Express \[ x^{13} + \frac{1}{x^{13}} \] as a polynomial in $y = x + \frac{1}{x}$.

a) Show that $\forall n \in \mathbb{N}_0, \exists A \in \mathbb{R}^{n\times n}: A^3=A+I$. b) Show that $\det(A)>0, \forall A$ fulfilling the above condition.

Let \[ f(x) = \sum_{i=0}^{i=n} a_i x^{n - i}\] be a polynomial of degree $n$ with integral coefficients. If $a_0, a_n,$ and $f(1)$ are odd, prove that $f(x) = 0$ has no rational roots.

The polynomial $Q(x)=x^3-21x+35$ has three different real roots. Find real numbers $a$ and $b$ such that the polynomial $x^2+ax+b$ cyclically permutes the roots of $Q$, that is, if $r$, $s$ and $t$ are the roots of $Q$ (in some order) then $P(r)=s$, $P(s)=t$ and $P(t)=r$.

Let $P(x)$ be a polynomial with real coefficients. Prove that there exist positive integers $n$ and $k$ such that $k$ has $n$ digits and more than $P(n)$ positive divisors.

$P(x)$ is polynomial with degree $n\geq 2$ and nonnegative coefficients. $a,b,c$ - sides for some triangle. Prove, that $\sqrt[n]{P(a)},\sqrt[n]{P(b)},\sqrt[n]{P(c)}$ are sides for some triangle too.

Determine all pairs of positive integers $ (m,n)$ such that $ (1\plus{}x^n\plus{}x^{2n}\plus{}\cdots\plus{}x^{mn})$ is divisible by $ (1\plus{}x\plus{}x^2\plus{}\cdots\plus{}x^{m})$.

Let $n$ be a positive integer. Suppose $S$ is a set of ordered $n-\mbox{tuples}$ of nonnegative integers such that, whenever $(a_1,\dots,an)\in S$ and $b_i$ are nonnegative integers with$b_i\le a_i$, the $n-\text{tuple}$ $(b_1,\dots,b_n)$ is also in $S$. If $h_m$ is the number of elements of $S$ with the sum of components equal to$m$, prove that $h_m$ is a polynomial in $m$ for all sufficiently large$m$.

[b]p1.[/b] Len's Spanish class has four tests in the first term. Len scores $72$, $81$, and $78$ on the first three tests. If Len wants to have an 80 average for the term, what is the minimum score he needs on the last test? [b]p2.[/b] In $1824$, the Electoral College had $261$ members. Andrew Jackson won $99$ Electoral College votes and John Quincy Adams won $84$ votes. A plurality occurs when no candidate has more than $50\%$ of the votes. Should a plurality occur, the vote goes to the House of Representatives to break the tie. How many more votes would Jackson have needed so that a plurality would not have occurred? [b]p3.[/b] $\frac12 + \frac16 + \frac{1}{12} + \frac{1}{20} + \frac{1}{30}= 1 - \frac{1}{n}$. Find $n$. [b]p4.[/b] How many ways are there to sit Samuel, Esun, Johnny, and Prat in a row of $4$ chairs if Prat and Johnny refuse to sit on an end? [b]p5.[/b] Find an ordered quadruple $(w, x, y, z)$ that satisfies the following: $$3^w + 3^x + 3^y = 3^z$$ where $w + x + y + z = 2017$. [b]p6.[/b] In rectangle $ABCD$, $E$ is the midpoint of $CD$. If $AB = 6$ inches and $AE = 6$ inches, what is the length of $AC$? [b]p7.[/b] Call an integer interesting if the integer is divisible by the sum of its digits. For example, $27$ is divisible by $2 + 7 = 9$, so $27$ is interesting. How many $2$-digit interesting integers are there? [b]p8.[/b] Let $a\#b = \frac{a^3-b^3}{a-b}$ . If $a, b, c$ are the roots of the polynomial $x^3 + 2x^2 + 3x + 4$, what is the value of $a\#b + b\#c + c\#a$? [b]p9.[/b] Akshay and Gowri are examining a strange chessboard. Suppose $3$ distinct rooks are placed into the following chessboard. Find the number of ways that one can place these rooks so that they don't attack each other. Note that two rooks are considered attacking each other if they are in the same row or the same column. [img]https://cdn.artofproblemsolving.com/attachments/f/1/70f7d68c44a7a69eb13ce12291c0600d11027c.png[/img] [b]p10.[/b] The Earth is a very large sphere. Richard and Allen have a large spherical model of Earth, and they would like to (for some strange reason) cut the sphere up with planar cuts. If each cut intersects the sphere, and Allen holds the sphere together so it does not fall apart after each cut, what is the maximum number of pieces the sphere can be cut into after $6$ cuts? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $f$ be a polynomial with real coefficients such that for each positive integer n the equation $f(x) = n$ has at least one rational solution. Find $f$.

Find the self-intersection index of the surface $x^4+y^4=1$ in the projective plane $\text{CP}^2$.

Let $T$ be a finite set of squarefree integers. (a) Show that there exists an integer polynomial $P(x)$ such that the set of squarefree numbers in the range of $P(n)$ across all $n \in \mathbb{Z}$ is exactly $T$. (b) Suppose that $T$ is allowed to be infinite. Is it still true that for all choices of $T$, such an integer polynomial $P(x)$ exists? [i]Allen Wang[/i]

Math teacher wrote in a table a polynomial $P(x)$ with integer coefficients and he said: "Today my daughter have a birthday.If in polynomial $P(x)$ we have $x=a$ where $a$ is the age of my daughter we have $P(a)=a$ and $P(0)=p$ where $p$ is a prime number such that $p>a$." How old is the daughter of math teacher?

Let $p(x)$ be a monic integer polynomial of degree $n$ that has $n$ real roots, $\alpha_1,\alpha_2,\ldots, \alpha_n$. Let $q(x)$ be an arbitrary integer polynomial that is relatively prime to polynomial $p(x)$. Prove that \[\sum_{i=1}^n \left|q(\alpha_i)\right|\ge n.\] [i]Submitted by Dávid Matolcsi, Berkeley[/i]

A line passing through $(20,21)$ intersects the curve $y = x^3-2x^2-3x+5$ at three distinct points $A, B,$ and $C,$ such that $B$ is the midpoint of $\overline{AC}$. The slope of this line is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.