Determine $3x_4+2x_5$ if $x_1$, $x_2$, $x_3$, $x_4$, and $x_5$ satisfy the system of equations below. \[ \begin{array}{l} 2x_1+x_2+x_3+x_4+x_5=6 \\ x_1+2x_2+x_3+x_4+x_5=12 \\ x_1+x_2+2x_3+x_4+x_5=24 \\ x_1+x_2+x_3+2x_4+x_5=48 \\ x_1+x_2+x_3+x_4+2x_5=96 \\ \end{array} \]

For each polynomial $Q(x)$ let $M(Q)$ be the set of non-negative integers $x$ with $0 < Q(x) < 2004.$ We consider polynomials $P_n(x)$ of the form \[P_n(x) = x^n + a_1 \cdot x^{n-1} + \ldots + a_{n-1} \cdot x + 1\] with coefficients $a_i \in \{ \pm1\}$ for $i = 1, 2, \ldots, n-1.$ For each $n = 3^k, k > 0$ determine: a.) $m_n$ which represents the maximum of elements in $M(P_n)$ for all such polynomials $P_n(x)$ b.) all polynomials $P_n(x)$ for which $|M(P_n)| = m_n.$

Let $f:\mathbb{R}\to\mathbb{R}$ be a convex function. a) Prove that $f$ is continous; b) Prove that there exists an unique function $g:[0,\infty)\to\mathbb{R}$ such that for all $x\geq 0$ we have \[ f(x+g(x)) = f(g(x)) - g(x) . \]

Solve the following system in positive numbers $\begin{cases} x+y\le 1 \\ \frac{2}{xy} +\frac{1}{x^2+y^2}=10\end{cases}$

Let $ \left( a_n \right)_{n\ge 1} $ be a non-constant arithmetic progression of positive numbers and $ \left( g_n \right)_{n\ge 1} $ be a non-constant geometric progression of positive numbers satisfying $ a_1=g_1 $ and $ a_{2019} =g_{2019} . $ Specify the set $ \left\{ k\in\mathbb{N} \big| a_k\le g_k \right\} $ and prove that it bijects the natural numbers. [i]Gheorghe Rotariu[/i]

Let $a, b, c$, be real numbers such that $$a^2b^2 + 18 abc > 4b^3+4a^3c+27c^2 .$$ Prove that $a^2>3b$.

Determine whether the polynomial $P(x)=(x^2-2x+5)(x^2-4x+20)+1$ is irreducible over $\mathbb{Z}[X]$.

Positive integer $l$ and positive real numbers $a_1, a_2, ..., a_l$ are given. For every positive integer $n$ we define $$c_n=\sum_{k_1+k_2+...+k_l=n}\frac{(2n)!}{(2k_1)!(2k_2)!...(2k_l)!}a_1^{k_1}a_2^{k_2}...a_l^{k_l}.$$ Prove that for every positive integer $n$ the inequality $\sqrt[n]{c_n}\leq \sqrt[n+1]{c_{n+1}}$ holds.

Prove that for any positive integer $ m$ there exist an infinite number of pairs of integers $(x,y)$ such that $(\text{i})$ $x$ and $y$ are relatively prime; $(\text{ii})$ $x$ divides $y^2+m;$ $(\text{iii})$ $y$ divides $x^2+m.$

[b]p1.[/b] a) Given four points in the plane, no three of which lie on the same line, each subset of three points determines the vertices of a triangle. Can all these triangles have equal areas? If so, give an example of four points (in the plane) with this property, and then describe all arrangements of four joints (in the plane) which permit this. If no such arrangement exists, prove this. b) Repeat part a) with "five" replacing "four" throughout. [b]p2.[/b] Three people at the base of a long stairway begin a race up the stairs. Person A leaps five steps with each stride (landing on steps $5$, $10$, $15$, etc.). Person B leaps a little more slowly but covers six steps with each stride. Person C leaps seven steps with each stride. A picture taken near the end of the race shows all three landing simultaneously, with Person A twenty-one steps from the top, person B seven steps from the top, and Person C one step from the top. How many steps are there in the stairway? If you can find more than one answer, do so. Justify your answer. [b]p3. [/b]Let $S$ denote the sum of an infinite geometric series. Suppose the sum of the squares of the terms is $2S$, and that df the cubes is $64S/13$. Find the first three terms of the original series. [b]p4.[/b] $A$, $B$ and $C$ are three equally spaced points on a circular hoop. Prove that as the hoop rolls along the horizontal line $\ell$, the sum of the distances of the points $A, B$, and $C$ above line $\ell$ is constant. [img]https://cdn.artofproblemsolving.com/attachments/3/e/a1efd0975cf8ff3cf6acb1da56da1dce35d81e.png[/img] [b]p5.[/b] A set of $n$ numbers $x_1,x_2,x_3,...,x_n$ (where $n>1$) has the property that the $k^{th}$ number (that is, $x_k$ ) is removed from the set, the remaining $(n-1)$ numbers have a sum equal to $k$ (the subscript o $x_k$ ), and this is true for each $k = 1,2,3,...,n$. a) SoIve for these $n$ numbers b) Find whether at least one of these $n$ numbers can be an integer. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

For a polynomial $P(x)$ with integer coefficients and a prime $p$, if there is no $n \in \mathbb{Z}$ such that $p|P(n)$, we say that polynomial $P$ [i]excludes[/i] $p$. Is there a polynomial with integer coefficients such that having degree of 5, excluding exactly one prime and not having a rational root?

Determine all polynomials P(x) with real coefficients such that [(x + 1)P(x − 1) − (x − 1)P(x)] is a constant polynomial.

Consider the following $50$-term sums: $S=\frac{1}{1\cdot 2}+\frac{1}{3\cdot 4}+...+\frac{1}{99\cdot 100}$, $T=\frac{1}{51\cdot 100}+\frac{1}{52\cdot 99}+...+\frac{1}{99\cdot 52}+\frac{1}{100\cdot 51}$. Express $\frac{S}{T}$ as an irreducible fraction.

Find all functions $f: \mathbb{R}\rightarrow \mathbb{R}$ such that \[ f(xy-1) + f(x)f(y) = 2xy-1 \] for all x and y

[b]a)[/b] Prove that for every natural numbers $n$ and $k$, we have monic polynomials of degree $n$, with integer coefficients like $A=\{P_1(x),.....,P_k(x)\}$ such that no two of them have a common factor and for every subset of $A$, the sum of elements of $A$ has all its roots real. [b]b)[/b] Are there infinitely many monic polynomial of degree $n$ with integer coefficients like $P_1(x),P_2(x),....$ such that no two of them have a common factor and the sum of a finite number of them has all it's roots real? [i]proposed by Mohammad Mansouri[/i]

Suppose that the roots $p,q$ of the equation $x^2-x+c=0$ where $c \in \mathbb{R}$, are rational numbers. Prove that the roots of the equation $x^2+px-q=0$ are also rational numbers.

One member of an infinite arithmetic sequence in the set of natural numbers is a perfect square. Show that there are infinitely many members of this sequence having this property.

Given positive numbers $a,b,c,d$. Prove that the set of inequalities $$a+b<c+d$$ $$(a+b)(c+d)<ab+cd$$ $$(a+b)cd<ab(c+d)$$ contain at least one wrong.

Let $\mathbb R^+$ denote the set of positive real numbers.Find all functions $f:\mathbb R^+\rightarrow \mathbb R^+$ satisfying the following equation for all $x,y\in \mathbb R^+$: $$f(xy+f(y)^2)=f(x)f(y)+yf(y)$$

Do there exist polynomials $p(x), q(x), r(x)$ whose coefficients are positive integers such that $p(x) = (x^{2}-3x+3) q(x)$ and $q(x) = (\frac{x^{2}}{20}-\frac{x}{15}+\frac{1}{12}) r(x)$?

Find all functions $f : R \to R$ satisfying the condition $f(x- f(y)) = 1+x-y$ for all $x,y \in R$.

Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for any $(x, y)$ real numbers we have $f(xf(y)+f(x))+f(y^2)=f(x)+yf(x+y)$

Prove that all subsets of a finite set can be arranged in a sequence in which every two successive subsets differ in exactly one element.

A frog starts at $0$ on a number line and plays a game. On each turn the frog chooses at random to jump $1$ or $2$ integers to the right or left. It stops moving if it lands on a nonpositive number or a number on which it has already landed. If the expected number of times it will jump is $\tfrac{p}{q}$ for relatively prime positive integers $p$ and $q$, find $p+q$. [i]Proposed by Michael Kural[/i]

Let $2\mathbb{Z} + 1$ denote the set of odd integers. Find all functions $f:\mathbb{Z} \mapsto 2\mathbb{Z} + 1$ satisfying \[ f(x + f(x) + y) + f(x - f(x) - y) = f(x+y) + f(x-y) \] for every $x, y \in \mathbb{Z}$.