Olympiad problems

2016 CMIMC, 1

Construction Mayhem University has been on a mission to expand and improve its campus! The university has recently adopted a new construction schedule where a new project begins every two days. Each project will take exactly one more day than the previous one to complete (so the first project takes 3, the second takes 4, and so on.) Suppose the new schedule starts on Day 1. On which day will there first be at least $10$ projects in place at the same time?

2016 CMIMC, 1

Tags: 2016 , CMIMC , computer science , Tiebreaker

A $\emph{planar}$ graph is a connected graph that can be drawn on a sphere without edge crossings. Such a drawing will divide the sphere into a number of faces. Let $G$ be a planar graph with $11$ vertices of degree $2$, $5$ vertices of degree $3$, and $1$ vertex of degree $7$. Find the number of faces into which $G$ divides the sphere.

2016 CMIMC, 3

Tags: 2016 , CMIMC , algebra , Tiebreaker , system of equations , YUH.

Suppose $x$ and $y$ are real numbers which satisfy the system of equations \[x^2-3y^2=\frac{17}x\qquad\text{and}\qquad 3x^2-y^2=\frac{23}y.\] Then $x^2+y^2$ can be written in the form $\sqrt[m]{n}$, where $m$ and $n$ are positive integers and $m$ is as small as possible. Find $m+n$.

2016 CMIMC, 6

Tags: 2016 , CMIMC , computer science

Aaron is trying to write a program to compute the terms of the sequence defined recursively by $a_0=0$, $a_1=1$, and \[a_n=\begin{cases}a_{n-1}-a_{n-2}&n\equiv0\pmod2\\2a_{n-1}-a_{n-2}&\text{else}\end{cases}\] However, Aaron makes a typo, accidentally computing the recurrence by \[a_n=\begin{cases}a_{n-1}-a_{n-2}&n\equiv0\pmod3\\2a_{n-1}-a_{n-2}&\text{else}\end{cases}\] For how many $0\le k\le2016$ did Aaron coincidentally compute the correct value of $a_k$?

2021 CMIMC Integration Bee, 6

Tags: calculus , integration , CMIMC , integration bee

$$\int_0^{20\pi}|x\sin(x)|\,dx$$ [i]Proposed by Connor Gordon[/i]

2016 CMIMC, 3

Tags: 2016 , algebra , CMIMC

Let $\ell$ be a real number satisfying the equation $\tfrac{(1+\ell)^2}{1+\ell^2}=\tfrac{13}{37}$. Then \[\frac{(1+\ell)^3}{1+\ell^3}=\frac mn,\] where $m$ and $n$ are positive coprime integers. Find $m+n$.

2021 CMIMC Integration Bee, 11

Tags: calculus , integration , CMIMC , integration bee

$$\int_0^\frac{\pi}{2}\frac{1}{4-3\cos^2(x)}\,dx$$ [i]Proposed by Connor Gordon[/i]

2021 CMIMC Integration Bee, 5

Tags: CMIMC , integration bee

$$\int\frac{\ln 2}{1+2^{-x}}\,dx$$ [i]Proposed by Connor Gordon[/i]

2016 CMIMC, 2

Tags: CMIMC , geometry , 2016

Let $ABCD$ be an isosceles trapezoid with $AD=BC=15$ such that the distance between its bases $AB$ and $CD$ is $7$. Suppose further that the circles with diameters $\overline{AD}$ and $\overline{BC}$ are tangent to each other. What is the area of the trapezoid?

2021 CMIMC Integration Bee, 10

Tags: calculus , integration , CMIMC , integration bee

$$\int_{-\infty}^\infty\frac{x\arctan(x)}{x^4+1}\,dx$$ [i]Proposed by Connor Gordon[/i]

2021 CMIMC Integration Bee, 4

Tags: CMIMC , integration bee

$$\int_1^2\left(x^5+6x^4+14x^3+16x^2+9x+2\right)dx$$ [i]Proposed by Connor Gordon[/i]

2021 CMIMC Integration Bee, 12

Tags: calculus , integration , CMIMC , integration bee

$$\int_1^\infty \frac{1 + 2x \ln 2}{x\sqrt{x 4^x - 1}}\,dx$$ [i]Proposed by Vlad Oleksenko[/i]

2016 CMIMC, 7

Tags: 2016 , CMIMC , algebra

Suppose $a$, $b$, $c$, and $d$ are positive real numbers that satisfy the system of equations \begin{align*}(a+b)(c+d)&=143,\\(a+c)(b+d)&=150,\\(a+d)(b+c)&=169.\end{align*} Compute the smallest possible value of $a^2+b^2+c^2+d^2$.

2016 CMIMC, 1

Tags: CMIMC , 2016 , combinatorics

The phrase "COLORFUL TARTAN'' is spelled out with wooden blocks, where blocks of the same letter are indistinguishable. How many ways are there to distribute the blocks among two bags of different color such that neither bag contains more than one of the same letter?