Let $a, b > 0$ be reals such that \[ a^3=a+1\\ b^6=b+3a \] Show that $a>b$

Find all functions $f:\mathbb{Q}\to\mathbb{Q}$ such that $$f(xy)+f(x+y)=f(x)f(y)+f(x)+f(y)$$ for all $x,y\in\mathbb{Q}$.

Let $(F_n)_{n\in{N^*}}$ be the Fibonacci sequence defined by $F_1=1$, $F_2=1$, $F_{n+1}=F_n+F_{n-1}$ for every $n\geq{2}$. Find the limit: \[ \lim_{n \to \infty}(\sum_{i=1}^n{\frac{F_i}{2^i}}) \]

Given real numbers $a,c,d$ show that there exists at most one function $f:\mathbb{R}\rightarrow\mathbb{R}$ which satisfies: \[f(ax+c)+d\le x\le f(x+d)+c\quad\text{for any}\ x\in\mathbb{R}\]

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]

Determine all ordered triples $(a, b, c)$ of real numbers such that whenever a function $f : \mathbb{R} \to \mathbb{R}$ satisfies $$|f(x) - f(y)| \le a(x - y)^2 + b(x - y) + c$$ for all real numbers $x$ and $y$, then $f$ must be a constant function.

Determine all continuous functions $f: \mathbb R \to \mathbb R$ such that \[f(x + y)f(x - y) = (f(x)f(y))^2, \quad \forall(x, y) \in\mathbb{R}^2.\]

Find all monotonically increasing or monotonically decreasing functions $f: \mathbb{R}_+\to\mathbb{R}_+$ which satisfy the equation $f\left(xy\right)\cdot f\left(\frac{f\left(y\right)}{x}\right)=1$ for any two numbers $x$ and $y$ from $\mathbb{R}_+$. Hereby, $\mathbb{R}_+$ is the set of all positive real numbers. [i]Note.[/i] A function $f: \mathbb{R}_+\to\mathbb{R}_+$ is called [i]monotonically increasing[/i] if for any two positive numbers $x$ and $y$ such that $x\geq y$, we have $f\left(x\right)\geq f\left(y\right)$. A function $f: \mathbb{R}_+\to\mathbb{R}_+$ is called [i]monotonically decreasing[/i] if for any two positive numbers $x$ and $y$ such that $x\geq y$, we have $f\left(x\right)\leq f\left(y\right)$.

Let $p$ be a prime number of the form $4k+1$. Show that \[\sum^{p-1}_{i=1}\left( \left \lfloor \frac{2i^{2}}{p}\right \rfloor-2\left \lfloor \frac{i^{2}}{p}\right \rfloor \right) = \frac{p-1}{2}.\]

Let $ S \equal{} \{0,1,2,\ldots,1999\}$ and $ T \equal{} \{0,1,2,\ldots \}.$ Find all functions $ f: T \mapsto S$ such that [b](i)[/b] $ f(s) \equal{} s \quad \forall s \in S.$ [b](ii)[/b] $ f(m\plus{}n) \equal{} f(f(m)\plus{}f(n)) \quad \forall m,n \in T.$

Let $Q(x)$ be a non-zero polynomial and $k$ be a natural number. Prove that the polynomial $P(x) = (x-1)^kQ(x)$ has at least $k+1$ non-zero coefficients.

Find all functions $f:R \rightarrow R$, which satisfy the equality for any $x,y \in R$: $f(xf(y)+y)+f(xy+x)=f(x+y)+2xy$,

Let $ \left( c_n \right)_{n\ge 1} $ be a sequence of real numbers. Prove that the sequences $ \left( c_n\sin n \right)_{n\ge 1} ,\left( c_n\cos n \right)_{n\ge 1} $ are both convergent if and only if $ \left( c_n \right)_{n\ge 1} $ converges to $ 0. $ [i]Mihai Piticari[/i] and [i]Vladimir Cerbu[/i]

Line $l$ tangents unit circle $S$ in point $P$. Point $A$ and circle $S$ are on the same side of $l$, and the distance from $A$ to $l$ is $h$ ($h > 2$). Two tangents of circle $S$ are drawn from $A$, and intersect line $l$ at points $B$ and $C$ respectively. Find the value of $PB \cdot PC$.

Let $\, a_1, a_2, a_3, \ldots \,$ be a sequence of positive real numbers satisfying $\, \sum_{j=1}^n a_j \geq \sqrt{n} \,$ for all $\, n \geq 1$. Prove that, for all $\, n \geq 1, \,$ \[ \sum_{j=1}^n a_j^2 > \frac{1}{4} \left( 1 + \frac{1}{2} + \cdots + \frac{1}{n} \right). \]

Let $p$ be a fixed prime. Determine all the integers $m$, as function of $p$, such that there exist $a_1, a_2, \ldots, a_p \in \mathbb{Z}$ satisfying \[m \mid a_1^p + a_2^p + \cdots + a_p^p - (p+1).\]

Let $ \mathbb{N}_0$ denote the set of nonnegative integers. Find all functions $ f$ from $ \mathbb{N}_0$ to itself such that \[ f(m \plus{} f(n)) \equal{} f(f(m)) \plus{} f(n)\qquad \text{for all} \; m, n \in \mathbb{N}_0. \]

Let be two functions $ f,g:\mathbb{R}\longrightarrow\mathbb{R} $ such that $ g $ has infinite limit at $ \infty . $ [b]a)[/b] Prove that if $ g $ continuous then $ \lim_{x\to\infty } f(x) =\lim_{x\to\infty } f(g(x)) $ [b]b)[/b] Provide an example of what $ f,g $ could be if $ f $ has no limit at $ \infty $ and $ \lim_{x\to\infty } f(g(x)) =0. $

It is possible to choose $ x > \frac {2}{3}$ in such a way that the value of $ \log_{10}(x^2 \plus{} 3) \minus{} 2 \log_{10}x$ is $ \textbf{(A)}\ \text{negative} \qquad \textbf{(B)}\ \text{zero} \qquad \textbf{(C)}\ \text{one}$ $ \textbf{(D)}\ \text{smaller than any positive number that might be specified}$ $ \textbf{(E)}\ \text{greater than any positive number that might be specified}$

Suppose that $f(x,y)$ is a continuous real-valued function on the unit square $0\le x\le1,0\le y\le1.$ Show that $\int_0^1\left(\int_0^1f(x,y)dx\right)^2dy + \int_0^1\left(\int_0^1f(x,y)dy\right)^2dx$ $\le\left(\int_0^1\int_0^1f(x,y)dxdy\right)^2 + \int_0^1\int_0^1\left[f(x,y)\right]^2dxdy.$

Let $ P(x)$ be the real polynomial function, $ P(x) \equal{} ax^3 \plus{} bx^2 \plus{} cx \plus{} d.$ Prove that if $ |P(x)| \leq 1$ for all $ x$ such that $ |x| \leq 1,$ then \[ |a| \plus{} |b| \plus{} |c| \plus{} |d| \leq 7.\]

For the function $f(x)=\int_0^x \frac{dt}{1+t^2}$, answer the questions as follows. Note : Please solve the problems without using directly the formula $\int \frac{1}{1+x^2}\ dx=\tan^{-1}x +C$ for Japanese High School students those who don't study arc sin x, arc cos x, arc tanx. (1) Find $f(\sqrt{3})$ (2) Find $\int_0^{\sqrt{3}} xf(x)\ dx$ (3) Prove that for $x>0$. $f(x)+f\left(\frac{1}{x}\right)$ is constant, then find the value.

Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that \[f(x + y) + y \le f(f(f(x)))\] holds for all $x, y \in \mathbb{R}$.

Let $S$ be a finite set, and let $\mathcal{A}$ be the set of all functions from $S$ to $S$. Let $f$ be an element of $\mathcal{A}$, and let $T=f(S)$ be the image of $S$ under $f$. Suppose that $f\circ g\circ f\ne g\circ f\circ g$ for every $g$ in $\mathcal{A}$ with $g\ne f$. Show that $f(T)=T$.

Find all real functions $f$ defined on the real numbers except zero, satisfying $f(2019) = 1$ and $f(x)f(y)+ f\left(\frac{2019}{x}\right) f\left(\frac{2019}{y}\right) =2f(xy)$ for all $x,y \ne 0$