Let $f(x) = \frac{x^2}8$. Starting at the point $(7,3)$, what is the length of the shortest path that touches the graph of $f$, and then the $x$-axis? [i]Proposed by Sam Delatore[/i]

[b]p1.[/b] Twelve people, three of whom are in the Mafia and one of whom is a police inspector, randomly sit around a circular table. What is the probability that the inspector ends up sitting next to at least one of the Mafia? [b]p2.[/b] Of the positive integers between $1$ and $1000$, inclusive, how many of them contain neither the digit “$4$” nor the digit “$7$”? [b]p3.[/b] You are really bored one day and decide to invent a variation of chess. In your variation, you create a new piece called the “krook,” which, on any given turn, can move either one square up or down, or one square left or right. If you have a krook at the bottom-left corner of the chessboard, how many different ways can the krook reach the top-right corner of the chessboard in exactly $17$ moves? [b]p4.[/b] Let $p$ be a prime number. What is the smallest positive integer that has exactly $p$ different positive integer divisors? Write your answer as a formula in terms of $p$. [b]p5.[/b] You make the square $\{(x, y)| - 5 \le x \le 5, -5 \le y \le 5\}$ into a dartboard as follows: (i) If a player throws a dart and its distance from the origin is less than one unit, then the player gets $10$ points. (ii) If a player throws a dart and its distance from the origin is between one and three units, inclusive, then the player gets awarded a number of points equal to the number of the quadrant that the dart landed on. (The player receives no points for a dart that lands on the coordinate axes in this case.) (iii) If a player throws a dart and its distance from the origin is greater than three units, then the player gets $0$ points. If a person throws three darts and each hits the board randomly (i.e with uniform distribution), what is the expected value of the score that they will receive? [b]p6.[/b] Teddy works at Please Forget Meat, a contemporary vegetarian pizza chain in the city of Gridtown, as a deliveryman. Please Forget Meat (PFM) has two convenient locations, marked with “$X$” and “$Y$ ” on the street map of Gridtown shown below. Teddy, who is currently at $X$, needs to deliver an eggplant pizza to $\nabla$ en route to $Y$ , where he is urgently needed. There is currently construction taking place at $A$, $B$, and $C$, so those three intersections will be completely impassable. How many ways can Teddy get from $X$ to $Y$ while staying on the roads (Traffic tickets are expensive!), not taking paths that are longer than necessary (Gas is expensive!), and that let him pass through $\nabla$ (Losing a job is expensive!)? [img]https://cdn.artofproblemsolving.com/attachments/e/0/d4952e923dc97596ad354ed770e80f979740bc.png[/img] [b]p7.[/b] $x, y$, and $z$ are positive real numbers that satisfy the following three equations: $$x +\frac{1}{y}= 4 \,\,\,\,\, y +\frac{1}{z}= 1\,\,\,\,\, z +\frac{1}{x}=\frac73.$$ Compute $xyz$. [b]p8.[/b] Alan, Ben, and Catherine will all start working at the Duke University Math Department on January $1$st, $2009$. Alan’s work schedule is on a four-day cycle; he starts by working for three days and then takes one day off. Ben’s work schedule is on a seven-day cycle; he starts by working for five days and then takes two days off. Catherine’s work schedule is on a ten-day cycle; she starts by working for seven days and then takes three days off. On how many days in $2009$ will none of the three be working? [b]p9.[/b] $x$ and $y$ are complex numbers such that $x^3 + y^3 = -16$ and $(x + y)^2 = xy$. What is the value of $|x + y|$? [b]p10.[/b] Call a four-digit number “well-meaning” if (1) its second digit is the mean of its first and its third digits and (2) its third digit is the mean of its second and fourth digits. How many well-meaning four-digit numbers are there? (For a four-digit number, its first digit is its thousands [leftmost] digit and its fourth digit is its units [rightmost] digit. Also, four-digit numbers cannot have “$0$” as their first digit.) [b]p11.[/b] Suppose that $\theta$ is a real number such that $\sum^{\infty}{k=2} \sin \left(2^k\theta \right)$ is well-defined and equal to the real number $a$. Compute: $$\sum^{\infty}{k=0} \left(\cot^3 \left(2^k\theta \right)-\cot \left(2^k\theta \right) \right) \sin^4 \left(2^k\theta \right).$$ Write your answer as a formula in terms of $a$. [b]p12.[/b] You have $13$ loaded coins; the probability that they come up as heads are $\cos\left( \frac{0\pi}{24 }\right)$,$ \cos\left( \frac{1\pi}{24 }\right)$, $\cos\left( \frac{2\pi}{24 }\right)$, $...$, $\cos\left( \frac{11\pi}{24 }\right)$ and $\cos\left( \frac{12\pi}{24 }\right)$, respectively. You throw all $13$ of these coins in the air at once. What is the probability that an even number of them come up as heads? [b]p13.[/b] Three married couples sit down on a long bench together in random order. What is the probability that none of the husbands sit next to their respective wives? [b]p14.[/b] What is the smallest positive integer that has at least $25$ different positive divisors? [b]p15.[/b] Let $A_1$ be any three-element set, $A_2 = \{\emptyset\}$, and $A_3 = \emptyset$. For each $i \in \{1, 2, 3\}$, let: (i) $B_i = \{\emptyset,A_i\}$, (ii) $C_i$ be the set of all subsets of $B_i$, (iii) $D_i = B_i \cup C_i$, and (iv) $k_i$ be the number of different elements in $D_i$. Compute $k_1k_2k_3$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

What is the value of $$(2^2-2) - (3^2-3) + (4^2-4)?$$ $\textbf{(A) } 1 \qquad \textbf{(B) } 2 \qquad \textbf{(C) } 5 \qquad \textbf{(D) } 8 \qquad \textbf{(E) } 12$

Let $x,y\in\mathbb{R}$ such that $x>2, y>3$. Find the minimum value of $\frac{(x+y)^2}{\sqrt{x^2-4}+\sqrt{y^2-9}}$

The graph below shows a portion of the curve defined by the quartic polynomial $ P(x) \equal{} x^4 \plus{} ax^3 \plus{} bx^2 \plus{} cx \plus{} d$. Which of the following is the smallest? $ \textbf{(A)}\ P( \minus{} 1)$ $ \textbf{(B)}\ \text{The product of the zeros of }P$ $ \textbf{(C)}\ \text{The product of the non \minus{} real zeros of }P$ $ \textbf{(D)}\ \text{The sum of the coefficients of }P$ $ \textbf{(E)}\ \text{The sum of the real zeros of }P$ [asy] size(170); defaultpen(linewidth(0.7)+fontsize(7));size(250); real f(real x) { real y=1/4; return 0.2125(x*y)^4-0.625(x*y)^3-1.6125(x*y)^2+0.325(x*y)+5.3; } draw(graph(f,-10.5,19.4)); draw((-13,0)--(22,0)^^(0,-10.5)--(0,15)); int i; filldraw((-13,10.5)--(22,10.5)--(22,20)--(-13,20)--cycle,white, white); for(i=-3; i<6; i=i+1) { if(i!=0) { draw((4*i,0)--(4*i,-0.2)); label(string(i), (4*i,-0.2), S); }} for(i=-5; i<6; i=i+1){ if(i!=0) { draw((0,2*i)--(-0.2,2*i)); label(string(2*i), (-0.2,2*i), W); }} label("0", origin, SE);[/asy]

If $5$ times a number is $2$, then $100$ times the reciprocal of the number is $\text{(A)}\ 2.5 \qquad \text{(B)}\ 40 \qquad \text{(C)}\ 50 \qquad \text{(D)}\ 250 \qquad \text{(E)}\ 500$

A cubic block with dimensions $n$ by $n$ by $n$ is made up of a collection of $1$ by $1$ by $1$ unit cubes. What is the smallest value of $n$ so that if the outer two layers of unit cubes are removed from the block, more than half the original unit cubes will still remain?

Marisela is putting on a juggling show! She starts with $1$ ball, tossing it once per second. Lawrence tosses her another ball every five seconds, and she always tosses each ball that she has once per second. Compute the total number of tosses Marisela has made one minute after she starts juggling.

Let $p = (a_{1}, a_{2}, \ldots , a_{12})$ be a permutation of $1, 2, \ldots, 12$. We will denote \[S_{p} = |a_{1}-a_{2}|+|a_{2}-a_{3}|+\ldots+|a_{11}-a_{12}|\]We'll call $p$ $\textit{optimistic}$ if $a_{i} > \min(a_{i-1}, a_{i+1})$ $\forall i = 2, \ldots, 11$. $a)$ What is the maximum possible value of $S_{p}$. How many permutations $p$ achieve this maximum?$\newline$ $b)$ What is the number of $\textit{optimistic}$ permtations $p$? $c)$ What is the maximum possible value of $S_{p}$ for an $\textit{optimistic}$ $p$? How many $\textit{optimistic}$ permutations $p$ achieve this maximum?

Find the maximum and minimum of the expression \[ \max(a_1, a_2) + \max(a_2, a_3), + \dots + \max(a_{n-1}, a_n) + \max(a_n, a_1), \] where $(a_1, a_2, \dots, a_n)$ runs over the set of permutations of $(1, 2, \dots, n)$.

Find all integer numbers $n\ge 4$ which satisfy the following condition: from every $n$ different $3$-element subsets of $n$-element set it is possible to choose $2$ subsets, which have exactly one element in common.

Let $ n\ge 3$ be a natural number. Find all nonconstant polynomials with real coeficcietns $ f_{1}\left(x\right),f_{2}\left(x\right),\ldots,f_{n}\left(x\right)$, for which \[ f_{k}\left(x\right)f_{k+ 1}\left(x\right) = f_{k +1}\left(f_{k + 2}\left(x\right)\right), \quad 1\le k\le n,\] for every real $ x$ (with $ f_{n +1}\left(x\right)\equiv f_{1}\left(x\right)$ and $ f_{n + 2}\left(x\right)\equiv f_{2}\left(x\right)$).

For each positive integer $ n,$ let $ f(n)$ be the number of ways to make $ n!$ cents using an unordered collection of coins, each worth $ k!$ cents for some $ k,\ 1\le k\le n.$ Prove that for some constant $ C,$ independent of $ n,$ \[ n^{n^2/2\minus{}Cn}e^{\minus{}n^2/4}\le f(n)\le n^{n^2/2\plus{}Cn}e^{\minus{}n^2/4}.\]

Choose a point $A$ inside a circle of radius $R$. Construct a pair of perpendicular lines through $A$. Then rotate these lines through the same angle $V$ about $A$. The figure formed inside the circle, as the lines move from their initial to their final position, is in the form of a cross with its centre at $A$. Find the area of this cross. (Problem from Latvia)

In triangle $ABC$, points $P$ and $Q$ are projections of point $A$ onto the bisectors of angles $ABC$ and $ACB$, respectively. Prove that $PQ\parallel BC$.

Of all triangles with given perimeter, find the triangle with the maximum area. Justify your answer