If $x^5 - x ^3 + x = a,$ prove that $x^6 \geq 2a - 1$.

There exists a unique pair of polynomials $(P(x),Q(x))$ such that \begin{align*} P(Q(x))&= P(x)(x^2-6x+7) \\ Q(P(x))&= Q(x)(x^2-3x-2) \end{align*} Compute $P(10)+Q(-10)$. [i]Proposed by Connor Gordon[/i]

[b]p1.[/b] Fluffy the Dog is an extremely fluffy dog. Because of his extreme fluffiness, children always love petting Fluffy anywhere. Given that Fluffy likes being petted $1/4$ of the time, out of $120$ random people who each pet Fluffy once, what is the expected number of times Fluffy will enjoy being petted? [b]p2.[/b] Andy thinks of four numbers $27$, $81$, $36$, and $41$ and whispers the numbers to his classmate Cynthia. For each number she hears, Cynthia writes down every factor of that number on the whiteboard. What is the sum of all the different numbers that are on the whiteboard? (Don't include the same number in your sum more than once) [b]p3.[/b] Charles wants to increase the area his square garden in his backyard. He increases the length of his garden by $2$ and increases the width of his garden by $3$. If the new area of his garden is $182$, then what was the original area of his garden? [b]p4.[/b] Antonio is trying to arrange his flute ensemble into an array. However, when he arranges his players into rows of $6$, there are $2$ flute players left over. When he arranges his players into rows of $13$, there are $10$ flute players left over. What is the smallest possible number of flute players in his ensemble such that this number has three prime factors? [b]p5.[/b] On the AMC $9$ (Acton Math Competition $9$), $5$ points are given for a correct answer, $2$ points are given for a blank answer and $0$ points are given for an incorrect answer. How many possible scores are there on the AMC $9$, a $15$ problem contest? [b]p6.[/b] Charlie Puth produced three albums this year in the form of CD's. One CD was circular, the second CD was in the shape of a square, and the final one was in the shape of a regular hexagon. When his producer circumscribed a circle around each shape, he noticed that each time, the circumscribed circle had a radius of $10$. The total area occupied by $1$ of each of the different types of CDs can be expressed in the form $a + b\pi + c\sqrt{d}$ where $d$ is not divisible by the square of any prime. Find $a + b + c + d$. [b]p7.[/b] You are picking blueberries and strawberries to bring home. Each bushel of blueberries earns you $10$ dollars and each bushel of strawberries earns you $8$ dollars. However your cart can only fit $24$ bushels total and has a weight limit of $100$ lbs. If a bushel of blueberries weighs $8$ lbs and each bushel of strawberries weighs $6$ lbs, what is your maximum profit. (You can only pick an integer number of bushels) [b]p8.[/b] The number $$\sqrt{2218 + 144\sqrt{35} + 176\sqrt{55} + 198\sqrt{77}}$$ can be expressed in the form $a\sqrt5 + b\sqrt7 + c\sqrt{11}$ for positive integers $a, b, c$. Find $abc$. [b]p9.[/b] Let $(x, y)$ be a point such that no circle passes through the three points $(9,15)$, $(12, 20)$, $(x, y)$, and no circle passes through the points $(0, 17)$, $(16, 19)$, $(x, y)$. Given that $x - y = -\frac{p}{q}$ for relatively prime positive integers $p$, $q$, Find $p + q$. [b]p10.[/b] How many ways can Alfred, Betty, Catherine, David, Emily and Fred sit around a $6$ person table if no more than three consecutive people can be in alphabetical order (clockwise)? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Given an integer $ n\ge 2$ and a reular 2n-gon. Color all verices of the 2n-gon with n colors such that: [b](i)[/b] Each vertice is colored by exactly one color. [b](ii)[/b] Two vertices don't have the same color. Two ways of coloring, satisfying the conditions above, are called equilavent if one obtained from the other by a rotation whose center is the center of polygon. Find the total number of mutually non-equivalent ways of coloring. [i]Alternative statement:[/i] In how many ways we can color vertices of an regular 2n-polygon using n different colors such that two adjent vertices are colored by different colors. Two colorings which can be received from each other by rotation are considered as the same.

Find all triples $(f,g,h)$ of injective functions from the set of real numbers to itself satisfying \begin{align*} f(x+f(y)) &= g(x) + h(y) \\ g(x+g(y)) &= h(x) + f(y) \\ h(x+h(y)) &= f(x) + g(y) \end{align*} for all real numbers $x$ and $y$. (We say a function $F$ is [i]injective[/i] if $F(a)\neq F(b)$ for any distinct real numbers $a$ and $b$.) [i]Proposed by Evan Chen[/i]

The quadrilateral $ABCD$ is inscribed in circle $\omega$. $F$ is the intersection point of $AC$ and $BD$. $BA$ and $CD$ meet at $E$. Let the projection of $F$ on $AB$ and $CD$ be $G$ and $H$, respectively. Let $M$ and $N$ be the midpoints of $BC$ and $EF$, respectively. If the circumcircle of $\triangle MNG$ only meets segment $BF$ at $P$, and the circumcircle of $\triangle MNH$ only meets segment $CF$ at $Q$, prove that $PQ$ is parallel to $BC$.

Prove that every prime of the form $4k+1$ is the hypotenuse of a rectangular triangle with integer sides.

Determine all positive integers $ n$ for which there is a coloring of all points in space so that each of the following conditions is satisfied: (i) Each point is painted in exactly one color. (ii) Exactly $ n$ colors are used. (iii) Each line is painted in at most two different colors.

Let $ABC$ be an equiangular triangle with circumcircle $\omega$. Let point $F\in AB$ and point $E\in AC$ so that $\angle ABE+\angle ACF=60^{\circ}$. The circumcircle of triangle $AFE$ intersects the circle $\omega$ in the point $D$. The halflines $DE$ and $DF$ intersect the line through $B$ and $C$ in the points $X$ and $Y$. Prove that the incenter of the triangle $DXY$ is independent of the choice of $E$ and $F$. (The angles in the problem statement are not directed. It is assumed that $E$ and $F$ are chosen in such a way that the halflines $DE$ and $DF$ indeed intersect the line through $B$ and $C$.)

Prove that :$$\sum_{k=0}^{58}C_{2017+k}^{58-k}C_{2075-k}^{k}=\sum_{p=0}^{29}C_{4091-2p}^{58-2p}$$

There exists a unique real value of $x$ such that \[(x+\sqrt{x})^2=16.\] Compute $x$.

a) Suppose that $n$ is an odd integer. Prove that $k(n-k)$ is divisible by $2$ for all positive integers $k$. b) Find an integer $k$ such that $k(100-k)$ is not divisible by $11$. c) Suppose that $p$ is an odd prime, and $n$ is an integer. Prove that there is an integer $k$ such that $k(n-k)$ is not divisible by $p$. d) Suppose that $p,q$ are two different odd primes, and $n$ is an integer. Prove that there is an integer $k$ such that $k(n-k)$ is not divisible by any of $p,q$.

Connie multiplies a number by 2 and gets 60 as her answer. However, she should have divided the number by 2 to get the correct answer. What is the correct answer? $\textbf{(A)}\ 7.5 \qquad \textbf{(B)}\ 15 \qquad \textbf{(C)}\ 30 \qquad \textbf{(D)}\ 120 \qquad \textbf{(E)}\ 240$

Let $f(x)=\lim_{n\to\infty} (\cos ^ n x+\sin ^ n x)^{\frac{1}{n}}$ for $0\leq x\leq \frac{\pi}{2}.$ (1) Find $f(x).$ (2) Find the volume of the solid generated by a rotation of the figure bounded by the curve $y=f(x)$ and the line $y=1$ around the $y$-axis.

For integers $a, b$, call the lattice point with coordinates $(a,b)$ [b]basic[/b] if $gcd(a,b)=1$. A graph takes the basic points as vertices and the edges are drawn in such way: There is an edge between $(a_1,b_1)$ and $(a_2,b_2)$ if and only if $2a_1=2a_2\in \{b_1-b_2, b_2-b_1\}$ or $2b_1=2b_2\in\{a_1-a_2, a_2-a_1\}$. Some of the edges will be erased, such that the remaining graph is a forest. At least how many edges must be erased to obtain this forest? At least how many trees exist in such a forest?

Five equilateral triangles, each with side $2\sqrt{3}$, are arranged so they are all on the same side of a line containing one side of each. Along this line, the midpoint of the base of one triangle is a vertex of the next. The area of the region of the plane that is covered by the union of the five triangular regions is [asy] defaultpen(linewidth(0.7)+fontsize(10)); pair C=origin, N=dir(0), B=dir(20), A=dir(135), M=dir(180), P=(3/7)*dir(C--N); draw((0,0)--(1,sqrt(3))--(2,0)--(3,sqrt(3))--(4,0)--(5,sqrt(3))--(6,0)); draw((1,0)--(2,sqrt(3))--(3,0)--(4,sqrt(3))--(5,0)); draw((-1.5,0)--(7.5,0)); [/asy] $ \textbf{(A)}\ 10\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 15\qquad\textbf{(D)}\ 10\sqrt{3}\qquad\textbf{(E)}\ 12\sqrt{3} $

In a tetrahedron, the midpoints of all the edges lie on the same sphere. Prove that it's altitudes intersect at one point.

Prove that for every square-free integer $n>1$, there exists a prime number $p$ and an integer $m$ satisfying \[ p \mid n \quad \text{and} \quad n \mid p^2+p\cdot m^p. \]

Byan has $999,998,\dots,2,1$ balls in $999$ bins from left to right, respectively. In one move, he selects two adjacent bins where the left bin has an even number of balls and the right bin has an odd number of balls and moves one ball from the left bin to the right bin. Byan keeps making moves until he is unable to. Find the sum of all possible numbers of balls that can be left in the bin that initially had $500$ balls after Byan is finished. [i]Tiebreaker #3[/i]

Let $\vartriangle ABC$ be a triangle with $AB = 5$, $BC = 8$, and, $CA = 7$. Let the center of the $A$-excircle be $O$, and let the $A$-excircle touch lines $BC$, $CA$, and,$ AB$ at points $X, Y$ , and, $Z$, respectively. Let $h_1$, $h_2$, and, $h_3$ denote the distances from $O$ to lines $XY$ , $Y Z$, and, ZX, respectively. If $h^2_1+ h^2_2+ h^2_3$ can be written as $\frac{m}{n}$ for relatively prime positive integers $m, n$, find $m + n$.

The radii $r$ and $R$ of two non-intersecting circles are given. The common internal tangents of these circles are perpendicular. Find the area of the triangle bounded by these tangents, as well as the common external tangents.

Sequences $x_n$ and $y_n$ satisfy the simultaneous relationships $x_k = x_{k+1} + y_{k+1}$ and $x_k > y_k$ for all $k \ge 1$. Furthermore, either $y_k = y_{k+1}$ or $y_k = x_{k+1}$. If $x_1 = 3 + \sqrt2$, $x_3 = 5 -\sqrt2$, and $y_1 = y_5$, evaluate $$(y_1)^2 + (y_2)^2 + (y_3)^2 + . . .$$

The triangle $ABC$ is such that $\angle BAC=30^{\circ},\angle ABC=45^{\circ}$. Prove that if $X$ lies on the ray $AC$, $Y$ lies on the ray $BC$ and $OX=BY$, where $O$ is the circumcentre of triangle $ABC$, then $S_{XY}$ passes through a fixed point. [i]Emil Kolev [/i]

Let $ A, B$ and $ C$ be non-collinear points. Prove that there is a unique point $ X$ in the plane of $ ABC$ such that \[ XA^2 \plus{} XB^2 \plus{} AB^2 \equal{} XB^2 \plus{} XC^2 \plus{} BC^2 \equal{} XC^2 \plus{} XA^2 \plus{} CA^2.\]

The six-pointed star in the figure is regular: all interior angles of the small triangles are equal. To each of the thirteen points marked are assigned a color: green or red. Prove that there will always be three points of the same color that are vertices of an equilateral triangle. [img]https://cdn.artofproblemsolving.com/attachments/b/f/c50a1f8cb81ea861f16a6a47c3b758c5993213.png[/img]