Find the number of integer solutions to $||x| - 2020| < 5$.

Let $f\in\mathbb{Z}[X]$ be an irreducible polynomial over the ring of integer polynomials, such that $|f(0)|$ is not a perfect square. Prove that if the leading coefficient of $f$ is 1 (the coefficient of the term having the highest degree in $f$) then $f(X^2)$ is also irreducible in the ring of integer polynomials. [i]Mihai Piticari[/i]

Let $n$ be a positive integer and let $(a_1,a_2,\ldots ,a_{2n})$ be a permutation of $1,2,\ldots ,2n$ such that the numbers $|a_{i+1}-a_i|$ are pairwise distinct for $i=1,\ldots ,2n-1$. Prove that $\{a_2,a_4,\ldots ,a_{2n}\}=\{1,2,\ldots ,n\}$ if and only if $a_1-a_{2n}=n$.

Determine the sum of absolute values for the complex roots of $ 20 x^8 \plus{} 7i x^7 \minus{}7ix \plus{} 20.$

Calculate the following indefinite integrals. [1] $\int (x^2+4x-3)^2(x+2)dx$ [2] $\int \frac{\ln x}{x(\ln x+1)}dx$ [3] $\int \frac{\sin \ (\pi \log _2 x)}{x}dx$ [4] $\int \frac{dx}{\sin x\cos ^ 2 x}$ [5] $\int \sqrt{1-3x}\ dx$

Number $ 0$ is written on the board. Two players alternate writing signs and numbers to the right, where the first player always writes either $ \plus{}$ or $ \minus{}$ sign, while the second player writes one of the numbers $ 1, 2, ... , 1993$,writing each of these numbers exactly once. The game ends after $ 1993$ moves. Then the second player wins the score equal to the absolute value of the expression obtained thereby on the board. What largest score can he always win?

Let $S_n=1-2+3-4+\cdots +(-1)^{n-1}n,\ n=1, 2, \cdots$. Then $S_{17}+S_{33}+S_{50}$ equals: $\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ -1 \qquad\textbf{(E)}\ -2$

For all $x>0$ prove $$\frac{\sin^2x-x}{\ln\left(\frac{\sin^2x}x\right)^{\sqrt x}}+\frac{\cos^2x-x}{\ln\left(\frac{\cos^2x}x\right)^{\sqrt x}}>|\sin x|+|\cos x|$$ [i]Proposed by Pirkulyiev Rovsen[/i]

Determine the lowest positive integer n such that following statement is true: If polynomial with integer coefficients gets value 2 for n different integers, then it can't take value 4 for any integer.

Find all triples $(x, y, z)$ of real numbers that satisfy $x^2 + y^2 + z^2 + 1 = xy + yz + zx +|x - 2y + z|$.

Let $f : \mathbb{C} \to \mathbb{C} $ be defined by $ f(z) = z^2 + iz + 1 $. How many complex numbers $z $ are there such that $ \text{Im}(z) > 0 $ and both the real and the imaginary parts of $f(z)$ are integers with absolute value at most $ 10 $? ${ \textbf{(A)} \ 399 \qquad \textbf{(B)} \ 401 \qquad \textbf{(C)} \ 413 \qquad \textbf{(D}} \ 431 \qquad \textbf{(E)} \ 441 $

Let $ a,b,c\in\mathbb{C}^* $ pairwise distinct, having the same absolute value, and satisfying: $$ a^2+b^2+c^2-ab-bc-ca=0. $$ Prove that $ a,b,c $ represents the affixes of the vertices of a right or equilateral triangle.

Prove that there exists a nonzero complex number $c$ and a real number $d$ such that \[\left|\left|\dfrac1{1+z+z^2}\right|-\left|\dfrac1{1+z+z^2}-c\right|\right|=d\] for all $z$ with $|z|=1$ and $1+z+z^2\neq 0$. (Here, $|z|$ denotes the absolute value of the complex number $z$, so that $|a+bi|=\sqrt{a^2+b^2}$ for real numbers $a,b$.)

Let $f(x)$ be a third-degree polynomial with real coefficients satisfying \[|f(1)|=|f(2)|=|f(3)|=|f(5)|=|f(6)|=|f(7)|=12.\] Find $|f(0)|$.

The integers $-1$, $2$, $-3$, $4$, $-5$, $6$ are written on a blackboard. At each move, we erase two numbers $a$ and $b$, then we re-write $2a+b$ and $2b+a$. How many of the sextuples $(0,0,0,3,-9,9)$, $(0,1,1,3,6,-6)$, $(0,0,0,3,-6,9)$, $(0,1,1,-3,6,-9)$, $(0,0,2,5,5,6)$ can be gotten? $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 5 $

Prove that if $|x| < 1$ and $|y| < 1$, then $\left|\frac{x - y}{1 -xy}\right|< 1$.

Let $a_1,a_2,a_3,\dots$ be a sequence of real numbers satisfying the inequality \[ |a_{k+m}-a_k-a_m| \leq 1 \quad \text{for all} \ k,m \in \mathbb{Z}_{>0}. \] Show that the following inequality holds for all positive integers $k,m$ \[ \left| \frac{a_k}{k}-\frac{a_m}{m} \right| < \frac{1}{k}+\frac{1}{m}. \]

Let $n \geq 5$ be a positive integer and let $A$ and $B$ be sets of integers satisfying the following conditions: i) $|A| = n$, $|B| = m$ and $A$ is a subset of $B$ ii) For any distinct $x,y \in B$, $x+y \in B$ iff $x,y \in A$ Determine the minimum value of $m$.

Given are $10$ distinct real numbers. Kyle wrote down the square of the difference for each pair of those numbers in his notebook, while Peter wrote in his notebook the absolute value of the differences of the squares of these numbers. Is it possible for the two boys to have the same set of $45$ numbers in their notebooks?

In a $ 19\times 19$ board, a piece called [i]dragon[/i] moves as follows: It travels by four squares (either horizontally or vertically) and then it moves one square more in a direction perpendicular to its previous direction. It is known that, moving so, a dragon can reach every square of the board. The [i]draconian distance[/i] between two squares is defined as the least number of moves a dragon needs to move from one square to the other. Let $ C$ be a corner square, and $ V$ the square neighbor of $ C$ that has only a point in common with $ C$. Show that there exists a square $ X$ of the board, such that the draconian distance between $ C$ and $ X$ is greater than the draconian distance between $ C$ and $ V$.

Given real numbers $x,y,z$ which satisfies $$|x+y+z|+|xy+yz+zx|+|xyz| \le 1$$ Show that $max\{ |x|,|y|,|z|\} \le 1$.

The set of all real solutions of the inequality \[ |x \minus{} 1| \plus{} |x \plus{} 2| < 3\] is $ \textbf{(A)}\ x \in ( \minus{} 3,2) \qquad \textbf{(B)}\ x \in ( \minus{} 1,2) \qquad \textbf{(C)}\ x \in ( \minus{} 2,1) \qquad$ $ \textbf{(D)}\ x \in \left( \minus{} \frac32,\frac72\right) \qquad \textbf{(E)}\ \O \text{ (empty})$ Note: I updated the notation on this problem.

Let $P(x), Q(x), R(x)$ be polynomials with integer coefficients, such that $P(x) = Q(x)R(x)$. Let's denote by $a$ and $b$ the largest absolute values of coefficients of $P, Q$ correspondingly. Does $b \le 2023a$ always hold? [i]Proposed by Dmytro Petrovsky[/i]

Let $A\subset \{x|0\le x<1\}$ with the following properties: 1. $A$ has at least 4 members. 2. For all pairwise different $a,b,c,d\in A$, $ab+cd\in A$ holds. Prove: $A$ has infinetly many members.

Given integer $n \geq 3$ and $x_1$, $x_2$, …, $x_n$ be $n$ real numbers satisfying $|x_1|+|x_2|+…+|x_n|=1$. Find the minimum of \[|x_1+x_2|+|x_2+x_3|+…+|x_{n-1}+x_n|+|x_n+x_1|.\] [i]Proposed by snap7822[/i]