Let $n\ge 2$ be a positive integer. There are $n$ real coefficient polynomials $P_1(x),P_2(x),\cdots ,P_n(x)$ which is not all the same, and their leading coefficients are positive. Prove that $$\deg(P_1^n+P_2^n+\cdots +P_n^n-nP_1P_2\cdots P_n)\ge (n-2)\max_{1\le i\le n}(\deg P_i)$$ and find when the equality holds.

For real numbers $a, b$, and $c$ the polynomial $p(x) = 3x^7 - 291x^6 + ax^5 + bx^4 + cx^2 + 134x - 2$ has $7$ real roots whose sum is $97$. Find the sum of the reciprocals of those $7$ roots.

[u]Round 5[/u] [b]p13.[/b] Sally is at the special glasses shop, where there are many different optical lenses that distort what she sees and cause her to see things strangely. Whenever she looks at a shape through lens $A$, she sees a shape with $2$ more sides than the original (so a square would look like a hexagon). When she looks through lens $B$, she sees the shape with $3$ fewer sides (so a hexagon would look like a triangle). How many sides are in the shape that has $200$ more diagonals when looked at from lense $A$ than from lense $B$? [b]p14.[/b] How many ways can you choose $2$ cells of a $5$ by $5$ grid such that they aren't in the same row or column? [b]p15.[/b] If $a + \frac{1}{b} = (2015)^{-1}$ and $b + \frac{1}{a} = (2016)^2$ then what are all the possible values of $b$? [u]Round 6[/u] [b]p16.[/b] In Canadian football, linebackers must wear jersey numbers from $30 -35$ while defensive linemen must wear numbers from $33 -38$ (both intervals are inclusive). If a team has $5$ linebackers and $4$ defensive linemen, how many ways can it assign jersey numbers to the $9$ players such that no two people have the same jersey number? [b]p17.[/b] What is the maximum possible area of a right triangle with hypotenuse $8$? [b]p18.[/b] $9$ people are to play touch football. One will be designated the quarterback, while the other eight will be divided into two (indistinct) teams of $4$. How many ways are there for this to be done? [u]Round 7[/u] [b]p19.[/b] Express the decimal $0.3$ in base $7$. [b]p20.[/b] $2015$ people throw their hats in a pile. One at a time, they each take one hat out of the pile so that each has a random hat. What is the expected number of people who get their own hat? [b]p21.[/b] What is the area of the largest possible trapezoid that can be inscribed in a semicircle of radius $4$? [u]Round 8[/u] [b]p22.[/b] What is the base $7$ expression of $1211_3 \cdot 1110_2 \cdot 292_{11} \cdot 20_3$ ? [b]p23.[/b] Let $f(x)$ equal the ratio of the surface area of a sphere of radius $x$ to the volume of that same sphere. Let $g(x)$ be a quadratic polynomial in the form $x^2 + bx + c$ with $g(6) = 0$ and the minimum value of $g(x)$ equal to $c$. Express $g(x)$ as a function of $f(x)$ (e.g. in terms of $f(x)$). [b]p24.[/b] In the country of Tahksess, the income tax code is very complicated. Citizens are taxed $40\%$ on their first $\$20, 000$ and $45\%$ on their next $\$40, 000$ and $50\%$ on their next $\$60, 000$ and so on, with each $5\%$ increase in tax rate aecting $\$20, 000$ more than the previous tax rate. The maximum tax rate, however, is $90\%$. What is the overall tax rate (percentage of money owed) on $1$ million dollars in income? PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3157009p28696627]here [/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3158564p28715928]here[/url]. .Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Allen and Alan play a game. A nonconstant polynomial $P(x,y)$ with real coefficients and a positive integer $d$ greater than the degree of $P$ are known to both Allen and Alan. Alan thinks of a polynomial $Q(x,y)$ with real coefficients and degree at most $d$ and keeps it secret. Allen can make queries of the form $(s,t)$, where $s$ and $t$ are real numbers such that $P(s,t)\neq0$. Alan must respond with the value $Q(s,t)$. Allen's goal is to determine whether $P$ divides $Q$. Find (in terms of $P$ and $d$) the smallest positive integer, $g$, such that Allen can always achieve this goal making no more than $g$ queries. [i]Linus Tang[/i]

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]

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)$).

An eighth degree polynomial funtion $ y \equal{} ax^8 \plus{} bx^7 \plus{} cx^6 \plus{} dx^5 \plus{} ex^4 \plus{} fx^3 \plus{} gx^2\plus{}hx\plus{}i\ (a\neq 0)$ touches the line $ y \equal{} px \plus{} q$ at $ x \equal{} \alpha ,\ \beta ,\ \gamma ,\ \delta \ (\alpha < \beta < \gamma <\delta).$ Find the area of the region bounded by these graphs in terms of $ a,\ \alpha ,\ \beta ,\gamma ,\ \delta .$

Let $P_1,P_2,P_3,P_4$ be the graphs of four quadratic polynomials drawn in the coordinate plane. Suppose that $P_1$ is tangent to $P_2$ at the point $q_2,P_2$ is tangent to $P_3$ at the point $q_3,P_3$ is tangent to $P_4$ at the point $q_4$, and $P_4$ is tangent to $P_1$ at the point $q_1$. Assume that all the points $q_1,q_2,q_3,q_4$ have distinct $x$-coordinates. Prove that $q_1,q_2,q_3,q_4$ lie on a graph of an at most quadratic polynomial.

Consider a polynomial in $n$ variables with real coefficients. We know that if every variable is $\pm1$, the value of the polynomial is positive, or negative if the number of $-1$'s is even, or odd, respectively. Prove that the degree of this polynomial is at least $n$.

Let's call a polynomial [i]mixed[/i] if it has both positive and negative coefficients ($0$ isn't considered positive or negative). Is the product of two mixed polynomials always mixed? [i]Proposed by Vadym Koval[/i]

Will and Lucas are playing a game. Will claims that he has a polynomial $f$ with integer coefficients in mind, but Lucas doesn’t believe him. To see if Will is lying, Lucas asks him on minute $i$ for the value of $f(i)$, starting from minute $ 1$. If Will is telling the truth, he will report $f(i)$. Otherwise, he will randomly and uniformly pick a positive integer from the range $[1,(i+1)!]$. Now, Lucas is able to tell whether or not the values that Will has given are possible immediately, and will call out Will if this occurs. If Will is lying, say the probability that Will makes it to round $20$ is $a/b$. If the prime factorization of $b$ is $p_1^{e_1}... p_k^{e_k}$ , determine the sum $\sum_{i=1}^{k} e_i$.

Let $P(x,y)=x^2y+xy^2$ and $Q(x,y)=x^2+xy+y^2.$ For $n=1,2,3,\dots,$ let \begin{align*}F_n(x,y)=(x+y)^n-x^n-y^n \text{ and,}\\ G_n(x,y)=(x+y)^n+x^n+y^n. \end{align*} One observes that $$G_2=2Q, F_3=3P, G_4=2Q^2, F_5=5PQ, G_6=2Q^3+3P^2.$$ Prove that, in fact, for each $n$ either $F_n$ or $G_n$ is expressible as a polynomial in $P$ and $Q$ with integer coefficients.

The degrees of polynomials $P$ and $Q$ with real coefficients do not exceed $n$. These polynomials satisfy the identity \[ P(x) x^{n + 1} + Q(x) (x+1)^{n + 1} = 1. \] Determine all possible values of $Q \left( - \frac{1}{2} \right)$.

Sum of all roots of the equation $$cos^{100} x + a_1 cos^{99} x + a_2cos^{98} x +... + a_99 cos x+ a_{100} = 0$$, in interval $\left[\pi, \frac{3\pi}{2} \right]$, is equal to $21\pi$, and the sum of all roots of the equation $$sin^{100} x + a_1 sin^{99} x + a_2sin ^{98} x +... + a_99sin x+ a_{100} = 0$$, in the same interval, is equal to $24\pi $. How many roots does the first equation have on the segment $\left[ \frac{\pi}{2}, \pi\right]$?

Let $Q(x)$ be a cubic polynomial with integer coefficients. Suppose that a prime $p$ divides $Q(x_j)$ for $j = 1$ ,$2$ ,$3$ ,$4$ , where $x_1 , x_2 , x_3 , x_4$ are distinct integers from the set $\{0,1,\cdots, p-1\}$. Prove that $p$ divides all the coefficients of $Q(x)$.