Found problems: 89
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.
2021 CMIMC Integration Bee, 3
$$\int_{0}^{\frac{\pi}{2}}\sin^2(x)\sin(2x)\,dx$$
[i]Proposed by Connor Gordon[/i]
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 $(243/32)^\alpha$.
2021 CMIMC Integration Bee, 15
$$\int_{-\infty}^\infty \frac{\sin(\pi x)}{x(1+x^2)}\,dx$$
[i]Proposed by Vlad Oleksenko[/i]
2021 CMIMC Integration Bee, 7
$$\int_0^\infty \frac{1}{(x^2+4)^{5/2}}\,dx$$
[i]Proposed by Connor Gordon[/i]
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 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 $c$.
2016 CMIMC, 3
Let $ABC$ be a triangle. The angle bisector of $\angle B$ intersects $AC$ at point $P$, while the angle bisector of $\angle C$ intersects $AB$ at a point $Q$. Suppose the area of $\triangle ABP$ is 27, the area of $\triangle ACQ$ is 32, and the area of $\triangle ABC$ is $72$. The length of $\overline{BC}$ can be written in the form $m\sqrt n$ where $m$ and $n$ are positive integers with $n$ as small as possible. What is $m+n$?
2016 CMIMC, 2
Determine the value of the sum \[\left|\sum_{1\leq i<j\leq 50}ij(-1)^{i+j}\right|.\]
2016 CMIMC, 4
Kevin colors three distinct squares in a $3\times 3$ grid red. Given that there exist two uncolored squares such that coloring one of them would create a horizontal or vertical red line, find the number of ways he could have colored the original three squares.
2016 CMIMC, 1
David, when submitting a problem for CMIMC, wrote his answer as $100\tfrac xy$, where $x$ and $y$ are two positive integers with $x<y$. Andrew interpreted the expression as a product of two rational numbers, while Patrick interpreted the answer as a mixed fraction. In this case, Patrick's number was exactly double Andrew's! What is the smallest possible value of $x+y$?
2016 CMIMC, 2
Suppose that some real number $x$ satisfies
\[\log_2 x + \log_8 x + \log_{64} x = \log_x 2 + \log_x 16 + \log_x 128.\] Given that the value of $\log_2 x + \log_x 2$ can be expressed as $\tfrac{a\sqrt{b}}{c}$, where $a$ and $c$ are coprime positive integers and $b$ is squarefree, compute $abc$.
2016 CMIMC, 3
Let $S$ be the set containing all positive integers whose decimal representations contain only 3’s and 7’s, have at most 1998 digits, and have at least one digit appear exactly 999 times. If $N$ denotes the number of elements in $S$, find the remainder when $N$ is divided by 1000.
2016 CMIMC, 10
Given $x_0\in\mathbb R$, $f,g:\mathbb R\to\mathbb R$, we define the $\emph{non-redundant binary tree}$ $T(x_0,f,g)$ in the following way:
[list=1]
[*]The tree $T$ initially consists of just $x_0$ at height $0$.
[*]Let $v_0,\dots,v_k$ be the vertices at height $h$. Then the vertices of height $h+1$ are added to $T$ by: for $i=0,1,\dots,k$, $f(v_i)$ is added as a child of $v_i$ if $f(v_i)\not\in T$, and $g(v_i)$ is added as a child of $v_i$ if $g(v_i)\not\in T$.
[/list]
For example, if $f(x)=x+1$ and $g(x)=x-1$, then the first three layers of $T(0,f,g)$ look like:
[asy]
size(100);
draw((-0.1,-0.2)--(-0.4,-0.8),EndArrow(size=3));
draw((0.1,-0.2)--(0.4,-0.8),EndArrow(size=3));
draw((-0.6,-1.2)--(-0.9,-1.8),EndArrow(size=3));
draw((0.6,-1.2)--(0.9,-1.8),EndArrow(size=3));
label("$0$",(0,0));
label("$1$",(-.5,-1));
label("$-1$",(.5,-1));
label("$2$",(-1,-2));
label("$-2$",(1,-2));[/asy]
If $f(x)=1024x-2047\lfloor x/2\rfloor$ and $g(x)=2x-3\lfloor x/2\rfloor+2\lfloor x/4\rfloor$, then how many vertices are in $T(2016,f,g)$?
2016 CMIMC, 10
Denote by $F_0(x)$, $F_1(x)$, $\ldots$ the sequence of Fibonacci polynomials, which satisfy the recurrence $F_0(x)=1$, $F_1(x)=x$, and $F_n(x)=xF_{n-1}(x)+F_{n-2}(x)$ for all $n\geq 2$. It is given that there exist unique integers $\lambda_0$, $\lambda_1$, $\ldots$, $\lambda_{1000}$ such that \[x^{1000}=\sum_{i=0}^{1000}\lambda_iF_i(x)\] for all real $x$. For which integer $k$ is $|\lambda_k|$ maximized?
2016 CMIMC, 5
Determine the sum of the positive integers $n$ such that there exist primes $p,q,r$ satisfying $p^{n} + q^{2} = r^{2}$.
2016 CMIMC, 2
Right isosceles triangle $T$ is placed in the first quadrant of the coordinate plane. Suppose that the projection of $T$ onto the $x$-axis has length $6$, while the projection of $T$ onto the $y$-axis has length $8$. What is the sum of all possible areas of the triangle $T$?
[asy]
import olympiad;
size(120);
defaultpen(linewidth(0.8));
pair A = (0.9,0.6), B = (1.7, 0.8), C = rotate(270, B)*A;
pair PAx = (A.x,0), PBx = (B.x,0), PAy = (0, A.y), PCy = (0, C.y);
draw(PAx--A--PAy^^PCy--C^^PBx--B, linetype("4 4"));
draw(rightanglemark(A,B,C,3));
draw(A--B--C--cycle);
draw((0,2)--(0,0)--(2,0),Arrows(size=8));
label("$6$",(PAx+PBx)/2,S);
label("$8$",(PAy+PCy)/2,W);
[/asy]
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 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 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 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 $S_{2016}$.