Let $p,q$ be two distinct odd primes. Calculate $\displaystyle \sum_{j=1}^{\frac{p-1}{2}}\left \lfloor \frac{qj}{p}\right \rfloor +\sum_{j=1}^{\frac{q-1}{2}}\left \lfloor \frac{pj}{q}\right\rfloor$.

Let $P(x)$ be given a polynomial of degree 4, having 4 positive roots. Prove that the equation \[(1-4 \cdot x) \cdot \frac{P(x)}{x^2} + (x^2 + 4 \cdot x - 1) \cdot \frac{P'(x)}{x^2} - P''(x) = 0\] has also 4 positive roots.

Prove that there is no polynomial $P(x)$ with real coefficients that satisfies $$P'(x)P''(x)>P(x)P'''(x)\qquad\text{for all }x\in\mathbb R.$$Is this statement true for all of the thrice differentiable real functions?

Let $ p(x)$ be a cubic polynomial with rational coefficients. $ q_1$, $ q_2$, $ q_3$, ... is a sequence of rationals such that $ q_n \equal{} p(q_{n \plus{} 1})$ for all positive $ n$. Show that for some $ k$, we have $ q_{n \plus{} k} \equal{} q_n$ for all positive $ n$.

There exists a polynomial $P$ of degree $5$ with the following property: if $z$ is a complex number such that $z^5+2004z=1$, then $P(z^2)=0$. Calculate the quotient $\tfrac{P(1)}{P(-1)}$.

Does there exist a two-variable polynomial $P(x, y)$ with real number coefficients such that $P(x, y)$ is positive exactly when $x$ and $y$ are both positive?

[b]p1.[/b] Find the last digit of $2022^{2022}$. [b]p2.[/b] Let $a_1 < a_2 <... < a_8$ be eight real numbers in an increasing arithmetic progression. If $a_1 + a_3 + a_5 + a_7 = 39$ and $a_2 + a_4 + a_6 + a_8 = 40$, determine the value of $a_1$. [b]p3.[/b] Patrick tries to evaluate the sum of the first $2022$ positive integers, but accidentally omits one of the numbers, $N$, while adding all of them manually, and incorrectly arrives at a multiple of $1000$. If adds correctly otherwise, find the sum of all possible values of $N$. [b]p4.[/b] A machine picks a real number uniformly at random from $[0, 2022]$. Andrew randomly chooses a real number from $[2020, 2022]$. The probability that Andrew’s number is less than the machine’s number is $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [b]p5.[/b] Let $ABCD$ be a square and $P$ be a point inside it such that the distances from $P$ to sides $AB$ and $AD$ respectively are $2$ and $4$, while $PC = 6$. If the side length of the square can be expressed in the form $a +\sqrt{b}$ for positive integers $a, b$, then determine $a + b$. [b]p6.[/b] Positive integers $a_1, a_2, ..., a_{20}$ sum to $57$. Given that $M$ is the minimum possible value of the quantity $a_1!a_2!...a_{20}!$, find the number of positive integer divisors of $M$. [b]p7.[/b] Jessica has $16$ balls in a box, where $15$ of them are red and one is blue. Jessica draws balls out the box three at a time until one of the three is blue. If she ever draws three red marbles, she discards one of them and shuffles the remaining two back into the box. The expected number of draws it takes for Jessica to draw the blue ball can be written as a common fraction $\frac{m}{n}$ where $m, n$ are relatively prime positive integers. Find $m + n$. [b]p8.[/b] The Lucas sequence is defined by these conditions: $L_0 = 2$, $L_1 = 1$, and $L_{n+2} =L_{n+1} +L_n$ for all $n \ge 0$. Determine the remainder when $L^2_{2019} +L^2_{2020}$ is divided by $L_{2023}$. [b]p9.[/b] Let $ABCD$ be a parallelogram. Point $P$ is selected in its interior such that the distance from $P$ to $BC$ is exactly $6$ times the distance from $P$ to $AD$, and $\angle APB = \angle CPD = 90^o$. Given that $AP = 2$ and $CP = 9$, the area of $ABCD$ can be expressed as $m\sqrt{n}$ where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Find $m + n$. [b]p10.[/b] Consider the polynomial $P(x) = x^{35} + ... + x + 1$. How many pairs $(i, j)$ of integers are there with $0 \le i < j \le 35$ such that if we flip the signs of the $x^i$ and $x^j$ terms in $P(x)$ to form a new polynomial $Q(x)$, then there exists a nonconstant polynomial $R(x)$ with integer coefficients dividing both $P(x)$ and $Q(x)$? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Given the equation $x^2+ax+1=0$, determine: a) The interval of possible values for $a$ where the solutions to the previous equation are not real. b) The loci of the roots of the polynomial, when $a$ is in the previous interval.

Find the coefficient of $x^2$ in the following polynomial $$(1 -x)^2(1 + 2x)^2(1 - 3x)^2... (1 -11x)^2.$$

Does there exist a finite set $M$ of nonzero real numbers, such that for any natural number $n$ a polynomial of degree no less than $n$ with coeficients in $M$, all of whose roots are real and belong to $M$? [i]E. Malinnikova[/i]

Let $P(X)$ be a polynomial with real coefficients such that $\{P(n)\}\leq\frac{1}{n}, \forall n\in\mathbb{N}$, where $\{a\}$ is the fractional part of the number $a$. Show that $P(n)\in\mathbb{Z}, \forall n\in\mathbb{N}$.

There is a unique degree-$10$ monic polynomial with integer coefficients $f(x)$ such that $$f \left( \sum^9_{j=0}\sqrt[10]{2021^j}\right)= 0.$$ Find the remainder when $f(1)$ is divided by $1000$.

Alice, Bob, and Carol each independently roll a fair six-sided die and obtain the numbers $a, b, c$, respectively. They then compute the polynomial $f(x)=x^{3}+p x^{2}+q x+r$ with roots $a, b, c$. If the expected value of the sum of the squares of the coefficients of $f(x)$ is $\frac{m}{n}$ for relatively prime positive integers $m, n$, find the remainder when $m+n$ is divided by 1000 .

Consider polynomial functions $ax^2 -bx +c$ with integer coefficients which have two distinct zeros in the open interval $(0,1).$ Exhibit with proof the least positive integer value of $a$ for which such a polynomial exists.

Let $\triangle ABC$ be an equilateral triangle with side length $s$ and $P$ a point in the interior of this triangle. Suppose that $PA$, $PB$, and $PC$ are the roots of the polynomial $t^3-18t^2+91t-89$. Then $s^2$ can be written in the form $m+\sqrt n$ where $m$ and $n$ are positive integers. Find $100m+n$. [i]Proposed by David Altizio[/i]