Positive real numbers $x,y,z$ have the sum $1$. Prove that $\sqrt{7x+3}+ \sqrt{7y+3}+\sqrt{7z+3} \le 7$. Can number $7$ on the right hand side be replaced with a smaller constant?

A weird calculator has a numerical display and only two buttons, $\boxed{D\sharp}$ and $\boxed{D\flat}$. The first button doubles the displayed number and then adds $1$. The second button doubles the displayed number and then subtracts $1$. For example, if the display is showing $5$, then pressing the $\boxed{D\sharp}$ produces $11$. If the display shows $5$ and we press $\boxed{D\flat}$, we get $9$. If the display shows $5$ and we press the sequence $\boxed{D\sharp}$, $\boxed{D\flat}$, $\boxed{D\sharp}$, $\boxed{D\sharp}$, we get a display of $87$. [list=i] [*] Suppose the initial displayed number is $1$. Give a sequence of exactly eight button presses that will result in a display of $313$. [*] Suppose the initial displayed number is $1$, and we then perform exactly eight button presses. Describe all the numbers that can possibly result? Prove your answer by explaining how all these numbers can be produced and that no other numbers can be produced. [/list]

Define the polynomials $P_0, P_1, P_2 \cdots$ by: \[ P_0(x)=x^3+213x^2-67x-2000 \] \[ P_n(x)=P_{n-1}(x-n), n \in N \] Find the coefficient of $x$ in $P_{21}(x)$.

Let $x_i,y_i$ be positive real numbers, $i = 1,2,...,n$. Prove that $$\left( \sum_{i=1}^n (x_i +y_i)^2\right)\left( \sum_{i=1}^n\frac{1}{x_iy_i}\right)\ge 4n^2$$

The side lengths of a scalene triangle are roots of the polynomial $$x^3-20x^2+131x-281.3.$$ Find the square of the area of the triangle.

Determine all real $ x$ satisfying the equation \[ \sqrt[5]{x^3 \plus{} 2x} \equal{} \sqrt[3]{x^5\minus{}2x}.\] Odd roots for negative radicands shall be included in the discussion.

Find all pairs of real numbers $(x,y)$ satisfying $$(2x+1)^2+y^2+(y-2x)^2=\frac13.$$

Let be a natural number $ n. $ Prove that there is a polynomial $ P\in\mathbb{Z} [X,Y] $ such that $ a+b+c=0 $ implies $$ a^{2n+1}+b^{2n+1}+c^{2n+1}=abc\left( P(a,b)+P(b,c)+P(c,a)\right) $$ [i]Dan Brânzei[/i]

Let $x, y, z$ be positive real numbers. Prove that: $\frac{x + 2y}{z + 2x + 3y}+\frac{y + 2z}{x + 2y + 3z}+\frac{z + 2x}{y + 2z + 3x} \le \frac{3}{2}$

Solve the system $$\begin{cases} \displaystyle 4^{-x}+27^{-y}= \frac{5}{6} \\ \displaystyle 27^y-4^x \le 1 \\ \displaystyle \log_{27}y-\log_4 x \ge \frac{1}{6} \end{cases}.$$

Find all he positive integers $x, y, z, t, w$, such as: $x+\frac{1}{y+\frac{1}{z+\frac{1}{t+\frac{1}{w}}}}=\frac{1998}{115}$

Carl chooses a [i]functional expression[/i]* $E$ which is a finite nonempty string formed from a set $x_1, x_2, \dots$ of variables and applications of a function $f$, together with addition, subtraction, multiplication (but not division), and fixed real constants. He then considers the equation $E = 0$, and lets $S$ denote the set of functions $f \colon \mathbb R \to \mathbb R$ such that the equation holds for any choices of real numbers $x_1, x_2, \dots$. (For example, if Carl chooses the functional equation $$ f(2f(x_1)+x_2) - 2f(x_1)-x_2 = 0, $$ then $S$ consists of one function, the identity function. (a) Let $X$ denote the set of functions with domain $\mathbb R$ and image exactly $\mathbb Z$. Show that Carl can choose his functional equation such that $S$ is nonempty but $S \subseteq X$. (b) Can Carl choose his functional equation such that $|S|=1$ and $S \subseteq X$? *These can be defined formally in the following way: the set of functional expressions is the minimal one (by inclusion) such that (i) any fixed real constant is a functional expression, (ii) for any positive integer $i$, the variable $x_i$ is a functional expression, and (iii) if $V$ and $W$ are functional expressions, then so are $f(V)$, $V+W$, $V-W$, and $V \cdot W$. [i]Proposed by Carl Schildkraut[/i]

Find all functions $f: \mathbb{R}\rightarrow \mathbb{R}$ such that for $\forall$ $x,y\in \mathbb{R}$ : $f(x+f(x+y))+xy=yf(x)+f(x)+f(y)+x$.

Two sequences $(x_n)_{n\in N}$ and $(y_n)_{n\in N}$ are defined recursively as follows: $x_0 = 2015$ and $x_{n+1} =\left \lfloor x_n \cdot \frac{y_{n+1}}{y_{n-1}} \right \rfloor$ for all $n \ge 0$, $y_0 = 307$ and $y_{n+1} = y_n + 1$ for all $n \ge 0$. Compute $\lim_{n\to \infty} \frac{x_n}{(y_n)^2}$.

Triangle $ABC$ satisfies the property that $\angle A = a \log x$, $\angle B = a \log 2x$, and $\angle C = a \log 4x$ radians, for some real numbers $a$ and $x$. If the altitude to side $AB$ has length $8$ and the altitude to side $BC$ has length $9$, find the area of $\vartriangle ABC$.

For real numbers $ x_1, x_2, \dots, x_n $ and $ y_1, y_2, \dots, y_n $ , the inequalities hold $ x_1 \geq x_2 \geq \ldots \geq x_n> 0 $ and $$ y_1 \geq x_1, ~ y_1y_2 \geq x_1x_2, ~ \dots, ~ y_1y_2 \dots y_n \geq x_1x_2 \dots x_n. $$ Prove that $ ny_1 + (n-1) y_2 + \dots + y_n \geq x_1 + 2x_2 + \dots + nx_n $.

Show that [0,1] cannot be partitioned into two disjoints sets A and B such that B=A+a for some real a.

Let $ \left(a_{n}\right)$ be the sequence of reals defined by $ a_{1}=\frac{1}{4}$ and the recurrence $ a_{n}= \frac{1}{4}(1+a_{n-1})^{2}, n\geq 2$. Find the minimum real $ \lambda$ such that for any non-negative reals $ x_{1},x_{2},\dots,x_{2002}$, it holds \[ \sum_{k=1}^{2002}A_{k}\leq \lambda a_{2002}, \] where $ A_{k}= \frac{x_{k}-k}{(x_{k}+\cdots+x_{2002}+\frac{k(k-1)}{2}+1)^{2}}, k\geq 1$.

Let $a, b, c$ be real numbers such that $ab + bc + ca = 1$. Prove that $$\frac{(a + b)^2 + 1}{c^2+2}+\frac{(b + c)^2 + 1}{a^2+2}+ \frac{(c + a)^2 + 1}{b^2+2} \ge 3$$

An infinite road has traffic lights at intervals of 1500m. The lights are all synchronised and are alternately green for $\frac 32$ minutes and red for 1 minute. For which $v$ can a car travel at a constant speed of $v$m/s without ever going through a red light?

A sequence $a_n$ is defined by $$a_0=0,\qquad a_1=3;$$$$a_n=8a_{n-1}+9a_{n-2}+16\text{ for }n\ge2.$$Find the least positive integer $h$ such that $a_{n+h}-a_n$ is divisible by $1999$ for all $n\ge0$.

Find all continious $f: \mathbb R\longrightarrow\mathbb R$ that for any $x,y$ \[f(x)+f(y)+f(xy)=f(x+y+xy)\]

A pair $(f,g)$ of degree $2$ real polynomials is called [i]foolish[/i] if $f(g(x)) = f(x) \cdot g(x)$ for all real $x.$ How many positive integers less than $2023$ can be a root of $g(x)$ for some foolish pair $(f,g)$?

Let \[ f(x)\equal{}a_0\plus{}a_1x\plus{}a_2x^2\plus{}a_{10}x^{10}\plus{}a_{11}x^{11}\plus{}a_{12}x^{12}\plus{}a_{13}x^{13} \; (a_{13} \not\equal{}0) \] and \[ g(x)\equal{}b_0\plus{}b_1x\plus{}b_2x^2\plus{}b_{3}x^{3}\plus{}b_{11}x^{11}\plus{}b_{12}x^{12}\plus{}b_{13}x^{13} \; (b_{3} \not\equal{}0) \] be polynomials over the same field. Prove that the degree of their greatest common divisor is at least $ 6$. [i]L. Redei[/i]

Let $a \ge 2$ be an integer. Find all polynomials $f$ with real coefficients such that $$A = \{a^{n^2} | n \ge 1, n \in Z\} \subset \{f(n) | n \ge 1, n \in Z\} = B.$$