Prove that for every $n\in \mathbb N$, there exists a set $S$ of $n$ positive integers such that for any two distinct $a,b\in S$, $a-b$ divides $a$ and $b$ but none of the other elements of $S$. [i]Proposed by Iurie Boreico[/i]

A function $f_n(x)\ (n=1,\ 2,\ \cdots)$ is defined by $f_1(x)=x$ and \[f_n(x)=x+\frac{e}{2}\int_0^1 f_{n-1}(t)e^{x-t}dt\ (n=2,\ 3,\ \cdots)\]. Find $f_n(x)$.

Let $m\geq 2$ be an integer. A positive integer $n$ has the property that for any positive integer $a$ coprime with $n$, we have $a^m - 1\equiv 0 \pmod n$. Prove that $n \leq 4m(2^m-1)$. Created by Harazi, modified by Marian Andronache.

Find all $f: R \longrightarrow R$ such that \[f(xy+f(x))=xf(y)+f(x)\] for every pair of real numbers $x,y$.

Find all functions $f:(0,+\infty) \to (0,+\infty)$ that satisfy $(i)$ $f(xf(y))=yf(x), \forall x,y > 0,$ $(ii)$ $\displaystyle\lim_{x\to+\infty} f(x) = 0.$

Let $f$ be continuous and nowhere monotone on $[0,1]$. Show that the set of points on which $f$ obtains a local minimum is dense.

Let $f(x)=\frac{9^x}{9^x+3}$. Compute $\sum_{k} f \biggl( \frac{k}{2002} \biggr)$, where $k$ goes over all integers $k$ between $0$ and $2002$ which are coprime to $2002$.

$(a)$ Prove that for $a, b, c, d \in\mathbb{R}, m \in [1,+\infty)$ with $am + b =-cm + d = m$, \[(i)\sqrt{a^2 + b^2}+\sqrt{c^2 + d^2}+\sqrt{(a-c)^2 + (b-d)^2}\ge \frac{4m^2}{1+m^2},\text{ and}\] \[(ii) 2 \le \frac{4m^2}{1+m^2} < 4.\] $(b)$ Express $a, b, c, d$ as functions of $m$ so that there is equality in $(i).$

Find all functions $f : R \to R$ such that $f(x^2)-f(y^2) \le (f(x)+y) (x-f(y))$ for all $x, y \in R$.

Let \( n \) be a positive integer. Determine the largest possible value of \( k \) with the following property: there exists a bijective function \( \phi: [0, 1] \to [0, 1]^k \) and a constant \( C > 0 \) such that, for all \( x, y \in [0, 1] \), \[ \| \phi(x) - \phi(y) \| \leq C \| x - y \|^k. \] Note: \( \| \cdot \| \) denotes the Euclidean norm, that is, \( \| (a_1, \ldots, a_n) \| = \sqrt{a_1^2 + \cdots + a_n^2} \).

(1) Evaluate $ \int_0^{\frac{\pi}{2}} (x\cos x\plus{}\sin ^ 2 x)\sin x\ dx$. (2) For $ f(x)\equal{}\int_0^x e^t\sin (x\minus{}t)\ dt$, find $ f''(x)\plus{}f(x)$.

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 $n$.

A function $f(n)$ is defined on the set of positive integers is said to be multiplicative if $f(mn)=f(m)f(n)$ whenever $m$ and $n$ have no common factors greater than $1$. Are the following functions multiplicative? Justify your answer. (a) $g(n)=5^k$ where $k$ is the number of distinct primes which divide $n$. (b) $h(n)=\begin{cases} 0 & \text{if} \ n \ \text{is divisible by} \ k^2 \ \text{for some integer} \ k>1 \\ 1 & \text{otherwise} \end{cases}$

The function $f:\mathbb{N}\rightarrow \mathbb{N}$ is [b]peruvian[/b] if it satifies the following two properties: $\triangleright f$ is strictly increasing. $\triangleright$ The numbers $a_1,a_2,a_3,\dots$ where $a_1=f(1)$ and $a_{n+1}=f(a_n)$ for every $n\geq 1$, are in arithmetic progression. Determine all peruvian functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that $f(1)=3$.

Prove that there is exactly a function $ f:\mathbb{R}_{\ge 0}\longrightarrow\mathbb{R}_{\ge 0} $ satisfying the following two properties: $ \text{(i)} x\in\mathbb{R}_{> 0}\implies \left( f(x)+f(f(x)) =4018020x \wedge f(x)>0 \right) $ $ \text{(ii)} 0=f(0)+f(f(0)) $

Let $f(x)=\frac{\sin x}{x}$, for $x>0$, and let $n$ be a positive integer. Prove that $|f^{(n)}(x)|<\frac{1}{n+1}$, where $f^{(n)}$ denotes the $n^{\mathrm{th}}$ derivative of $f$. (Proposed by Alexander Bolbot, State University, Novosibirsk)

Prove that the function $f (x)$ satisfying the relation $|f (x) - f (y) | \le | x - y|^a$ for any real numbers $x, y$ and some number $a> 1$ is constant.

Let $f$ be a function defined on $\{(i,j): i,j \in \mathbb{N}\}$ satisfying (i) $f(i,i+1)=\frac{1}{3}$ for all $i$ (ii) $f(i,j)=f(i,k)+f(k,j)-2f(i,k)f(k,j)$ for all $k$ such that $i <k<j$. Find the value of $f(1,100)$.

Evaluate $ \int_0^1 \frac{x}{1\plus{}x}\sqrt{1\minus{}x^2}\ dx$.

Consider a charged capacitor made with two square plates of side length $L$, uniformly charged, and separated by a very small distance $d$. The EMF across the capacitor is $\xi$. One of the plates is now rotated by a very small angle $\theta$ to the original axis of the capacitor. Find an expression for the difference in charge between the two plates of the capacitor, in terms of (if necessary) $d$, $\theta$, $\xi$, and $L$. Also, approximate your expression by transforming it to algebraic form: i.e. without any non-algebraic functions. For example, logarithms and trigonometric functions are considered non-algebraic. Assume $ d << L $ and $ \theta \approx 0 $. $\emph{Hint}$: You may assume that $ \frac {\theta L}{d} $ is also very small. [i]Problem proposed by Trung Phan[/i] [hide="Clarification"] There are two possible ways to rotate the capacitor. Both were equally scored but this is what was meant: [asy]size(6cm); real h = 7; real w = 2; draw((-w,0)--(-w,h)); draw((0,0)--(0,h), dashed); draw((0,0)--h*dir(64)); draw(arc((0,0),2,64,90)); label("$\theta$", 2*dir(77), dir(77)); [/asy] [/hide]

Let $\mathbb X$ be the set of all bijective functions from the set $S=\{1,2,\cdots, n\}$ to itself. For each $f\in \mathbb X,$ define \[T_f(j)=\left\{\begin{aligned} 1, \ \ \ & \text{if} \ \ f^{(12)}(j)=j,\\ 0, \ \ \ & \text{otherwise}\end{aligned}\right.\] Determine $\sum_{f\in\mathbb X}\sum_{j=1}^nT_{f}(j).$ (Here $f^{(k)}(x)=f(f^{(k-1)}(x))$ for all $k\geq 2.$)

Let $ y$ be the function of $ x$ satisfying the differential equation $ y'' \minus{} y \equal{} 2\sin x$. (1) Let $ y \equal{} e^xu \minus{} \sin x$, find the differential equation with which the function $ u$ with respect to $ x$ satisfies. (2) If $ y(0) \equal{} 3,\ y'(0) \equal{} 0$, then determine $ y$.

Find all functions $f : \mathbb{Q} \to \mathbb{R}$ such that $f(x)f(y)f(x+y) = f(xy)(f(x) + f(y))$ for all $x,y\in\mathbb{Q}$. [i]Sammy Luo and Alex Zhu.[/i]

Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ from $C^2$ (id est, $f$ is twice differentiable and $f''$ is continuous.) such that for every real number $t$ we have $f(t)^2=f(t \sqrt{2})$.

Find all functions $f : R \to R$ such that $f(x^2)-f(y^2) \le (f(x)+y) (x-f(y))$ for all $x, y \in R$.