Let ${\mathbb Q}^{+}$ be the set of positive rational numbers. Construct a function $f:{\mathbb Q}^{+}\rightarrow{\mathbb Q}^{+}$ such that \[f(xf(y)) = \frac{f(x)}{y}\] for all $x, y \in{\mathbb Q}^{+}$.

The game of $pebbles$ is played on an infinite board of lattice points $(i,j)$. Initially there is a $pebble$ at $(0,0)$. A move consists of removing a $pebble$ from point $(i,j)$and placing a $pebble$ at each of the points $(i+1,j)$ and $(i,j+1)$ provided both are vacant. Show taht at any stage of the game there is a $pebble$ at some lattice point $(a,b)$ with $0 \leq a+b \leq 3$

Find all pairs of function $f : Q \rightarrow R$ and $g : Q \rightarrow R,$ for which equations \begin{align*} f(x+y) &= f(x) f(y) + g(x) g(y) \\ g(x+y) &= f(x)g(y) + g(x)f(y) + g(x)g(y) \end{align*} holds for all rational numbers $x$ and $y.$ [i]Proposed by Gurgen Asatryan, Armenia [/i]

Four faces of tetrahedron $ABCD$ are congruent triangles whose angles form an arithmetic progression. If the lengths of the sides of the triangles are $a < b < c$, determine the radius of the sphere circumscribed about the tetrahedron as a function on $a, b$, and $c$. What is the ratio $c/a$ if $R = a \ ?$

(1) Let $ f(x)$ be a continuous function defined on $ [a,\ b]$, it is known that there exists some $ c$ such that \[ \int_a^b f(x)\ dx \equal{} (b \minus{} a)f(c)\ (a < c < b)\] Explain the fact by using graph. Note that you don't need to prove the statement. (2) Let $ f(x) \equal{} a_0 \plus{} a_1x \plus{} a_2x^2 \plus{} \cdots\cdots \plus{} a_nx^n$, Prove that there exists $ \theta$ such that \[ f(\sin \theta) \equal{} a_0 \plus{} \frac {a_1}{2} \plus{} \frac {a_3}{3} \plus{} \cdots\cdots \plus{} \frac {a_n}{n \plus{} 1},\ 0 < \theta < \frac {\pi}{2}.\]

A function f has the following property: If $k > 1, j > 1$, and $\gcd(k, j) = m$, then $f(kj) = f(m) (f\left(\frac km\right) + f\left(\frac jm\right))$. What values can $f(1984)$ and $f(1985)$ take?

Find all real-valued continuously differentiable functions $f$ on the real line such that for all $x$, \[ \left( f(x) \right)^2 = \displaystyle\int_0^x \left[ \left( f(t) \right)^2 + \left( f'(t) \right)^2 \right] \, \mathrm{d}t + 1990. \]

Find all functions $f: \mathbb{R}^+ \to \mathbb{R}^+$ such that $$(z + 1)f(x + y) = f(xf(z) + y) + f(yf(z) + x),$$ for all positive real numbers $x, y, z$. [i]Fajar Yuliawan, Indonesia[/i]

Find all functions $ f :\mathbb{Z}\mapsto\mathbb{Z} $ such that following conditions holds: $a)$ $f(n) \cdot f(-n)=f(n^2)$ for all $n\in\mathbb{Z}$ $b)$ $f(m+n)=f(m)+f(n)+2mn$ for all $m,n\in\mathbb{Z}$

If $f(x) = ax^4-bx^2+x+5$ and $f(-3) = 2$, then $f(3) =$ $\textbf{(A)}\ -5 \qquad \textbf{(B)}\ -2 \qquad \textbf{(C)}\ 1 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 8$

Given a nonzero natural number $n$, let $f_n$ be the function of the closed interval $[0, 1]$ in $R$ defined like this: $$f_n(x) = \begin{cases}n^2x, \,\,\, if \,\,\, 0 \le x < 1/n\\ 3/n, \,\,\,if \,\,\,1/n \le x \le 1 \end{cases}$$ a) Represent the function graphically. b) Calculate $A_n =\int_0^1 f_n(x) dx$. c) Find, if it exists, $\lim_{n\to \infty} A_n$ .

If $ g(x)\equal{}1\minus{}x^2$ and $ f(g(x)) \equal{} \frac{1\minus{}x^2}{x^2}$ when $ x\neq0$, then $ f(1/2)$ equals $ \textbf{(A)}\ 3/4 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ \sqrt2/2 \qquad \textbf{(E)}\ \sqrt2$

Find all $f$ functions from real numbers to itself such that for all real numbers $x,y$ the equation \[f(f(y)+x^2+1)+2x=y+(f(x+1))^2\] holds.

Find all the functions $f: \mathbb{Z}\to \mathbb{Z}$ satisfying the following property: if $a$, $b$ and $c$ are integers such that $a+b+c=0$, then $$f(a)+f(b)+f(c)=a^2+b^2+c^2.$$

In the tribe of Zimmer, being able to hike long distances and knowing the roads through the forest are both extremely important, so a boy who reaches the age of manhood is not designated as a man by the tribe until he completes an interesting rite of passage. The man must go on a sequence of hikes. The first hike is a $5$ kilometer hike down the main road. The second hike is a $5\frac{1}{4}$ kilometer hike down a secondary road. Each hike goes down a different road and is a quarter kilometer longer than the previous hike. The rite of passage is completed at the end of the hike where the cumulative distance walked by the man on all his hikes exceeds $1000$ kilometers. So in the tribe of Zimmer, how many roads must a man walk down, before you call him a man?

Find all functions $f:\mathbb{R}^{+}\rightarrow \mathbb{R}^{+}$ such that $x,y\in \mathbb{R}^{+},$ \[ f\left(\frac{y}{f(x+1)}\right)+f\left(\frac{x+1}{xf(y)}\right)=f(y) \]

If a, b, and c are non-negative real numbers satisfying $a + b + c = 400$, find the maximum possible value of $\sqrt{2a+b}+\sqrt{2b+c}+\sqrt{2c+a}$.

Let $d(n)$ be the number of positive divisors of a natural number $n$.Find all $k\in \mathbb{N}$ such that there exists $n\in \mathbb{N}$ with $d(n^2)/d(n)=k$.

a) Prove that every function of the form $$f(x)=\frac{a_{0}}{2}+\cos(x)+\sum_{n=2}^{N}a_{n}\cos(nx)$$ with $|a_{0}|<1$ has positive as well as negative values in the period $[0,2\pi)$. b) Prove that the function $$F(x)=\sum_{n=1}^{100}\cos(n^{\frac{3}{2}}x)$$ has at least $40$ zeroes in the interval $(0,1000)$.

Prove that for any positive integer $n$ and any real numbers $a_1,a_2,\cdots,a_n,b_1,b_2,\cdots,b_n$ we have that the equation \[a_1 \sin(x) + a_2 \sin(2x) +\cdots+a_n\sin(nx)=b_1 \cos(x)+b_2\cos(2x)+\cdots +b_n \cos(nx)\] has at least one real root.

Prove that in an abelian ring $ A $ in which $ 1\neq 0, $ every element is idempotent if and only if the number of polynomial functions from $ A $ to $ A $ is equal to the square of the cardinal of $ A. $

Let be a natural number $ n, $ two $ \text{n-tuplets} $ of real numbers $ a:=\left( a_1,a_2,\ldots, a_n \right) , b:=\left( b_1,b_2,\ldots, b_n \right) , $ and the function $ f:\mathbb{R}\longrightarrow\mathbb{R}, f(x)=\sum_{i=1}^na_i\cos \left( b_ix \right) $. Prove that if the numbers of $ b $ are all positive and pairwise distinct, [b]a)[/b] then, $ f\ge 0 $ implies that the numbers of $ a $ are all equal. [b]b)[/b] if the numbers of $ a $ are all nonzero and $ f $ is periodic, then the ratio of any two numbers of $ b $ is rational. [i]Marin Tolosi[/i]

Known $f:\mathbb{N}_0 \to \mathbb{N}_0$ function for $\forall x,y\in \mathbb{N}_0$ the following terms are paid $(a). f(0,y)=y+1$ $(b). f(x+1,0)=f(x,1)$ $(c). f(x+1,y+1)=f(x,f(x+1,y)).$ Find the value if $f(4,1981)$

Let $Z^+$ denote the set of positive integers. Determine , with proof, if there exists a function $f:\mathbb{Z^+}\rightarrow\mathbb {Z^+}$ such that $f(f(f(f(f(n)))))$ = $2022n$ for all positive integers $n$.

Let $f (n)$ be a function satisfying the following three conditions for all positive integers $n$: (a) $f (n)$ is a positive integer, (b) $f (n + 1) > f (n)$, (c) $f ( f (n)) = 3n$. Find $f (2001)$.