Found problems: 54
2016 ASDAN Math Tournament, 25
Find the best rational approximation $x$ to $\sqrt[3]{2016}$ such that $|x-\sqrt[3]{2016}|$ is as small as possible. You may either find an $x=\tfrac{a}{b}$ where $a,b$ are coprime integers or find a decimal approximation. Let $C$ be the actual answer and $A$ be the answer you submit. Your score will be given by $\lceil10+\tfrac{16.5}{0.1+e^{30|A-C|}}\rceil$, where $\lceil x\rceil$ denote the smallest integer which is $\geq x$.
2017 ASDAN Math Tournament, 21
In trapezoid $ABCD$, we have $\overline{AD}\parallel\overline{BC}$, $BC=3$, and $CD=4$. In addition, $\cos\angle ADC=\tfrac{1}{3}$ and $\angle ABC=2\angle ADC$. Compute $AC$.
2016 ASDAN Math Tournament, 5
In the following diagram, the square and pentagon are both regular and share segment $AB$ as designated. What is the measure of $\angle CBD$ in degrees?
2016 ASDAN Math Tournament, 24
Alex, Bill, and Charlie want to play a game of DotA. They each come online at a uniformly random time between $8:00$ and $8:05\text{ }\text{PM}$, and each person queues for $2$ minutes. However, if any of them sees any other of them online while queuing, they merge parties and restart the queue, again waiting for $2$ minutes starting from the merger time.
For example, suppose that Alex logs in at $8:00\text{ PM}$, Bill logs in at $8:01\text{ PM}$, and Charlie logs in at $8:02:30\text{ PM}$ ($30$ seconds past $8:02\text{ PM}$). At $8:01\text{ PM}$, Alex and Bill would merge parties and queue for $2$ minutes starting at $8:01\text{ PM}$. At $8:02:30\text{ PM}$, Charlie would merge with Alex and Bill’s party, since Alex and Bill have waited together for only $1.5$ minutes.
What is the probability that they will play as a party of $3$?
2017 ASDAN Math Tournament, 1
Alice and Bob are racing. Alice runs at a rate of $2\text{ m/s}$. Bob starts $10\text{ m}$ ahead of Alice and runs at a rate of $1.5\text{ m/s}$. How many seconds after the race starts will Alice pass Bob?
2017 ASDAN Math Tournament, 6
You roll three six-sided dice. If the three dice and indistinguishable, how many combinations of numbers can result?
2017 ASDAN Math Tournament, 3
Four mathematicians, four physicists, and four programmers gather in a classroom. The $12$ people organize themselves into four teams, with each team having one mathematician, one physicist, and one programmer. How many possible arrangements of teams can exist?
2017 ASDAN Math Tournament, 17
For $\triangle ABC$, $AB=BC=5$, and $AC=6$. Circle $O$ is inscribed in $\triangle ABC$, and circle $P$ is tangent to circle $O$, $AB$, and $AC$. Compute the area of $\triangle ABC$ not covered by circles $O$ and $P$.
2016 ASDAN Math Tournament, 7
What is
$$\sum_{n=1996}^{2016}\lfloor\sqrt{n}\rfloor?$$
2016 ASDAN Math Tournament, 11
Ebeneezer is painting the edges of a cube. He wants to paint the edges so that the colored edges form a loop that does not intersect itself. For example, the loop should not look like a “figure eight” shape. If two colorings are considered equivalent if there is a rotation of the cubes so that the colored edges are the same, what is the number of possible edge colorings?
2016 ASDAN Math Tournament, 18
Compute the number of nonnegative integer triples $(x,y,z)$ which satisfy $4x+2y+z\leq36$.
2016 ASDAN Math Tournament, 23
Find all quadruples of real numbers $(a,b,c,d)$ that satisfy the system of equations:
\begin{align*}
a+4b+8c+4d&=53\\
3a^2+4b^2+12c^2+2d^2&=159\\
9a^3+4b^3+18c^3+d^3&=477.
\end{align*}
2016 ASDAN Math Tournament, 9
An equilateral triangle $\triangle ABC$ with side length $3$ has center $O$. A circle is drawn centered at $O$ with radius $1$. Find the area of the region contained inside both the triangle and circle.
2017 ASDAN Math Tournament, 26
A lattice point is a coordinate pair $(a,b)$ where both $a,b$ are integers. What is the number of lattice points $(x,y)$ that satisfy $\tfrac{x^2}{2017}+\tfrac{2y^2}{2017}<1$ and $y\equiv2x\pmod{7}$?
Let $C$ be the actual answer, $A$ be the answer you submit, and $D=|A-C|$. Your score will be rounded up from $\max(0,25-e^{D/100})$.
2016 ASDAN Math Tournament, 16
Let the notation $\underline{ABC}$ denote the number compromised of the digits $A$, $B$, and $C$ with $0\leq A,B,C\leq9$. That is, $\underline{ABC}=100A+10B+C$ and $\underline{CCAAC}=10000C+1000C+100A+10A+C$. Now, if $(\underline{ABC})^2=\underline{CCAAC}$, where $A$, $B$, and $C$ are distinct nonzero digits, find the $3$ digit number $\underline{ABC}$.
2017 ASDAN Math Tournament, 2
Let $5$ and $13$ be lengths of two sides of a right triangle. Compute the sum of all possible lengths of the third side.
2017 ASDAN Math Tournament, 27
How many primes between $2$ and $2^{30}$ are $1$ more than a multiple of $2017$? If $C$ is the correct answer and $A$ is your answer, then your score will be rounded up from $\max(0,25-15|\ln\tfrac{A}{C}|)$.
2017 ASDAN Math Tournament, 23
Ben creates an $8\times8$ grid of coins, where each coin faces heads with probability $\tfrac{1}{2}$, and tails with probability $\tfrac{1}{2}$. Ben then makes a series of moves; each move consists of selecting a coin in the grid and flipping over all coins in the same row and column as the selected coin. Suppose that in Ben’s current grid of coins, it is possible to make a series of moves so that all coins in the grid are heads, and that Ben will make the fewest number of moves to do so. What is the expected number of moves that Ben makes?
2016 ASDAN Math Tournament, 12
Find the number of real solutions $x$, in radians, to
$$\sin(x)=\frac{x}{1000}.$$
2017 ASDAN Math Tournament, 24
Consider all rational numbers of the form $\tfrac{p}{q}$ where $p,q$ are relatively prime positive integers less than or equal to $8$, and plot them on the $xy$-plane, where $\tfrac{p}{q}$ corresponds to point $(p,q)$. Arrange the rationals in increasing order $\{P_1,P_2,\dots,P_n\}$ and form a polygon by connecting points $P_i$ and $P_{i+1}$ for $1\le i<n$ and connecting both $P_1$ and $P_n$ to the origin. What is the area of the polygon?
2016 ASDAN Math Tournament, 19
Let $z\neq0$ be a complex number satisfying $z^2=z+i|z|$. ($|z|$ denotes the length between the origin and $z$ in the complex plane.) Find $z\cdot\overline{z}$, where $\overline{z}=a-bi$ is the complex conjugate of $z=a+bi$.
2016 ASDAN Math Tournament, 6
Suppose we have $3$ baskets and $4$ distinguishable balls. Each ball is placed into a randomly selected basket. Compute the probability that the basket with the most balls has at least $3$ balls.
2017 ASDAN Math Tournament, 7
Point $C$ is chosen on the arc of a semicircle with diameter $AB$. The two circles with diameters of $AC$ and $BC$ intersect again at point $D$. If $DA=20$ and $DB=16$, compute the length of $DC$.
2016 ASDAN Math Tournament, 13
Suppose $\{a_n\}_{n=1}^\infty$ is a sequence. The partial sums $\{s_n\}_{n=1}^\infty$ are defined by
$$s_n=\sum_{i=1}^na_i.$$
The Cesàro sums are then defined as $\{A_n\}_{n=1}^\infty$, where
$$A_n=\frac{1}{n}\cdot\sum_{i=1}^ns_i.$$
Let $a_n=(-1)^{n+1}$. What is the limit of the Cesàro sums of $\{a_n\}_{n=1}^\infty$ as $n$ goes to infinity?
2016 ASDAN Math Tournament, 26
The Euclidean Algorithm on inputs $a$ and $b$ is a way to find the greatest common divisor $\gcd(a,b)$. Suppose WLOG that $a>b$. On each step of the Euclidan Algorithm, we solve the equation $a=bq+r$ for integers $q,r$ such that $0\leq r<b$, and repeat on $b$ and $r$. Thus $\gcd(a,b)=\gcd(b,r)$, and we repeat. If $r=0$, we are done. For example, $\gcd(100,15)=\gcd(15,10)=\gcd(10,5)=5$, because $100=15\cdot6+10$, $15=10\cdot1+5$, and $10=5\cdot2+0$. Thus, the Euclidean Algorithm here takes $3$ steps. What is the largest number of steps that the Euclidean Algorithm can take on some integer inputs $a,b$ where $0<a,b<10^{2016}$?
Let $C$ be the actual answer and $A$ be the answer you submit. If $\tfrac{|A-C|}{C}>\tfrac{1}{2}$, then your score will be $0$. Otherwise, your score will be given by $\max\{0,\lceil25-2(\tfrac{|A-C|}{20})^{1/2.2}\rceil\}$.