$ for all $n\geq2$ and for some $\alpha>0$. Given that $a_0=1$ and $a_1=2$, compute the value of $\alpha$ for which
$$\sum_{n=0}^{\infty}\frac{a_n}{2^n}=10$$