A sequence of integers $\{f(n)\}$ for $n=0,1,2,\ldots$ is defined as follows: $f(0)=0$ and for $n>0$, $$\begin{matrix}f(n)=&f(n-1)+3,&\text{if }n=0\text{ or }1\pmod6,\\&f(n-1)+1,&\text{if }n=2\text{ or }5\pmod6,\\&f(n-1)+2,&\text{if }n=3\text{ or }4\pmod6.\end{matrix}$$Derive an explicit formula for $f(n)$ when $n\equiv0\pmod6$, showing all necessary details in your derivation.

Consider a function $f: R \to R$ that fulfills the following two properties: $f$ is periodic of period $5$ (that is, for all $x\in R$, $f (x + 5) = f (x)$), and by restricting $f$ to the interval $[-2,3]$, $f$ coincides to $x^2$. Determine the value of $f(2018).$

Find all functions $f:\mathbb{N}\rightarrow(0,\infty)$ such that $f(4)=4$ and \[\frac{1}{f(1)f(2)}+\frac{1}{f(2)f(3)}+\cdots+\frac{1}{f(n)f(n+1)}=\frac{f(n)}{f(n+1)},~\forall n\in\mathbb{N},\] where $\mathbb{N}=\{1,2,\dots\}$ is the set of positive integers.

$(NET 6)$ A curve determined by $y =\sqrt{x^2 - 10x+ 52}, 0\le x \le 100,$ is constructed in a rectangular grid. Determine the number of squares cut by the curve.

Prove that for any real numbers $ a,b,c, $ the function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ defined as $$ f(x)=\sqrt{(x-c)^2+b^2} +\sqrt{(x+c)^2+b^2} $$ is decreasing on $ (-\infty ,0] $ and increasing on $ [0,\infty ) . $

Each face of a tetrahedron can be placed in a circle of radius $ 1$. Show that the tetrahedron can be placed in a sphere of radius $ \frac{3}{2\sqrt2}$.

The given parabola $y=ax^2+bx+c$ doesn't intersect the $X$-axis and passes from the points $A(-2,1)$ and $B(2,9)$. Find all the possible values of the $x$ coordinates of the vertex of this parabola.

Let be a continuous function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ having the property that there exists a continuous and bounded function $ g:\mathbb{R}\longrightarrow\mathbb{R} $ that verifies the equality $$ f(x)=\int_0^x f(\xi )g(\xi )d\xi , $$ for any real number $ x. $ Prove that $ f=0. $ [i]Nelu Chichirim[/i]

Let $ f(n)$ be a function defined on the set of all positive integers and having its values in the same set. Suppose that $ f(f(n) \plus{} f(m)) \equal{} m \plus{} n$ for all positive integers $ n,m.$ Find the possible value for $ f(1988).$

There exists a function $G(x)$ such that $G(x)+G\left(\frac{x-\sqrt{3}}{x\sqrt{3}+1}\right)=\sqrt{2}+x$. Find $G(x)$. [i]Proposed by beanielove2.[/i]

Let $f(x,y)$ be a function from ordered pairs of positive integers to real numbers such that \[ f(1,x) = f(x,1) = \frac{1}{x} \quad\text{and}\quad f(x+1,y+1)f(x,y)-f(x,y+1)f(x+1,y) = 1 \] for all ordered pairs of positive integers $(x,y)$. If $f(100,100) = \frac{m}{n}$ for two relatively prime positive integers $m,n$, compute $m+n$. [i]David Yang[/i]

Find all surjective functions $ f: \mathbb{N} \to \mathbb{N}$ such that for every $ m,n \in \mathbb{N}$ and every prime $ p,$ the number $ f(m + n)$ is divisible by $ p$ if and only if $ f(m) + f(n)$ is divisible by $ p$. [i]Author: Mohsen Jamaali and Nima Ahmadi Pour Anari, Iran[/i]

Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $\forall x\in \mathbb{R} \ \ f(x) = max(2xy-f(y))$ where $y\in \mathbb{R}$.

On the number line, consider the point $x$ that corresponds to the value $10$. Consider $24$ distinct integer points $y_1$, $y_2$, $\ldots$, $y_{24}$ on the number line such that for all $k$ such that $1\leq k\leq 12$, we have that $y_{2k-1}$ is the reflection of $y_{2k}$ across $x$. Find the minimum possible value of \[\textstyle\sum_{n=1}^{24}(|y_n-1|+|y_n+1|).\]

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

Find the solution of the functional equation $P(x)+P(1-x)=1$ with power $2015$ P.S: $P(y)=y^{2015}$ is also a function with power $2015$

Let $a,b,c$ be reals and \[f(a, b, c) = \left| \frac{ |b-a|}{|ab|} +\frac{b+a}{ab} -\frac 2c \right| +\frac{ |b-a|}{|ab|} +\frac{b+a}{ab} +\frac 2c\] Prove that $f(a, b, c) = 4 \max \{\frac 1a, \frac 1b,\frac 1c \}.$

Consider the two functions \[f(x) = x^2+2bx+1\quad\text{and}\quad g(x) = 2a(x+b),\] where the variable $x$ and the constants $a$ and $b$ are real numbers. Each such pair of the constants $a$ and $b$ may be considered as a point $(a,b)$ in an $ab-$plane. Let $S$ be the set of such points $(a,b)$ for which the graphs of $y = f(x)$ and $y = g(x)$ do NOT intersect (in the $xy-$ plane.). The area of $S$ is $\textbf{(A)} \ 1 \qquad \textbf{(B)} \ \pi \qquad \textbf{(C)} \ 4 \qquad \textbf{(D)} \ 4 \pi \qquad \textbf{(E)} \ \text{infinite}$

Find all functions $f: \mathbb{Q}^+ \to \mathbb{R}^+ $ with the property: \[f(xy)=f(x+y)(f(x)+f(y)) \,,\, \forall x,y \in \mathbb{Q}^+\] [i]Proposed by Nikolay Nikolov[/i]

Suppose $ a, b,$ and $ c$ are positive integers with $ a \plus{} b \plus{} c \equal{} 2006$, and $ a!b!c! \equal{} m\cdot10^n$, where $ m$ and $ n$ are integers and $ m$ is not divisible by 10. What is the smallest possible value of $ n$? $ \textbf{(A) } 489 \qquad \textbf{(B) } 492 \qquad \textbf{(C) } 495 \qquad \textbf{(D) } 498 \qquad \textbf{(E) } 501$

Let $ 0\leq \theta\leq \frac{\pi}{2}$ . Prove that $\sin \theta \geq \frac{2\theta}{\pi}$.

A simple pendulum experiment is constructed from a point mass $m$ attached to a pivot by a massless rod of length $L$ in a constant gravitational field. The rod is released from an angle $\theta_0 < \frac{\pi}{2}$ at rest and the period of motion is found to be $T_0$. Ignore air resistance and friction. The experiment is repeated with a new pendulum with a rod of length $4L$, using the same angle $\theta_0$, and the period of motion is found to be $T$. Which of the following statements is correct? $\textbf{(A) } T = 2T_0 \text{ regardless of the value of } \theta_0\\ \textbf{(B) } T > 2T_0 \text{ with } T \approx 2T_0 \text{ if } \theta_0 \ll 1\\ \textbf{(C) } T < 2T_0 \text{ with } T \approx 2T_0 \text{ if } \theta_0 \ll 1\\ \textbf{(D) } T < 2T_0 \text{ with some values of } \theta_0 \text{ and } T > 2T_0 \text{ for other values of } \theta_0\\ \textbf{(E) } T \text{ and } T_0 \text{ are not defined because the motion is not periodic unless } \theta_0 \ll 1$

The integers $ c_{m,n}$ with $ m \geq 0, \geq 0$ are defined by \[ c_{m,0} \equal{} 1 \quad \forall m \geq 0, c_{0,n} \equal{} 1 \quad \forall n \geq 0,\] and \[ c_{m,n} \equal{} c_{m\minus{}1,n} \minus{} n \cdot c_{m\minus{}1,n\minus{}1} \quad \forall m > 0, n > 0.\] Prove that \[ c_{m,n} \equal{} c_{n,m} \quad \forall m > 0, n > 0.\]

Let $\phi(n,m), m \neq 1$, be the number of positive integers less than or equal to $n$ that are coprime with $m.$ Clearly, $\phi(m,m) = \phi(m)$, where $\phi(m)$ is Euler’s phi function. Find all integers $m$ that satisfy the following inequality: \[\frac{\phi(n,m)}{n} \geq \frac{\phi(m)}{m}\] for every positive integer $n.$

If $\sin x+\cos x=1/5$ and $0\le x<\pi$, then $\tan x$ is $\textbf{(A) }-\frac{4}{3}\qquad\textbf{(B) }-\frac{3}{4}\qquad\textbf{(C) }\frac{3}{4}\qquad\textbf{(D) }\frac{4}{3}\qquad$ $\textbf{(E) }\text{not completely determined by the given information}$