Found problems: 36
2015 ASDAN Math Tournament, 15
Let $ABCD$ be a regular tetrahedron of side length $12$. Select points $E,F,G,H$ on $AC,BC,BD,AD$, respectively, such that $AE=BF=BG=AH=3$. Compute the area of quadrilateral $EFGH$.
2015 ASDAN Math Tournament, 34
Compute the number of natural numbers $1\leq n\leq10^6$ such that the least prime divisor of $n$ is $17$. Your score will be given by $\lfloor26\min\{(\tfrac{A}{C})^2,(\tfrac{C}{A})^2\}\rfloor$, where $A$ is your answer and $C$ is the actual answer.
2015 ASDAN Math Tournament, 23
Two regular hexagons of side length $2$ are laid on top of each other such that they share the same center point and one hexagon is rotated $30^\circ$ about the center from the other. Compute the area of the union of the two hexagons.
2015 ASDAN Math Tournament, 24
Trains $A$ and $B$ are on the same track a distance $100$ miles apart heading towards one another, each at a speed of $50$ miles per hour. A fly starting out at the front of train $A$ flies towards train $B$ at a speed of $75$ miles per hour. Upon reaching train $B$, the fly turns around and flies towards train $A$, again at $75$ miles per hour. The fly continues flying back and forth between the two trains at $75$ miles per hour until the two trains hit each other. How many minutes does the fly spend closer to train $A$ than to train $B$ before getting squashed?
2015 ASDAN Math Tournament, 13
A three-digit number $x$ in base $10$ has a units-digit of $6$. When $x$ is written is base $9$, the second digit of the number is $4$, and the first and third digit are equal in value. Compute $x$ in base $10$.
2015 ASDAN Math Tournament, 9
Compute the sum of the digits of $101^6$.
2015 ASDAN Math Tournament, 29
Suppose that the following equations hold for positive integers $x$, $y$, and $n$, where $n>18$:
\begin{align*}
x+3y&\equiv7\pmod{n}\\
2x+2y&\equiv18\pmod{n}\\
3x+y&\equiv7\pmod{n}
\end{align*}
Compute the smallest nonnegative integer $a$ such that $2x\equiv a\pmod{n}$.
2015 ASDAN Math Tournament, 14
A standard deck of $52$ cards is shuffled and randomly arranged in a queue, with each card having a suit $(\diamondsuit,\clubsuit,\heartsuit,\spadesuit)$ and a rank $(\text{Ace},2,3,4,5,6,7,8,9,10,\text{Jack},\text{Queen},\text{ King})$. For example, a card with the $\diamondsuit$ suit and the $7$ rank would be denoted as $\diamondsuit7$, and a card with the $\spadesuit$ and the $\text{Ace}$ rank would be denoted as $\spadesuit\text{Ace}$. In the queue, there exists a card with a rank of $\text{Ace}$ that appears for the first time in the queue. Let the card immediately following the above card be denoted as card $C$. Is the probability that $C$ is a $\spadesuit\text{A}$ higher than, equal to, or lower than the probability that $C$ is a $\clubsuit2$?
2015 ASDAN Math Tournament, 36
A blue square of side length $10$ is laid on top of a coordinate grid with corners at $(0,0)$, $(0,10)$, $(10,0)$, and $(10,10)$. Red squares of side length $2$ are randomly placed on top of the grid, changing the color of a $2\times2$ square section red. Each red square when placed lies completely within the blue square, and each square's four corners take on integral coordinates. In addition, randomly placed red squares may overlap, keeping overlapped regions red. Compute the expected value of the number of red squares necessary to turn the entire blue square red, rounded to the nearest integer. Your score will be given by $\lfloor25\min\{(\tfrac{A}{C})^2,(\tfrac{C}{A})^2\}\rfloor$, where $A$ is your answer and $C$ is the actual answer.
2015 ASDAN Math Tournament, 27
In triangle $ABC$, $D$ is a point on $AB$ between $A$ and $B$, $E$ is a point on $AC$ between $A$ and $C$, and $F$ is a point on $BC$ between $B$ and $C$ such that $AF$, $BE$, and $CD$ all meet inside $\triangle ABC$ at a point $G$. Given that the area of $\triangle ABC$ is $15$, the area of $\triangle ABE$ is $5$, and the area of $\triangle ACD$ is $10$, compute the area of $\triangle ABF$.
2015 ASDAN Math Tournament, 35
Let $S$ be the set of positive integers less than $10^6$ that can be written as the sum of two perfect squares. Compute the number of elements in $S$. Your score will be given by $\max\{\lfloor75(\min\{(\tfrac{A}{C})^2,(\tfrac{C}{A})^2\}-\tfrac{2}{3})\rfloor,0\}$, where $A$ is your answer and $C$ is the actual answer.
2015 ASDAN Math Tournament, 19
Compute the number of $0\leq n\leq2015$ such that $6^n+8^n$ is divisible by $7$.
2015 ASDAN Math Tournament, 32
Let $ABC$ be a triangle with $AB=8$, $BC=7$, and $AC=11$. Let $\Gamma_1$ and $\Gamma_2$ be the two possible circles that are tangent to $AB$, $AC$, and $BC$ when $AC$ and $BC$ are extended, with $\Gamma_1$ having the smaller radius. $\Gamma_1$ and $\Gamma_2$ are tangent to $AB$ to $D$ and $E$, respectively, and $CE$ intersects the perpendicular bisector of $AB$ at a point $F$. What is $\tfrac{CF}{FD}$?
2015 ASDAN Math Tournament, 33
Compute the number of digits is $2015!$. Your score will be given by $\max\{\lfloor125(\min\{\tfrac{A}{C},\tfrac{C}{A}\}-\tfrac{1}{5})\rfloor,0\}$, where $A$ is your answer and $C$ is the actual answer.
2015 ASDAN Math Tournament, 11
In the following diagram, each circle has radius $6$ and each circle passes through the center of the other two circles. Compute the area of the white center region and express your answer in terms of $\pi$.
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2015 ASDAN Math Tournament, 25
Let $a_n$ be a sequence with $a_0=1$ and defined recursively by
$$a_{n+1}=\begin{cases}a_n+2&\text{if }n\text{ is even},\\2a_n&\text{if }n\text{ is odd.}\end{cases}$$
What are the last two digits of $a_{2015}$?
2015 ASDAN Math Tournament, 5
Compute the number of zeros at the end of $2015!$.
2015 ASDAN Math Tournament, 22
You flip a fair coin which results in heads ($\text{H}$) or tails ($\text{T}$) with equal probability. What is the probability that you see the consecutive sequence $\text{THH}$ before the sequence $\text{HHH}$?
2015 ASDAN Math Tournament, 4
Compute the number of positive integers less than or equal to $2015$ that are divisible by $5$ or $13$, but not both.
2015 ASDAN Math Tournament, 1
Rio likes fruit, and one day she decides to pick persimmons. She picks a total of $12$ persimmons from the first $5$ trees she sees. Rio has $5$ more trees to pick persimmons from. If she wants to pick an average of $4$ persimmons per tree overall, what is the average number of persimmons that she must pick from each of the last $5$ trees for her goal?
2015 ASDAN Math Tournament, 8
Lynnelle and Moor love toy cars, and together, they have $27$ red cars, $27$ purple cars, and $27$ green cars. The number of red cars Lynnelle has individually is the same as the number of green cars Moor has individually. In addition, Lynnelle has $17$ more cars of any color than Moor has of any color. How many purple cars does Lynnelle have?
2015 ASDAN Math Tournament, 18
Andrew takes a square sheet of paper $ABCD$ of side length $1$ and folds a kite shape. To do this, he takes the corners at $B$ and $D$ and folds the paper such that both corners now rest at a point $E$ on $AC$. This fold results in two creases $CF$ and $CG$, respectively, where $F$ lies on $AB$ and $G$ lies on $AD$. Compute the length of $FG$.
2015 ASDAN Math Tournament, 16
Find the maximum value of $c$ such that
\begin{align*}
1&=-cx+y\\
-7&=x^2+y^2+8y
\end{align*}
has a unique real solution $(x,y)$.
2015 ASDAN Math Tournament, 17
How many ways are there to write $91$ as the sum of at least $2$ consecutive positive integers?
2015 ASDAN Math Tournament, 7
The Yamaimo family is moving to a new house, so they’ve packed their belongings into boxes, which weigh $100\text{ kg}$ in total. Mr. Yamaimo realizes that $99\%$ of the weight of the boxes is due to books. Later, the family unpacks some of the books (and nothing else). Mr. Yamaimo notices that now only $95\%$ of the weight of the boxes is due to books. How much do the boxes weigh now in kilograms?