[b]p7[/b] Cindy cuts regular hexagon $ABCDEF$ out of a sheet of paper. She folds $B$ over $AC$, resulting in a pentagon. Then, she folds $A$ over $CF$, resulting in a quadrilateral. The area of $ABCDEF$ is $k$ times the area of the resulting folded shape. Find $k$. [b]p8[/b] Call a sequence $\{a_n\} = a_1, a_2, a_3, . . .$ of positive integers [i]Fib-o’nacci[/i] if it satisfies $a_n = a_{n-1}+a_{n-2}$ for all $n \ge 3$. Suppose that $m$ is the largest even positive integer such that exactly one [i]Fib-o’nacci[/i] sequence satisfies $a_5 = m$, and suppose that $n$ is the largest odd positive integer such that exactly one [i]Fib-o’nacci[/i] sequence satisfies $a_5 = n$. Find $mn$. [b]p9[/b] Compute the number of ways there are to pick three non-empty subsets $A$, $B$, and $C$ of $\{1, 2, 3, 4, 5, 6\}$, such that $|A| = |B| = |C|$ and the following property holds: $$A \cap B \cap C = A \cap B = B \cap C = C \cap A.$$

Three points of a triangle are among 8 vertex of a cube. So the number of such acute triangles is $\text{(A)}0\qquad\text{(B)}6\qquad\text{(C)}8\qquad\text{(D)}24$

In a non equilateral triangle $ABC$ the sides $a,b,c$ form an arithmetic progression. Let $I$ be the incentre and $O$ the circumcentre of the triangle $ABC$. Prove that (1) $IO$ is perpendicular to $BI$; (2) If $BI$ meets $AC$ in $K$, and $D$, $E$ are the midpoints of $BC$, $BA$ respectively then $I$ is the circumcentre of triangle $DKE$.

Let $ x$, $ y$, and $ z$ be integers satisfying the equation \[ \dfrac{2008}{41y^2}\equal{}\dfrac{2z}{2009}\plus{}\dfrac{2007}{2x^2}.\] Determine the greatest value that $ z$ can take. [i]Budi Surodjo, Jogjakarta[/i]

$P(x)$ is a polynomial with real coefficients such that $P(a_1) = 0, P(a_{i+1}) = a_i$ ($i = 1, 2,\ldots$) where $\{a_i\}_{i=1,2,\ldots}$ is an infinite sequence of distinct natural numbers. Determine the possible values of degree of $P(x)$.

Given an integer ${n>1}$, denote by $P_{n}$ the product of all positive integers $x$ less than $n$ and such that $n$ divides ${x^2-1}$. For each ${n>1}$, find the remainder of $P_{n}$ on division by $n$. [i]Proposed by John Murray, Ireland[/i]

Find, with proof, the maximum positive integer $k$ for which it is possible to color $6k$ cells of $6 \times 6$ grid such that, for any choice of three distinct rows $R_1$, $R_2$, $R_3$ and three distinct columns $C_1$, $C_2$, $C_3$, there exists an uncolored cell $c$ and integers $1 \le i, j \le 3$ so that $c$ lies in $R_i$ and $C_j$

Let $(a_{n})_{n\geq 1}$ and $(b_{n})_{n\geq 1}$ be the sequence of positive integers defined by $a_{n+1}=na_{n}+1$ and $b_{n+1}=nb_{n}-1$ for $n\geq 1$. Show that the two sequence cannot have infinitely many common terms. [i]Laurentiu Panaitopol[/i]

Prove that number $1$ has infinitely many representations of the form $$1 =\frac{1}{5}+\frac{1}{a_1}+\frac{1}{a_2}+ ...+\frac{1}{a_n}$$ , where$ n$ and $a_i $ are positive integers with $5 < a_1 < a_2 < ... < a_n$.

Let $a,\ b$ be positive real numbers. Consider the circle $C_1: (x-a)^2+y^2=a^2$ and the ellipse $C_2: x^2+\frac{y^2}{b^2}=1.$ (1) Find the condition for which $C_1$ is inscribed in $C_2$. (2) Suppose $b=\frac{1}{\sqrt{3}}$ and $C_1$ is inscribed in $C_2$. Find the coordinate $(p,\ q)$ of the point of tangency in the first quadrant for $C_1$ and $C_2$. (3) Under the condition in (1), find the area of the part enclosed by $C_1,\ C_2$ for $x\geq p$. 60 point

Let $f(x)=x^2-ax+b$, where $a$ and $b$ are positive integers. (a) Suppose that $a=2$ and $b=2$. Determine the set of real roots of $f(x)-x$, and the set of real roots of $f(f(x))-x$. (b) Determine the number of positive integers $(a,b)$ with $1\le a,b\le 2011$ for which every root of $f(f(x))-x$ is an integer.

Let $ m$ and $ n$ be two positive integers. Let $ a_1$, $ a_2$, $ \ldots$, $ a_m$ be $ m$ different numbers from the set $ \{1, 2,\ldots, n\}$ such that for any two indices $ i$ and $ j$ with $ 1\leq i \leq j \leq m$ and $ a_i \plus{} a_j \leq n$, there exists an index $ k$ such that $ a_i \plus{} a_j \equal{} a_k$. Show that \[ \frac {a_1 \plus{} a_2 \plus{} ... \plus{} a_m}{m} \geq \frac {n \plus{} 1}{2}. \]

Prove that: a) there are infinitely many natural numbers of the form 3p + 1, p is positive integer , which can be represented as the difference of 2 cubes of positive integers; b) there are infinitely many natural numbers of the form 5q + 1, q is positive integer , which can be represented as the difference of two cubes of positive integers.

Given $2017$ positive numbers $x_1,\dots,x_{2017}$ such that $$\sum_{i=1}^{2017}x_i=\sum_{i=1}^{2017}\frac{1}{x_i}=2018,$$ compute the maximum possible value of $x_1+\frac{1}{x_1}$.

In 3-dimensional hyperbolic space, we are given a plane $P$ and four distinct straight lines: the lines $a_1$ and $a_2$ are perpendicular to $P$; while the lines $r_1$ and $r_2$ do not intersect $P$, and their distances from $P$ are equal. Denote by $S_i$ the surface of revolution obtained by rotating $r_i$ around $a_i$. Show that the common points of $S_1$ and $S_2$ can be covered by two planes.

Let $f(x) =x^4+14x^3+52x^2+56x+16.$ Let $z_1, z_2, z_3, z_4$ be the four roots of $f$. Find the smallest possible value of $|z_az_b+z_cz_d|$ where $\{a,b,c,d\}=\{1,2,3,4\}$.

[b]p1.[/b] Suppose $a_1 \cdot 2 = a_2 \cdot 3 = a_3$ and $a_1 + a_2 + a_3 = 66$. What is $a_3$? [b]p2.[/b] Ankit buys a see-through plastic cylindrical water bottle. However, in coming home, he accidentally hits the bottle against a wall and dents the top portion of the bottle (above the $7$ cm mark). Ankit now wants to determine the volume of the bottle. The area of the base of the bottle is $20$ cm$^2$ . He fills the bottle with water up to the $5$ cm mark. After flipping the bottle upside down, he notices that the height of the empty space is at the $7$ cm mark. Find the total volume (in cm$^3$) of this bottle. [img]https://cdn.artofproblemsolving.com/attachments/1/9/f5735c77b056aaf31b337ea1b777a591807819.png[/img] [b]p3.[/b] If $P$ is a quadratic polynomial with leading coefficient $ 1$ such that $P(1) = 1$, $P(2) = 2$, what is $P(10)$? [b]p4.[/b] Let ABC be a triangle with $AB = 1$, $AC = 3$, and $BC = 3$. Let $D$ be a point on $BC$ such that $BD =\frac13$ . What is the ratio of the area of $BAD$ to the area of $CAD$? [b]p5.[/b] A coin is flipped $ 12$ times. What is the probability that the total number of heads equals the total number of tails? Express your answer as a common fraction in lowest terms. [b]p6.[/b] Moor pours $3$ ounces of ginger ale and $ 1$ ounce of lime juice in cup $A$, $3$ ounces of lime juice and $ 1$ ounce of ginger ale in cup $B$, and mixes each cup well. Then he pours $ 1$ ounce of cup $A$ into cup $B$, mixes it well, and pours $ 1$ ounce of cup $B$ into cup $A$. What proportion of cup $A$ is now ginger ale? Express your answer as a common fraction in lowest terms. [b]p7.[/b] Determine the maximum possible area of a right triangle with hypotenuse $7$. Express your answer as a common fraction in lowest terms. [b]p8.[/b] Debbie has six Pusheens: $2$ pink ones, $2$ gray ones, and $2$ blue ones, where Pusheens of the same color are indistinguishable. She sells two Pusheens each to Alice, Bob, and Eve. How many ways are there for her to do so? [b]p9.[/b] How many nonnegative integer pairs $(a, b)$ are there that satisfy $ab = 90 - a - b$? [b]p10.[/b] What is the smallest positive integer $a_1...a_n$ (where $a_1, ... , a_n$ are its digits) such that $9 \cdot a_1 ... a_n = a_n ... a_1$, where $a_1$, $a_n \ne 0$? [b]p11.[/b] Justin is growing three types of Japanese vegetables: wasabi root, daikon and matsutake mushrooms. Wasabi root needs $2$ square meters of land and $4$ gallons of spring water to grow, matsutake mushrooms need $3$ square meters of land and $3$ gallons of spring water, and daikon need $ 1$ square meter of land and $ 1$ gallon of spring water to grow. Wasabi sell for $60$ per root, matsutake mushrooms sell for $60$ per mushroom, and daikon sell for $2$ per root. If Justin has $500$ gallons of spring water and $400$ square meters of land, what is the maximum amount of money, in dollars, he can make? [b]p12.[/b] A [i]prim [/i] number is a number that is prime if its last digit is removed. A [i]rime [/i] number is a number that is prime if its first digit is removed. Determine how many numbers between $100$ and $999$ inclusive are both prim and rime numbers. [b]p13.[/b] Consider a cube. Each corner is the intersection of three edges; slice off each of these corners through the midpoints of the edges, obtaining the shape below. If we start with a $2\times 2\times 2$ cube, what is the volume of the resulting solid? [img]https://cdn.artofproblemsolving.com/attachments/4/8/856814bf99e6f28844514158344477f6435a3a.png[/img] [b]p14.[/b] If a parallelogram with perimeter $14$ and area $ 12$ is inscribed in a circle, what is the radius of the circle? [b]p15.[/b] Take a square $ABCD$ of side length $1$, and draw $\overline{AC}$. Point $E$ lies on $\overline{BC}$ such that $\overline{AE}$ bisects $\angle BAC$. What is the length of $BE$? [b]p16.[/b] How many integer solutions does $f(x) = (x^2 + 1)(x^2 + 2) + (x^2 + 3)(x + 4) = 2017$ have? [b]p17.[/b] Alice, Bob, Carol, and Dave stand in a circle. Simultaneously, each player selects another player at random and points at that person, who must then sit down. What is the probability that Alice is the only person who remains standing? [b]p18.[/b] Let $x$ be a positive integer with a remainder of $2$ when divided by $3$, $3$ when divided by $4$, $4$ when divided by $5$, and $5$ when divided by $6$. What is the smallest possible such $x$? [b]p19[/b]. A circle is inscribed in an isosceles trapezoid such that all four sides of the trapezoid are tangent to the circle. If the radius of the circle is $ 1$, and the upper base of the trapezoid is $ 1$, what is the area of the trapezoid? [b]p20.[/b] Ray is blindfolded and standing $ 1$ step away from an ice cream stand. Every second, he has a $1/4$ probability of walking $ 1$ step towards the ice cream stand, and a $3/4$ probability of walking $ 1$ step away from the ice cream stand. When he is $0$ steps away from the ice cream stand, he wins. What is the probability that Ray eventually wins? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

There are $ n$ markers, each with one side white and the other side black. In the beginning, these $ n$ markers are aligned in a row so that their white sides are all up. In each step, if possible, we choose a marker whose white side is up (but not one of the outermost markers), remove it, and reverse the closest marker to the left of it and also reverse the closest marker to the right of it. Prove that, by a finite sequence of such steps, one can achieve a state with only two markers remaining if and only if $ n \minus{} 1$ is not divisible by $ 3$. [i]Proposed by Dusan Dukic, Serbia[/i]

Let $a_0,a_1,a_2,\dots $ be a sequence of real numbers such that $a_0=0, a_1=1,$ and for every $n\geq 2$ there exists $1 \leq k \leq n$ satisfying \[ a_n=\frac{a_{n-1}+\dots + a_{n-k}}{k}. \]Find the maximum possible value of $a_{2018}-a_{2017}$.

Ana and Banana play a game. First, Ana picks a real number $p$ with $0 \le p \le 1$. Then, Banana picks an integer $h$ greater than $1$ and creates a spaceship with $h$ hit points. Now every minute, Ana decreases the spaceship's hit points by $2$ with probability $1-p$, and by $3$ with probability $p$. Ana wins if and only if the number of hit points is reduced to exactly $0$ at some point (in particular, if the spaceship has a negative number of hit points at any time then Ana loses). Given that Ana and Banana select $p$ and $h$ optimally, compute the integer closest to $1000p$. [i]Proposed by Lewis Chen[/i]

Prove that for any tetrahedron the radius of the inscribed sphere $r <\frac{ ab}{ 2(a + b)}$, where $a$ and $b$ are the lengths of any pair of opposite edges.

Let $A = {1, 2, 3, 4, 5}$. Find the number of functions $f$ from the nonempty subsets of $A$ to $A$, such that $f(B) \in B$ for any $B \subset A$, and $f(B \cup C)$ is either $f(B)$ or $f(C)$ for any $B$, $C \subset A$

The sum of an infinite geometric series is a positive number $S$, and the second term in the series is $1$. What is the smallest possible value of $S?$ $\textbf{(A)}\ \frac{1+\sqrt{5}}{2} \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ \sqrt{5} \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 4$

Prove that the simultaneous equations $$x^4 -x^2 =y^4 -y^2 =z^4 -z^2$$ are satisfied by the points of $4$ straight lines and $6$ ellipses, and by no other points.

Are there any real numbers $a,b,c$ such that $a+b+c=6$, $ab+bc+ca=9$ and $a^4+b^4+c^4=260$? What about if we let $a^4+b^4+c^4=210$? [i] (Andrei Bâra)[/i]