For each integer $n$ ($n \ge 2$), let $f(n)$ denote the sum of all positive integers that are at most $n$ and not relatively prime to $n$. Prove that $f(n+p) \neq f(n)$ for each such $n$ and every prime $p$.

Let $\mathbb{R}^{+}$ denote the set of positive real numbers. Find all functions $f : \mathbb{R}^{+} \to \mathbb{R}^{+}$ satisfying for all $x, y \in \mathbb{R}^{+}$ the equality \[f(xf(y))=f(xy)+x\]

Find all $a\in\mathbb{R}$ for which there exists a non-constant function $f: (0,1]\rightarrow\mathbb{R}$ such that \[a+f(x+y-xy)+f(x)f(y)\leq f(x)+f(y)\] for all $x,y\in(0,1].$

Given the vector spaces $V,W$ with coefficients over a field $K$ and function $ \phi :V\to W$ satisfying the relation : $$\varphi(\lambda x+y)= \lambda \varphi(x)+\phi (y)$$ for all $x,y \in V, \lambda \in K$. Such a function is called linear. Let $L\varphi=\{x\in V/\varphi(x)=0\}$ , and$M=\varphi(V)$ , prove that : (i) $L\varphi$ is subspace of $V$ and $M$ is subspace of $W$ (ii) $L\varphi={O}$ iff $\varphi$ is $1-1$ (iii) Dimension of $V$ equals to dimension of $L\varphi$ plus dimension of $M$ (iv) If $\theta : \mathbb{R}^3\to\mathbb{R}^3$ with $\theta(x,y,z)=(2x-z,x-y,x-3y+z)$, prove that $\theta$ is linear function . Find $L\theta=\{x\in {R}^3/\theta(x)=0\}$ and dimension of $M=\theta({R}^3)$.

Let's call a function $f : R \to R$ [i]cool[/i] if there are real numbers $a$ and $b$ such that $f(x + a)$ is an even function and $f(x + b)$ is an odd function. (a) Prove that every cool function is periodic. (b) Give an example of a periodic function that is not cool.

Alice and Bob play a game in which they take turns removing stones from a heap that initially has $n$ stones. The number of stones removed at each turn must be one less than a prime number. The winner is the player who takes the last stone. Alice plays first. Prove that there are infinitely many such $n$ such that Bob has a winning strategy. (For example, if $n=17,$ then Alice might take $6$ leaving $11;$ then Bob might take $1$ leaving $10;$ then Alice can take the remaining stones to win.)

Let $ f: \mathbb{R}^2 \rightarrow \mathbb{R} $ be a function such that for every regular $ n $-gon $ A_1A_2 \ldots A_n $ we have $ f(A_1)+f(A_2)+\cdots +f(A_n)=0 $. Prove that $ f(x)=0 $ for all reals $ x $.

Determine all functions $ f$ defined in the set of rational numbers and taking their values in the same set such that the equation $ f(x + y) + f(x - y) = 2f(x) + 2f(y)$ holds for all rational numbers $x$ and $y$.

In the Euclidean three-dimensional space, we call [i]folding[/i] of a sphere $S$ every partition of $S \setminus \{x,y\}$ into disjoint circles, where $x$ and $y$ are two points of $S$. A folding of $S$ is called [b]linear[/b] if the circles of the [i]folding[/i] are obtained by the intersection of $S$ with a family of parallel planes or with a family of planes containing a straight line $D$ exterior to $S$. Is every [i]folding[/i] of a sphere $S$ [b]linear[/b]?

Let $\mathbb N$ be the set of positive integers. Let $f: \mathbb N \to \mathbb N$ be a function satisfying the following two conditions: (a) $f(m)$ and $f(n)$ are relatively prime whenever $m$ and $n$ are relatively prime. (b) $n \le f(n) \le n+2012$ for all $n$. Prove that for any natural number $n$ and any prime $p$, if $p$ divides $f(n)$ then $p$ divides $n$.

Let $f(x)$ be a continuous real-valued function defined on the interval $[0, 1]$. Show that \[\int_0^1\int_0^1|f(x)+f(y)|dx \; dy \ge \int_0^1 |f(x)|dx\]

Let $k$ be a positive integer. In a circle with radius $1$, finitely many chords are drawn. You know that every diameter of the circle intersects at most $k$ of these chords. Prove that the sum of the lengths of all these chords is less than $k \cdot \pi$.

Prove that there exist two functions $f,g \colon \mathbb{R} \to \mathbb{R}$, such that $f\circ g$ is strictly decreasing and $g\circ f$ is strictly increasing. [i](Poland) Andrzej Komisarski and Marcin Kuczma[/i]

Let $ f$ be a strictly increasing, continuous function mapping $ I=[0,1]$ onto itself. Prove that the following inequality holds for all pairs $ x,y \in I$: \[ 1-\cos (xy) \leq \int_0^xf(t) \sin (tf(t))dt + \int_0^y f^{-1}(t) \sin (tf^{-1}(t)) dt .\] [i]Zs. Pales[/i]

Suppose that $n\ge2$ and $a_1,a_2,...,a_n$ are natural numbers that $ (a_1,a_2,...,a_n)=1$. Find all strictly increasing function $f: \mathbb{Z} \to \mathbb{R} $ that: $$ \forall x_1,x_2,...,x_n \in \mathbb{Z} : f(\sum_{i=1}^{n} {x_ia_i}) = \sum_{i=1}^{n} {f(x_ia_i})$$ [i]Proposed by Navid Safaei and Ali Mirzaei [/i]

Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that $$ (x+y)f(2yf(x)+f(y))=x^{3}f(yf(x)), \ \ \ \forall x,y\in \mathbb{R}^{+}.$$ (Albania)

Let $\mathbb{N}$ be the set of positive integers. For every $n \in \mathbb{N}$, define $d(n)$ as the number of positive divisors of $n$. Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that: a) $d(f(x)) = x$ for all $x \in \mathbb{N}$ b) $f(xy)$ divides $(x-1)y^{xy-1}f(x)$ for all $x,y \in \mathbb{N}$

Let $\mathbb{F}_p$ be the set of integers modulo $p$. Call a function $f : \mathbb{F}_p^2 \to \mathbb{F}_p$ [i]quasiperiodic[/i] if there exist $a,b \in \mathbb{F}_p$, not both zero, so that $f(x + a, y + b) = f(x, y)$ for all $x,y \in \mathbb{F}_p$. Find the number of functions $\mathbb{F}_p^2 \to \mathbb{F}_p$ that can be written as the sum of some number of quasiperiodic functions.

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

Let $f: N \to N$ be a function satisfying (a) $1\le f(x)-x \le 2019$ $\forall x \in N$ (b) $f(f(x))\equiv x$ (mod $2019$) $\forall x \in N$ Show that $\exists x \in N$ such that $f^k(x)=x+2019 k, \forall k \in N$

Evaluate \[ \int_{\minus{}1}^1 \frac{1\plus{}2x^2\plus{}3x^4\plus{}4x^6\plus{}5x^8\plus{}6x^{10}\plus{}7x^{12}}{\sqrt{(1\plus{}x^2)(1\plus{}x^4)(1\plus{}x^6)}}dx.\]

Does every monotone increasing function $f : \mathbb[0,1] \rightarrow \mathbb[0,1]$ have a fixed point? What about every monotone decreasing function?

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a continuous and bounded function such that \[x\int_{x}^{x+1}f(t)\, \text{d}t=\int_{0}^{x}f(t)\, \text{d}t,\quad\text{for any}\ x\in\mathbb{R}.\] Prove that $f$ is a constant function.

Let $a(x)$ and $b(x)$ be continuous functions on $[0,1]$ and let $0 \leq a(x) \leq a <1$ on that range. Under what other conditions (if any) is the solution of the equation for $u,$ $$ u= \max_{0 \leq x \leq 1} b(x) +a(x)u$$ given by $$u = \max_{0 \leq x \leq 1} \frac{b(x)}{1-a(x)}.$$

Let $f(x)$ be a continuous function defined on $0\leq x\leq \pi$ and satisfies $f(0)=1$ and \[\left\{\int_0^{\pi} (\sin x+\cos x)f(x)dx\right\}^2=\pi \int_0^{\pi}\{f(x)\}^2dx.\] Evaluate $\int_0^{\pi} \{f(x)\}^3dx.$