Found problems: 158
2017 ASDAN Math Tournament, 3
Triangle $ABC$ has $AB=4,BC=6,CA=5$. Let $M$ be the midpoint of $\overline{BC}$ and $P$ the point on the circumcircle of $\triangle ABC$ such that $\angle MPA=90^\circ$. Let points $D$ and $E$ lie on $\overline{AC}$ and $\overline{AB}$ respectively such that $\overline{BD}\perp\overline{AC}$ and $\overline{CE}\perp\overline{AB}$. Find $\tfrac{PD}{PE}$.
2017 CMIMC Number Theory, 3
For how many triples of positive integers $(a,b,c)$ with $1\leq a,b,c\leq 5$ is the quantity \[(a+b)(a+c)(b+c)\] not divisible by $4$?
2017 CMIMC Computer Science, 9
Alice thinks of an integer $1 \le n \le 2048$. Bob asks $k$ true or false questions about Alice's integer; Alice then answers each of the questions, but she may lie on at most one question. What is the minimum value of $k$ for which Bob can guarantee he knows Alice's integer after she answers?
2017 ASDAN Math Tournament, 10
Compute
$$\lim_{k\rightarrow\infty}\left(\frac{2017^{1/k}}{k+1}+\frac{2017^{2/k}}{k+\frac{1}{2}}+\dots+\frac{2017^{k/k}}{k+\frac{1}{k}}\right).$$
2017 CMIMC Number Theory, 8
Let $N$ be the number of ordered triples $(a,b,c) \in \{1, \ldots, 2016\}^{3}$ such that $a^{2} + b^{2} + c^{2} \equiv 0 \pmod{2017}$. What are the last three digits of $N$?
2017 ASDAN Math Tournament, 1
Two arbitrary distinct lattice points are selected on the coordinate plane within the square marked by the points $(0,0)$, $(3,0)$, $(0,3)$, and $(3,3)$ (the lattice points may lie on a side or a corner of the square). What is the probability that the distance between the two points is at most $\sqrt{2}$?
2017 ASDAN Math Tournament, 15
Each face of a regular tetrahedron can be colored one of red, purple, blue, or orange. How many distinct ways can we color the faces of the tetrahedron? Colorings are considered distinct if they cannot reach one another by rotation.
2017 ASDAN Math Tournament, 7
Three identical circles are packed into a unit square. Each of the three circles are tangent to each other and tangent to at least one side of the square. If $r$ is the maximum possible radius of the circle, what is $(2-\tfrac{1}{r})^2$?
2017 CMIMC Computer Science, 7
You are presented with a mystery function $f:\mathbb N^2\to\mathbb N$ which is known to satisfy \[f(x+1,y)>f(x,y)\quad\text{and}\quad f(x,y+1)>f(x,y)\] for all $(x,y)\in\mathbb N^2$. I will tell you the value of $f(x,y)$ for \$1. What's the minimum cost, in dollars, that it takes to compute the $19$th smallest element of $\{f(x,y)\mid(x,y)\in\mathbb N^2\}$? Here, $\mathbb N=\{1,2,3,\dots\}$ denotes the set of positive integers.
2017 CMIMC Individual Finals, 1
Cody has an unfair coin that flips heads with probability either $\tfrac13$ or $\tfrac23$, but he doesn't know which one it is. Using this coin, what is the fewest number of independent flips needed to simulate a coin that he knows will flip heads with probability $\tfrac13$?
2017 ASDAN Math Tournament, 3
Let $f(x)=x^4+2x+1$. Find the slope of the tangent line to the curve at $(0,1)$.
2017 ASDAN Math Tournament, 11
If $a+b+c=12$ and $a^2+b^2+c^2=62$, what is $ab+bc+ac$?
2017 CMIMC Team, 4
Say an odd positive integer $n > 1$ is $\textit{twinning}$ if $p - 2 \mid n$ for every prime $p \mid n$. Find the number of twinning integers less than 250.
2017 ASDAN Math Tournament, 12
Anna has a magical compass which can point only in four directions: North, East, South, West. Initially, the compass points North. After each minute, the compass can either turn left, turn right, or stay at its current orientation, with each action occurring with equal probability. What is the probability that the compass points South after $6$ minutes?
2017 ASDAN Math Tournament, 16
Let $x$ and $y$ be real numbers satisfying $9x^2+16y^2=144$. What is the maximum possible value of $xy$?
2017 CMIMC Team, 3
Suppose Pat and Rick are playing a game in which they take turns writing numbers from $\{1, 2, \dots, 97\}$ on a blackboard. In each round, Pat writes a number, then Rick writes a number; Rick wins if the sum of all the numbers written on the blackboard after $n$ rounds is divisible by 100. Find the minimum positive value of $n$ for which Rick has a winning strategy.
2017 CMIMC Computer Science, 5
Given a list $A$ of $n$ real numbers, the following algorithm, known as $\textit{insertion sort}$, sorts the elements of $A$ from least to greatest.
\begin{tabular}{l}
1: \textbf{FUNCTION} $IS(A)$ \\
2: $\quad$ \textbf{FOR} $i=0,\ldots, n-1$: \\
3: $\quad\quad$ $j \leftarrow i$\\
4: $\quad\quad$ \textbf{WHILE} $j>0$ \& $A[j-1]>A[j]:$\\
5: $\quad\quad\quad$ \textbf{SWAP} $A[j], A[j-1]$\\
6: $\quad\quad\quad$ $j \leftarrow j-1$\\
7: \textbf{RETURN} $A$
\end{tabular}
As $A$ ranges over all permutations of $\{1, 2, \ldots, n\}$, let $f(n)$ denote the expected number of comparisons (i.e., checking which of two elements is greater) that need to be made when sorting $A$ with insertion sort. Evaluate $f(13) - f(12)$.
2017 ASDAN Math Tournament, 5
Compute the maximum value attained by $f(x)=x^{1/x^2}$.
2017 ASDAN Math Tournament, 1
If $a$, $6$, and $b$, in that order, form an arithmetic sequence, compute $a+b$.
2017 ASDAN Math Tournament, 22
Let $x=2\sin8^\circ+2\sin16^\circ+\dots+2\sin176^\circ$. What is $\arctan(x)$?
2017 CMIMC Individual Finals, 3
The parabola $\mathcal P$ given by equation $y=x^2$ is rotated some acute angle $\theta$ clockwise about the origin such that it hits both the $x$ and $y$ axes at two distinct points. Suppose the length of the segment $\mathcal P$ cuts the $x$-axis is $1$. What is the length of the segment $\mathcal P$ cuts the $y$-axis?
2017 CMIMC Computer Science, 6
Define a self-balanced tree to be a tree such that for any node, the size of the left subtree is within 1 of the size of the right subtree. How many balanced trees are there of size 2046?
2017 ASDAN Math Tournament, 4
Alice and Bob are painting a house. Alice can paint a house in $20$ hours by herself. Bob can paint a house in $40$ hours by himself. Both people start at the same time, paint at their own constant rate, and work together to paint one house. When the house is fully painted, what fraction of the house was painted by Alice?
2017 ASDAN Math Tournament, 8
How many integer solutions are there to $y^2=x^2-2017$?
2017 ASDAN Math Tournament, 2
Find the remainder of $7^{1010}+8^{2017}$ when divided by $57$.