Found problems: 190
2016 CMIMC, 5
Let $\mathcal{S}$ be a regular 18-gon, and for two vertices in $\mathcal{S}$ define the $\textit{distance}$ between them to be the length of the shortest path along the edges of $\mathcal{S}$ between them (e.g. adjacent vertices have distance 1). Find the number of ways to choose three distinct vertices from $\mathcal{S}$ such that no two of them have distance 1, 8, or 9.
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 $\geq x$.
2016 ASDAN Math Tournament, 3
Moor has $2016$ white rabbit candies. He and his $n$ friends split the candies equally amongst themselves, and they find that they each have an integer number of candies. Given that $n$ is a positive integer (Moor has at least $1$ friend), how many possible values of $n$ exist?
2016 ASDAN Math Tournament, 13
Ash writes the positive integers from $1$ to $2016$ inclusive as a single positive integer $n=1234567891011\dots2016$. What is the result obtained by successively adding and subtracting the digits of $n$? (In other words, compute $1-2+3-4+5-6+7-8+9-1+0-1+\dots$.)
2016 NIMO Summer Contest, 12
Let $p$ be a prime. It is given that there exists a unique nonconstant function $\chi:\{1,2,\ldots, p-1\}\to\{-1,1\}$ such that $\chi(1) = 1$ and $\chi(mn) = \chi(m)\chi(n)$ for all $m, n \not\equiv 0 \pmod p$ (here the product $mn$ is taken mod $p$). For how many positive primes $p$ less than $100$ is it true that \[\sum_{a=1}^{p-1}a^{\chi(a)}\equiv 0\pmod p?\] Here as usual $a^{-1}$ denotes multiplicative inverse.
[i]Proposed by David Altizio[/i]
2016 ASDAN Math Tournament, 12
Let
$$f(x)=\frac{2016^x}{2016^x+\sqrt{2016}}.$$
Evaluate
$$\sum_{k=0}^{2016}f\left(\frac{k}{2016}\right).$$
2016 Macedonia JBMO TST, 4
Let $x$, $y$, and $z$ be positive real numbers. Prove that
$\sqrt {\frac {xy}{x^2 + y^2 + 2z^2}} + \sqrt {\frac {yz}{y^2 + z^2 + 2x^2}}+\sqrt {\frac {zx}{z^2 + x^2 + 2y^2}} \le \frac{3}{2}$.
When does equality hold?
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 $3$?
2016 CMIMC, 2
The $\emph{Stooge sort}$ is a particularly inefficient recursive sorting algorithm defined as follows: given an array $A$ of size $n$, we swap the first and last elements if they are out of order; we then (if $n\ge3$) Stooge sort the first $\lceil\tfrac{2n}3\rceil$ elements, then the last $\lceil\tfrac{2n}3\rceil$, then the first $\lceil\tfrac{2n}3\rceil$ elements again. Given that this runs in $O(n^\alpha)$, where $\alpha$ is minimal, find the value of $(243/32)^\alpha$.
2016 ASDAN Math Tournament, 9
Compute
$$\int_0^\infty\frac{\ln\left(\frac{1+x^{11}}{1+x^3}\right)}{(1+x^2)\ln x}dx.$$
2016 ASDAN Math Tournament, 9
A cyclic quadrilateral $ABCD$ has side lengths $AB=14$, $BC=19$, $CD=26$, and $DA=29$. Compute the sine of the smaller angle between diagonals $AC$ and $BD$.
2016 ASDAN Math Tournament, 6
For what positive value $k$ does the equation $\ln x=kx^2$ have exactly one solution?
2016 ASDAN Math Tournament, 1
If $x=14$ and $y=6$, then compute $\tfrac{x^2-y^2}{x-y}$.
2016 Macedonia JBMO TST, 5
Solve the following equation in the set of positive integers
$x + y^2 + (GCD(x, y))^2 = xy \cdot GCD(x, y)$.
2017 ASDAN Math Tournament, 6
If $x+y^{-99}=3$ and $x+y=-2$, find the sum of all possible values of $x$.
2016 CMIMC, 5
The parabolas $y=x^2+15x+32$ and $x = y^2+49y+593$ meet at one point $(x_0,y_0)$. Find $x_0+y_0$.
2016 CMIMC, 9
Let $\lfloor x\rfloor$ denote the greatest integer function and $\{x\}=x-\lfloor x\rfloor$ denote the fractional part of $x$. Let $1\leq x_1<\ldots<x_{100}$ be the $100$ smallest values of $x\geq 1$ such that $\sqrt{\lfloor x\rfloor\lfloor x^3\rfloor}+\sqrt{\{x\}\{x^3\}}=x^2.$ Compute \[\sum_{k=1}^{50}\dfrac{1}{x_{2k}^2-x_{2k-1}^2}.\]
2016 CMIMC, 9
Compute the number of positive integers $n \leq 50$ such that there exist distinct positive integers $a,b$ satisfying
\[
\frac{a}{b} +\frac{b}{a} = n \left(\frac{1}{a} + \frac{1}{b}\right).
\]
2016 CMIMC, 9
Ryan has three distinct eggs, one of which is made of rubber and thus cannot break; unfortunately, he doesn't know which egg is the rubber one. Further, in some 100-story building there exists a floor such that all normal eggs dropped from below that floor will not break, while those dropped from at or above that floor will break and cannot be dropped again. What is the minimum number of times Ryan must drop an egg to determine the floor satisfying this property?
2016 ASDAN Math Tournament, 2
A pet shop sells cats and two types of birds: ducks and parrots. In the shop, $\tfrac{1}{12}$ of animals are ducks, and $\tfrac{1}{4}$ of birds are ducks. Given that there are $56$ cats in the pet shop, how many ducks are there in the pet shop?
2016 NIMO Summer Contest, 14
Find the smallest positive integer $n$ such that $n^2+4$ has at least four distinct prime factors.
[i]Proposed by Michael Tang[/i]
2016 ASDAN Math Tournament, 7
What is
$$\sum_{n=1996}^{2016}\lfloor\sqrt{n}\rfloor?$$
2016 CMIMC, 10
Let $f:\mathbb{N}\mapsto\mathbb{R}$ be the function \[f(n)=\sum_{k=1}^\infty\dfrac{1}{\operatorname{lcm}(k,n)^2}.\] It is well-known that $f(1)=\tfrac{\pi^2}6$. What is the smallest positive integer $m$ such that $m\cdot f(10)$ is the square of a rational multiple of $\pi$?
2016 CMIMC, 6
Suppose integers $a < b < c$ satisfy \[ a + b + c = 95\qquad\text{and}\qquad a^2 + b^2 + c^2 = 3083.\] Find $c$.