Find all functions $f:\mathbb{R^+} \to \mathbb{R^+}$ such that for all $x,y,z\in \mathbb{R^+}$: $$\biggl\{\frac{f(x)}{f(y)}\biggl\}+\biggl\{\frac{f(y)}{f(z)}\biggl\}+ \biggl\{\frac{f(z)}{f(x)}\biggl\}= \biggl\{\frac{x}{y}\biggl\} +\biggl\{\frac{y}{z}\biggl\}+ \biggl\{\frac{z}{x}\biggl\}$$ Note: For any real number $x$, let $\{x\}$ denote the fractional part of $x$, defined as For example, $\{2,7\}=0,7$ .

Let $f: \mathbb{R}\to\mathbb{R}$ be a function such that $(f(x))^{n}$ is a polynomial for every integer $n\geq 2$. Is $f$ also a polynomial?

Find the volume of the solid of the domain expressed by the inequality $x^2-x\leq y\leq x$, generated by a rotation about the line $y=x.$

a) Prove that the function $F:\mathbb{R}\rightarrow \mathbb{R},\ F(x)=2\lfloor x\rfloor-\cos(3\pi\{x\})$ is continuous over $\mathbb{R}$ and for any $y\in \mathbb{R}$, the equation $F(x)=y$ has exactly three solutions. b) Let $k$ a positive even integer. Prove that there is no function $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $f$ is continuous over $\mathbb{R}$ and that for any $y\in \text{Im}\ f$, the equation $f(x)=y$ has exactly $k$ solutions $(\text{Im}\ f=f(\mathbb{R}))$.

Let $\mathbb{R}$ be the set of real numbers. Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function such that \[f(x+y)f(x-y)\geqslant f(x)^2-f(y)^2\] for every $x,y\in\mathbb{R}$. Assume that the inequality is strict for some $x_0,y_0\in\mathbb{R}$. Prove that either $f(x)\geqslant 0$ for every $x\in\mathbb{R}$ or $f(x)\leqslant 0$ for every $x\in\mathbb{R}$.

Let $ a,b$ and $ c $ be positive real numbers with $ abc = 1 $. Prove that \[ \sqrt{ 9 + 16a^2}+\sqrt{ 9 + 16b^2}+\sqrt{ 9 + 16c^2} \ge 3 +4(a+b+c)\]

Let $f:\mathbb{N}\rightarrow \mathbb{N}$ be a strictly increasing function such that $f(2)=2$ and $f(mn)=f(m)f(n)$ for every pair of relatively prime positive integers $m$ and $n$. Prove that $f(n)=n$ for every positive integer $n$.

Find all functions $ f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying: $ x^2 f(x)\plus{}f(1\minus{}x)\equal{}2x\minus{}x^4$ for all $ x \in \mathbb{R}$.

The real numbers $x_1, x_2, ... , x_n$ belong to the interval $(0,1)$ and satisfy $x_1 + x_2 + ... + x_n = m + r$, where $m$ is an integer and $r \in [0,1)$. Show that $x_1 ^2 + x_2 ^2 + ... + x_n ^2 \leq m + r^2$.

Let $x_i \in \{0,1\}(i=1,2,\cdots ,n)$,if the value of function $f=f(x_1,x_2, \cdots ,x_n)$ can only be $0$ or $1$,then we call $f$ a $n$-var Boole function,and we denote $D_n(f)=\{(x_1,x_2, \cdots ,x_n)|f(x_1,x_2, \cdots ,x_n)=0\}.$ $(1)$ Find the number of $n$-var Boole function; $(2)$ Let $g$ be a $n$-var Boole function such that $g(x_1,x_2, \cdots ,x_n) \equiv 1+x_1+x_1x_2+x_1x_2x_3 +\cdots +x_1x_2 \cdots x_n \pmod 2$, Find the number of elements of the set $D_n(g)$,and find the maximum of $n \in \mathbb{N}_+$ such that $\sum_{(x_1,x_2, \cdots ,x_n) \in D_n(g)}(x_1+x_2+ \cdots +x_n) \le 2017.$

Find the range of the function $f: \mathbb{R} \to \mathbb{R},$ $$f(x)=\frac{3+2\sin x}{\sqrt{1+\cos x}+\sqrt{1-\cos x}}.$$

Given an equilateral triangle $ABC$ of side $a$ in a plane, let $M$ be a point on the circumcircle of the triangle. Prove that the sum $s = MA^4 +MB^4 +MC^4$ is independent of the position of the point $M$ on the circle, and determine that constant value as a function of $a$.

You use a lock with four dials, each of which is set to a number between 0 and 9 (inclusive). You can never remember your code, so normally you just leave the lock with each dial one higher than the correct value. Unfortunately, last night someone changed all the values to 5. All you remember about your code is that none of the digits are prime, 0, or 1, and that the average value of the digits is 5. How many combinations will you have to try?

A function $f:\mathbb{N} \to \mathbb{N}$ is given. If $a,b$ are coprime, then $f(ab)=f(a)f(b)$. Also, if $m,k$ are primes (not necessarily different), then $$f(m+k-3)=f(m)+f(k)-f(3).$$ Find all possible values of $f(11)$.

Which of the following are true? $\textbf{(A)}~\exists f:\mathbb N\to\mathbb Z\text{ onto and increasing}$ $\textbf{(B)}~\exists f:\mathbb Z\to\mathbb Q\text{ onto and increasing}$ $\textbf{(C)}~\exists f:\mathbb Q\to\mathbb Z\text{ onto and increasing and bounded}$ $\textbf{(D)}~\text{None of the above}$

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]

Let $ a_1$, $ a_2$, ..., $ a_n$ and $ b_1$, $ b_2$, ..., $ b_n$ be $ 2 \cdot n$ real numbers. Prove that the following two statements are equivalent: [b]i)[/b] For any $ n$ real numbers $ x_1$, $ x_2$, ..., $ x_n$ satisfying $ x_1 \leq x_2 \leq \ldots \leq x_ n$, we have $ \sum^{n}_{k \equal{} 1} a_k \cdot x_k \leq \sum^{n}_{k \equal{} 1} b_k \cdot x_k,$ [b]ii)[/b] We have $ \sum^{s}_{k \equal{} 1} a_k \leq \sum^{s}_{k \equal{} 1} b_k$ for every $ s\in\left\{1,2,...,n\minus{}1\right\}$ and $ \sum^{n}_{k \equal{} 1} a_k \equal{} \sum^{n}_{k \equal{} 1} b_k$.

The number of maps $f$ from $1,2,3$ into the set $1,2,3,4,5$ such that $f(i)\le f(j)$ whenever $i\le j$ is $\textbf{(A)}~60$ $\textbf{(B)}~50$ $\textbf{(C)}~35$ $\textbf{(D)}~30$

Let $N$ be the number of functions $f$ from $\{1,2,\ldots, 8\}$ to $\{1,2,3,\ldots, 255\}$ with the property that: [list] [*] $f(k)=1$ for some $k \in \{1,2,3,4,5,6,7,8\}$ [*] If $f(a) =f(b)$, then $a=b$. [*] For all $n \in \{1,2,3,4,5,6,7,8\}$, if $f(n) \neq 1$, then $f(k)+1>\frac{f(n)}{2} \geq f(k)$ for some $k \in \{1,2,\ldots, 7,8\}$. [*] For all $k,n \in \{1,2,3,4,5,6,7,8\}$, if $f(n)=2f(k)+1$, then $k<n$. [/list] Compute the number of positive integer divisors of $N$. [i]2021 CCA Math Bonanza Individual Round #15[/i]

Consider an $n$-by-$n$ board of unit squares for some odd positive integer $n$. We say that a collection $C$ of identical dominoes is a [i]maximal grid-aligned configuration[/i] on the board if $C$ consists of $(n^2-1)/2$ dominoes where each domino covers exactly two neighboring squares and the dominoes don't overlap: $C$ then covers all but one square on the board. We are allowed to slide (but not rotate) a domino on the board to cover the uncovered square, resulting in a new maximal grid-aligned configuration with another square uncovered. Let $k(C)$ be the number of distinct maximal grid-aligned configurations obtainable from $C$ by repeatedly sliding dominoes. Find the maximum value of $k(C)$ as a function of $n$. [i]Proposed by Holden Mui[/i]

Let $\mathcal{F}$ be the set of the differentiable functions $f: \mathbb{R} \to \mathbb{R}$ satisfying $f(x) \ge f(x+ \sin x)$ for any $x \in \mathbb{R}.$ a) Prove that there exist nonconstant functions in $\mathcal{F}.$ b) Prove that if $f \in \mathcal{F},$ then the set of solutions of the equation $f'(x)=0$ is infinite.

Show that $\sum^{\infty}_{n=1}\frac{(-1)^{n-1}\sin(\log n)}{n^\alpha}$ converges iff $\alpha>0$.

Let $N$ be the set of all non-negative integers and for each $n \in N$ denote $n'= n +1$. The function $A : N^3 \to N$ is defined as follows: (i) $A(0, m, n) = m'$ for all $m, n \in N$ (ii) $A(k', 0, n) =\left\{ \begin{array}{ll} n & if \, \, k = 0 \\ 0 & if \, \,k = 1, \\ 1 & if \, \, k > 1 \end{array} \right.$ for all $k, n \in N$ (iii) $A(k', m', n) = A(k, A(k',m,n), n)$ for all $k,m, n \in N$. Compute $A(5, 3, 2)$. (H. Nestra)

Let $f(x)$ be a linear function and $k,l,m$ - pairwise different real numbers. It is known that $f(k)=l^3+m^3$, $f(l)=m^3+k^3$ and $f(m)=k^3+l^3$. Find the value of $k+l+m$.

A sequence of real numbers $a_1,a_2,\ldots$ satisfies the relation $$a_n=-\max_{i+j=n}(a_i+a_j)\qquad\text{for all}\quad n>2017.$$ Prove that the sequence is bounded, i.e., there is a constant $M$ such that $|a_n|\leq M$ for all positive integers $n$.