Found problems: 116
2018 MOAA, 2
If $x > 0$ and $x^2 +\frac{1}{x^2}= 14$, find $x^5 +\frac{1}{x^5}$.
2018 ASDAN Math Tournament, 3
Each day, the city of Berkeley is either rainy or foggy, with a $\tfrac{3}{4}$ chance of the weather remaining the same as that of the previous day. If we only know that it is rainy today, what is the probability that it is rainy in $7$ days?
MOAA Team Rounds, 2018.2
If $x > 0$ and $x^2 +\frac{1}{x^2}= 14$, find $x^5 +\frac{1}{x^5}$.
2018 CMIMC Combinatorics, 1
Ninety-eight apples who always lie and one banana who always tells the truth are randomly arranged along a line. The first fruit says "One of the first forty fruit is the banana!'' The last fruit responds "No, one of the $\emph{last}$ forty fruit is the banana!'' The fruit in the middle yells "I'm the banana!'' In how many positions could the banana be?
MOAA Team Rounds, 2018.10
Vincent is playing a game with Evil Bill. The game uses an infinite number of red balls, an infinite number of green balls, and a very large bag. Vincent first picks two nonnegative integers $g$ and $k$ such that $g < k \le 2016$, and Evil Bill places $g$ green balls and $2016 - g$ red balls in the bag, so that there is a total of $2016$ balls in the bag. Vincent then picks a ball of either color and places it in the bag. Evil Bill then inspects the bag. If the ratio of green balls to total balls in the bag is ever exactly $\frac{k}{2016}$ , then Evil Bill wins. If the ratio of green balls to total balls is greater than $\frac{k}{2016}$ , then Vincent wins. Otherwise, Vincent and Evil Bill repeat the previous two actions (Vincent picks a ball and Evil Bill inspects the bag). If $S$ is the sum of all possible values of $k$ that Vincent could choose and be able to win, determine the largest prime factor of $S$.
2018 CMIMC CS, 3
You are given the existence of an unsorted sequence $a_1,\ldots, a_5$ of five distinct real numbers. The Erdos-Szekeres theorem states that there exists a subsequence of length $3$ which is either strictly increasing or strictly decreasing. You do not have access to the $a_i$, but you do have an oracle which, when given two indexes $1\leq i < j\leq 5$, will tell you whether $a_i < a_j$ or $a_i > a_j$. What is the minimum number of calls to the oracle needed in order to identify an ordered triple of integers $(r,s,t)$ such that $a_r,a_s,a_t$ is one such sequence?
2018 CMIMC CS, 9
Consider the following modified algorithm for binary search, which we will call $\textit{weighted binary search}$:
\begin{tabular}{l}
01: \textbf{FUNCTION} SEARCH($L$, value) \\
02:$\qquad$ hi $\leftarrow$ $\operatorname{len}(L) - 1$ \\
03:$\qquad$ lo $\leftarrow$ 0 \\
04:$\qquad$ \textbf{WHILE} hi $\geq$ lo \\
05:$\qquad\qquad$ guess $\leftarrow$ $\lfloor w \cdot\text{lo} + (1-w) \cdot \text{hi}\rfloor$ \\
06:$\qquad\qquad$ mid $\leftarrow$ $L[\text{guess}]$ \\
07:$\qquad\qquad$ \textbf{IF} mid $> \text{value}$ \\
08: $\qquad\qquad\qquad$ hi $\leftarrow$ $\text{guess} - 1$ \\
09: $\qquad\qquad$ \textbf{ELSE IF} mid $< \text{value}$ \\
10: $\qquad\qquad\qquad$ lo $\leftarrow$ $\text{guess} + 1$ \\
11: $\qquad\qquad$ \textbf{ELSE} \\
12: $\qquad\qquad\qquad$ \textbf{RETURN} guess \\
13:$\qquad$ \textbf{RETURN} -1 (not found)
\end{tabular}\\
Assume $L$ is a list of the integers $\{1,2,\ldots,100\}$, in that order. Further assume that accessing the $k$th index of $L$ costs $k+1$ tokens (e.g. $L[0]$ costs $1$ token). Let $S$ be the set of all $w\in[\tfrac12,1)$ which minimize the average cost when $\texttt{value}$ is an integer selected at random in the range $[1,50]$. Given that $S=\left(x,\tfrac {74}{99}\right]$, determine $x$.
2018 MOAA, 10
Vincent is playing a game with Evil Bill. The game uses an infinite number of red balls, an infinite number of green balls, and a very large bag. Vincent first picks two nonnegative integers $g$ and $k$ such that $g < k \le 2016$, and Evil Bill places $g$ green balls and $2016 - g$ red balls in the bag, so that there is a total of $2016$ balls in the bag. Vincent then picks a ball of either color and places it in the bag. Evil Bill then inspects the bag. If the ratio of green balls to total balls in the bag is ever exactly $\frac{k}{2016}$ , then Evil Bill wins. If the ratio of green balls to total balls is greater than $\frac{k}{2016}$ , then Vincent wins. Otherwise, Vincent and Evil Bill repeat the previous two actions (Vincent picks a ball and Evil Bill inspects the bag). If $S$ is the sum of all possible values of $k$ that Vincent could choose and be able to win, determine the largest prime factor of $S$.
2018 ASDAN Math Tournament, 6
Square $ABCD$ has side length $5$. Draw E on $BC$ and $F$ on $AD$ such that $BE < AF$. Next, flip $ABCD$ across $EF$ to a square $A'B'C'D'$ such that $C'$ lies in the interior of $ABCD$ and $C$ lies in the interior of $A'B'C'D'$. Suppose that $CC' = 4$ and $DD' = 2$. Compute $AA'$.
2018 MOAA, 6
Consider an $m \times n$ grid of unit squares. Let $R$ be the total number of rectangles of any size, and let $S$ be the total number of squares of any size. Assume that the sides of the rectangles and squares are parallel to the sides of the $m \times n$ grid. If $\frac{R}{S} =\frac{759}{50}$ , then determine $mn$.
2018 CMIMC Algebra, 2
Suppose $x>1$ is a real number such that $x+\tfrac 1x = \sqrt{22}$. What is $x^2-\tfrac1{x^2}$?
2018 MOAA, 3
Let $BE$ and $CF$ be altitudes in triangle $ABC$. If $AE = 24$, $EC = 60$, and $BF = 31$, determine $AF$.
MOAA Team Rounds, 2018.3
Let $BE$ and $CF$ be altitudes in triangle $ABC$. If $AE = 24$, $EC = 60$, and $BF = 31$, determine $AF$.
2018 CMIMC CS, 7
I give you a function $\textbf{rand}$ that returns a number chosen uniformly at random from $[0,T]$ for some number $T$ that you don't know. Your task is to approximate $T$. You do this by calling $\textbf{rand}$ $100$ times, recording the results as $X_1,X_2,\dots,X_{100}$, and guessing \[\hat{T}=\alpha\cdot\max\{X_1,X_2,\dots,X_{100}\}\] for some $\alpha$. Which value of $\alpha$ ensures that $\mathbb{E}[\hat{T}]=T$?
2018 ASDAN Math Tournament, 2
What are the last $2$ digits of the number $2018^{2018}$ when written in base $7$?
2018 Cyprus IMO TST, Source
[url=https://artofproblemsolving.com/community/c677808][b]Cyprus IMO TST 2018[/b][/url]
[url=https://artofproblemsolving.com/community/c6h1666662p10591751][b]Problem 1.[/b][/url] Determine all integers $n \geq 2$ for which the number $11111$ in base $n$ is a perfect square.
[url=https://artofproblemsolving.com/community/c6h1666663p10591753][b]Problem 2.[/b][/url] Consider a trapezium $AB \Gamma \Delta$, where $A\Delta \parallel B\Gamma$ and $\measuredangle A = 120^{\circ}$. Let $E$ be the midpoint of $AB$ and let $O_1$ and $O_2$ be the circumcenters of triangles $AE \Delta$ and $BE\Gamma$, respectively. Prove that the area of the trapezium is equal to six time the area of the triangle $O_1 E O_2$.
[url=https://artofproblemsolving.com/community/c6h1666660p10591747][b]Problem 3.[/b][/url] Find all triples $(\alpha, \beta, \gamma)$ of positive real numbers for which the expression
$$K = \frac{\alpha+3 \gamma}{\alpha + 2\beta + \gamma} + \frac{4\beta}{\alpha+\beta+2\gamma} - \frac{8 \gamma}{\alpha+ \beta + 3\gamma}$$obtains its minimum value.
[url=https://artofproblemsolving.com/community/c6h1666661p10591749][b]Problem 4.[/b][/url] Let $\Lambda= \{1, 2, \ldots, 2v-1,2v\}$ and $P=\{\alpha_1, \alpha_2, \ldots, \alpha_{2v-1}, \alpha_{2v}\}$ be a permutation of the elements of $\Lambda$.
(a) Prove that
$$\sum_{i=1}^v \alpha_{2i-1}\alpha_{2i} \leq \sum_{i=1}^v (2i-1)2i.$$(b) Determine the largest positive integer $m$ such that we can partition the $m\times m$ square into $7$ rectangles for which every pair of them has no common interior points and their lengths and widths form the following sequence:
$$1,2,3,4,5,6,7,8,9,10,11,12,13,14.$$
2018 ASDAN Math Tournament, 6
Given that $x > 1$, compute $x$ such that
$$\log_{16}(x) + \log_x(2)$$
is minimal.
2018 ASDAN Math Tournament, 10
Let $p$ be an odd prime. A degree $d$ polynomial $f$ with non-negative integer coefficients less than $p$ is called $p-floppy$ if the coefficients of $f(x)f(-x) - f(x^2)$ are all divisible by $p$ and if exactly $d$ entries in the sequence $(f(0), f(1), f(2), \dots , f(p-1))$ are divisible by $p$. How many non-constant $61$-floppy polynomials are there?
2018 ASDAN Math Tournament, 8
Compute the remainder when
$$\sum_{n=1}^{2018} n^4$$
is divided by $53$.
2018 CMIMC CS, 6
For integer $n\geq 2$ and real $0\leq p\leq 1$, define $\mathcal{W}_{n,p}$ to be the complete weighted undirected random graph with vertex set $\{1,2,\ldots,n\}$: the edge $(i,j)$ will have weight $\min(i,j)$ with probability $p$ and weight $\max(i,j)$ otherwise. Let $\mathcal{L}_{n,p}$ denote the total weight of the minimum spanning tree of $\mathcal{W}_{n,p}$. Find the largest integer less than the expected value of $\mathcal{L}_{2018,1/2}$.
2018 CMIMC Combinatorics, 9
Compute the number of rearrangements $a_1, a_2, \dots, a_{2018}$ of the sequence $1, 2, \dots, 2018$ such that $a_k > k$ for $\textit{exactly}$ one value of $k$.
2018 CMIMC Combinatorics, 10
Call a set $S \subseteq \{0,1,\dots,14\}$ $\textit{sparse}$ if $x+1 \pmod{15}$ is not in $S$ whenever $x \in S$. Find the number of sparse sets $T$ such that the sum of the elements of $T$ is a multiple of 15.
2018 CMIMC Algebra, 3
Let $P(x)=x^2+4x+1$. What is the product of all real solutions to the equation $P(P(x))=0$?
2018 CMIMC Combinatorics, 8
Fred and George play a game, as follows. Initially, $x = 1$. Each turn, they pick $r \in \{3,5,8,9\}$ uniformly at random and multiply $x$ by $r$. If $x+1$ is a multiple of 13, Fred wins; if $x+3$ is a multiple of 13, George wins; otherwise, they repeat. Determine the probability that Fred wins the game.
2018 MOAA, 9
Quadrilateral $ABCD$ with $AC = 800$ is inscribed in a circle, and $E, W, X, Y, Z$ are the midpoints of segments $BD$, $AB$, $BC$, $CD$, $DA$, respectively. If the circumcenters of $EW Z$ and $EXY$ are $O_1$ and $O_2$, respectively, determine $O_1O_2$.