Found problems: 190
2016 ASDAN Math Tournament, 1
Compute
$$\lim_{x\rightarrow1}\frac{x^3-1}{x-1}.$$
2016 CMIMC, 9
1007 distinct potatoes are chosen independently and randomly from a box of 2016 potatoes numbered $1, 2, \dots, 2016$, with $p$ being the smallest chosen potato. Then, potatoes are drawn one at a time from the remaining 1009 until the first one with value $q < p$ is drawn. If no such $q$ exists, let $S = 1$. Otherwise, let $S = pq$. Then given that the expected value of $S$ can be expressed as simplified fraction $\tfrac{m}{n}$, find $m+n$.
2016 ASDAN Math Tournament, 7
Let $x$, $y$, and $z$ be real numbers satisfying the equations
\begin{align*}
4x+2yz-6z+9xz^2&=4\\
xyz&=1.
\end{align*}
Find all possible values of $x+y+z$.
2016 CMIMC, 9
For how many permutations $\pi$ of $\{1,2,\ldots,9\}$ does there exist an integer $N$ such that \[N\equiv \pi(i)\pmod{i}\text{ for all integers }1\leq i\leq 9?\]
2016 NIMO Summer Contest, 11
A set $S$ of positive integers is $\textit{sum-complete}$ if there are positive integers $m$ and $n$ such that an integer $a$ is the sum of the elements of some nonempty subset of $S$ if and only if $m \le a \le n$.
Let $S$ be a sum-complete set such that $\{1, 3\} \subset S$ and $|S| = 8$. Find the greatest possible value of the sum of the elements of $S$.
[i]Proposed by Michael Tang[/i]
2016 ASDAN Math Tournament, 7
Compute
$$\int_0^{\frac{\pi}{2}}\frac{e^x(\sin x+\cos x-2)}{(\cos x-2)^2}dx.$$
2016 CMIMC, 1
Let \[f(x)=\dfrac{1}{1-\dfrac{1}{1-x}}\,.\] Compute $f^{2016}(2016)$, where $f$ is composed upon itself $2016$ times.
2016 ASDAN Math Tournament, 4
The radius $r$ of a circle is increasing at a rate of $2$ meters per minute. Find the rate of change, in $\text{meters}^2/\text{minute}$, of the area when $r$ is $6$ meters.
2016 CMIMC, 8
Consider the sequence of sets defined by $S_0=\{0,1\},S_1=\{0,1,2\}$, and for $n\ge2$, \[S_n=S_{n-1}\cup\{2^n+x\mid x\in S_{n-2}\}.\] For example, $S_2=\{0,1,2\}\cup\{2^2+0,2^2+1\}=\{0,1,2,4,5\}$. Find the $200$th smallest element of $S_{2016}$.
2016 ASDAN Math Tournament, 12
Find the number of real solutions $x$, in radians, to
$$\sin(x)=\frac{x}{1000}.$$
2016 CMIMC, 3
Let $\{x\}$ denote the fractional part of $x$. For example, $\{5.5\}=0.5$. Find the smallest prime $p$ such that the inequality \[\sum_{n=1}^{p^2}\left\{\dfrac{n^p}{p^2}\right\}>2016\] holds.
2016 CMIMC, 8
Given that
\[
\sum_{x=1}^{70} \sum_{y=1}^{70} \frac{x^{y}}{y} =
\frac{m}{67!}
\] for some positive integer $m$, find $m \pmod{71}$.
2016 CMIMC, 1
For all integers $n\geq 2$, let $f(n)$ denote the largest positive integer $m$ such that $\sqrt[m]{n}$ is an integer. Evaluate \[f(2)+f(3)+\cdots+f(100).\]
2016 CMIMC, 3
How many pairs of integers $(a,b)$ are there such that $0\leq a < b \leq 100$ and such that $\tfrac{2^b-2^a}{2016}$ is an integer?
2016 ASDAN Math Tournament, 2
Consider the curves with equations $x^n+y^n=1$ for $n=2,4,6,8,\dots$. Denote $L_{2k}$ the length of the curve with $n=2k$. Find $\lim_{k\rightarrow\infty}L_{2k}$.
2016 ASDAN Math Tournament, 15
Circles $\omega_1$ and $\omega_2$ have radii $r_1<r_2$ respectively and intersect at distinct points $X$ and $Y$. The common external tangents intersect at point $Z$. The common tangent closer to $X$ touches $\omega_1$ and $\omega_2$ at $P$ and $Q$ respectively. Line $ZX$ intersects $\omega_1$ and $\omega_2$ again at points $R$ and $S$ and lines $RP$ and $SQ$ intersect again at point $T$. If $XT=8$, $XZ=15$, and $XY=12$, then what is $\tfrac{r_1}{r_2}$?
2016 ASDAN Math Tournament, 7
The side lengths of triangle $ABC$ are $13$, $14$, and $15$. Let $I$ be the incenter of the triangle. Compute the product $AI\cdot BI\cdot CI$.
2016 CMIMC, 3
Let $\varepsilon$ denote the empty string. Given a pair of strings $(A,B)\in\{0,1,2\}^*\times\{0,1\}^*$, we are allowed the following operations:
\[\begin{cases}
(A,1)\to(A0,\varepsilon)\\
(A,10)\to(A00,\varepsilon)\\
(A,0B)\to(A0,B)\\
(A,11B)\to(A01,B)\\
(A,100B)\to(A0012,1B)\\
(A,101B)\to(A00122,10B)
\end{cases}\]
We perform these operations on $(A,B)$ until we can no longer perform any of them. We then iteratively delete any instance of $20$ in $A$ and replace any instance of $21$ with $1$ until there are no such substrings remaining. Among all binary strings $X$ of size $9$, how many different possible outcomes are there for this process performed on $(\varepsilon,X)$?
2016 CMIMC, 6
Shen, Ling, and Ru each place four slips of paper with their name on it into a bucket. They then play the following game: slips are removed one at a time, and whoever has all of their slips removed first wins. Shen cheats, however, and adds an extra slip of paper into the bucket, and will win when four of his are drawn. Given that the probability that Shen wins can be expressed as simplified fraction $\tfrac{m}{n}$, compute $m+n$.
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 CMIMC, 2
Let $a_1$, $a_2$, $\ldots$ be an infinite sequence of (positive) integers such that $k$ divides $\gcd(a_{k-1},a_k)$ for all $k\geq 2$. Compute the smallest possible value of $a_1+a_2+\cdots+a_{10}$.
2016 CMIMC, 1
Point $A$ lies on the circumference of a circle $\Omega$ with radius $78$. Point $B$ is placed such that $AB$ is tangent to the circle and $AB=65$, while point $C$ is located on $\Omega$ such that $BC=25$. Compute the length of $\overline{AC}$.
2016 ASDAN Math Tournament, 5
Let $f(x)$ be a real valued function. Recall that if the inverse function $f^{-1}(x)$ exists, then $f^{-1}(x)$ satisfies $f(f^{-1}(x))=f^{-1}(f(x))=x$. Given that the inverse of the function $f(x)=x^3-12x^2+48x-60$ exists, find all real $a$ that satisfy $f(a)=f^{-1}(a)$.
2016 CMIMC, 4
Andrew the Antelope canters along the surface of a regular icosahedron, which has twenty equilateral triangle faces and edge length 4. If he wants to move from one vertex to the opposite vertex, the minimum distance he must travel can be expressed as $\sqrt{n}$ for some integer $n$. Compute $n$.
2016 CMIMC, 9
Let $\triangle ABC$ be a triangle with $AB=65$, $BC=70$, and $CA=75$. A semicircle $\Gamma$ with diameter $\overline{BC}$ is constructed outside the triangle. Suppose there exists a circle $\omega$ tangent to $AB$ and $AC$ and furthermore internally tangent to $\Gamma$ at a point $X$. The length $AX$ can be written in the form $m\sqrt{n}$ where $m$ and $n$ are positive integers with $n$ not divisible by the square of any prime. Find $m+n$.