Find all polynomials $P(x)$ of the smallest possible degree with the following properties: (i) The leading coefficient is $200$; (ii) The coefficient at the smallest non-vanishing power is $2$; (iii) The sum of all the coefficients is $4$; (iv) $P(-1) = 0, P(2) = 6, P(3) = 8$.

Consider solutions to the equation \[x^2-cx+1 = \dfrac{f(x)}{g(x)},\] where $f$ and $g$ are polynomials with nonnegative real coefficients. For each $c>0$, determine the minimum possible degree of $f$, or show that no such $f,g$ exist. [i]Proposed by Linus Hamilton and Calvin Deng[/i]

Find all pairs of polynomials $p(x)$ and $q(x)$ with real coefficients for which \[p(x)q(x+1)-p(x+1)q(x)=1.\]

Show that the equation \[x(x+1)(x+2)\dots (x+2020)-1=0\] has exactly one positive solution $x_0$, and prove that this solution $x_0$ satisfies \[\frac{1}{2020!+10}<x_0<\frac{1}{2020!+6}.\]

Let $ f(x) $ and $ g(x) $ be polynomials with integer coefficients. Prove that if for every integer value $ n $ the number $ g(n) $ is divisible by the number $ f(n) $, then $ g(x) = f(x)\cdot h(x) $, where $ h(x) $ is a polynomial,. Show with an example that the coefficients of the polynomial $ h(x) $ do not have to be integer.

Let $u, v, w$ be the roots of $x^3 -5x^2 + 4x-3 = 0$. Find a cubic polynomial having $u^3, v^3, w^3$ as roots.

Find all polynomials $f(x)$ with integer coefficients and leading coefficient equal to 1, for which $f(0)=2010$ and for each irrational $x$, $f(x)$ is also irrational.

Find the number of quadruples $(a,b,c,d)$ of integers which satisfy both \begin{align*}\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} &= \frac{1}{2}\qquad\text{and}\\\\2(a+b+c+d) &= ab + cd + (a+b)(c+d) + 1.\end{align*}

Find the number of second-degree polynomials $ f(x)$ with integer coefficients and integer zeros for which $ f(0)\equal{}2010$.

(a) Let $n \geq 1$ be an integer. Prove that $X^n+Y^n+Z^n$ can be written as a polynomial with integer coefficients in the variables $\alpha=X+Y+Z$, $\beta= XY+YZ+ZX$ and $\gamma = XYZ$. (b) Let $G_n=x^n \sin(nA)+y^n \sin(nB)+z^n \sin(nC)$, where $x,y,z, A,B,C$ are real numbers such that $A+B+C$ is an integral multiple of $\pi$. Using (a) or otherwise show that if $G_1=G_2=0$, then $G_n=0$ for all positive integers $n$.

28. [b][15][/b] Find the shortest distance between the lines $\frac{x+2}{2}=\frac{y-1}{3}=\frac{z}{1}$ and $\frac{x-3}{-1}=\frac{y}{1}=\frac{z+1}{2}$ 29. [b][15][/b] Find the largest real number $k$ such that there exists a sequence of positive reals ${a_i}$ for which $\sum_{n=1}^{\infty}a_n$ converges but $\sum_{n=1}^{\infty}\frac{\sqrt{a_n}}{n^k}$ does not. 30. [b][15][/b] Find the largest integer $n$ such that the following holds: there exists a set of $n$ points in the plane such that, for any choice of three of them, some two are unit distance apart. 31. [b][17][/b] Two random points are chosen on a segment and the segment is divided at each of these two points. Of the three segments obtained, find the probability that the largest segment is more than three times longer than the smallest segment. 32. [b][17][/b] Find the sum of all positive integers $n\le 2015$ that can be expressed in the form $\left\lceil{\frac{x}{2}}\right \rceil +y+xy$, where $x$ and $y$ are positive integers. 33. [b][17][/b] How many ways are there to place four points in the plane such that the set of pairwise distances between the points consists of exactly $2$ elements? (Two configurations are the same if one can be obtained from the other via rotation and scaling.) 34. [b][20][/b] Let $n$ be the second smallest integer that can be written as the sum of two positive cubes in two different ways. Compute $n$. If your guess is $a$, you will receive $\max(25-5\cdot \max(\frac{a}{n},\frac{n}{a}),0)$, rounded up. 35. [b][20][/b] Let $n$ be the smallest positive integer such that any positive integer can be expressed as the sum of $n$ integer 2015th powers. Find $n$. If your answer is $a$, your score will be $\max(20-\frac{1}{5}|\log _{10} \frac{a}{n}|,0)$, rounded up. 36. [b][20][/b] Consider the following seven false conjectures with absurdly high counterexamples. Pick any subset of them, and list their labels in order of their smallest counterexample (the smallest $n$ for which the conjecture is false) from smallest to largest. For example, if you believe that the below list is already ordered by counterexample size, you should write ”PECRSGA”. - [b]P.[/b] (Polya’s conjecture) For any integer $n$, at least half of the natural numbers below $n$ have an odd number of prime factors. - [b]E.[/b] (Euler’s conjecture) There is no perfect cube $n$ that can be written as the sum of three positive cubes. - [b]C.[/b] (Cyclotomic) The polynomial with minimal degree whose roots are the primitive $n$th roots of unity has all coefficients equal to $-1$, $0$, or $1$. - [b]R.[/b] (Prime race) For any integer $n$, there are more primes below $n$ equal to $2(\mod 3)$ than there are equal to $1 (\mod 3)$. - [b]S.[/b] (Seventeen conjecture) For any integer $n$, $n^{17} + 9$ and $(n + 1)^{17} + 9$ are relatively prime. - [b]G.[/b] (Goldbach’s (other) conjecture) Any odd composite integer $n$ can be written as the sum of a prime and twice a square. - [b]A.[/b] (Average square) Let $a_1 = 1$ and $a_{k+1}=\frac{1+a_1^2+a_2^2+...+a_k^2}{k}$. Then $a_n$ is an integer for any n. If your answer is a list of $4\le n\le 7$ labels in the correct order, your score will be $(n-2)(n-3)$. Otherwise, your score will be $0$.

For a prime $p$, let $\mathbb{F}_p$ denote the integers modulo $p$, and let $\mathbb{F}_p[x]$ be the set of polynomials with coefficients in $\mathbb{F}_p$. Find all $p$ for which there exists a quartic polynomial $P(x) \in \mathbb{F}_p[x]$ such that for all integers $k$, there exists some integer $\ell$ such that $P(\ell) \equiv k \pmod p$. (Note that there are $p^4(p-1)$ quartic polynomials in $\mathbb{F}_p[x]$ in total.) [i]Aprameya Tripathy[/i]

[b]p1.[/b] In the picture below, the two parallel cuts divide the square into three pieces of equal area. The distance between the two parallel cuts is $d$. The square has length $s$. Find and prove a formula that expresses $s$ as a function of $d$. [img]https://cdn.artofproblemsolving.com/attachments/c/b/666074d28de50cdbf338a2c667f88feba6b20c.png[/img] [b]p2.[/b] Let $S$ be a subset of $\{1, 2, 3, . . . 10, 11\}$. We say that $S$ is lucky if no two elements of $S$ differ by $4$ or $7$. (a) Give an example of a lucky set with five elements. (b) Is it possible to find a lucky set with six elements? Explain why or why not.[/quote] [b]p3.[/b] Find polynomials $p(x)$ and $q(x)$ with real coefficients such that (a) $p(x) - q(x) = x^3 + x^2 - x - 1$ for all real $x$, (b) $p(x) > 0$ for all real $x$, (c) $q(x) > 0$ for all real $x$. [b]p4.[/b] A permutation on $\{1, 2, 3, …, n\}$ is a rearrangement of the symbols. For example $32154$ is a permutation on $\{1, 2, 3, 4, 5\}$. Given a permutation $a_1a_2a_3…a_n$, an inversion is a pair of $a_i$ and $a_j$ such that $a_i > a_j$ but $i < j$. For example, $32154$ has $4$ inversions. Suppose you are only allowed to exchange adjacent symbols. For any permutation, show that the minimum number of exchanges required to put all the symbols in their natural positions (that is, $123 …n$) is the number of inversions. [b]p5.[/b] We say a number $N$ is a nontrivial sum of consecutive positive integers if it can be written as the sum of $2$ or more consecutive positive integers. What is the set of numbers from $1000$ to $2000$ that are NOT nontrivial sums of consecutive positive integers? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all monic polynomials $P,Q$ which are non-constant, have real coefficients and they satisfy $2P(x)=Q(\frac{(x+1)^2}{2})-Q(\frac{(x-1)^2}{2})$ and $P(1)=1$ for all real $x$.

Let $P,Q,R$ be polynomials of degree at least $1$ with integer coefficients such that for any real number $x$ holds: $P(Q(x))\equal{}Q(R(x))\equal{}R(P(x))$. Show that the polynomials $P,Q,R$ are equal.