Find all $n\in \mathbb{N} $ so that 7 divides $5^n + 1$

The sequence $\{x_n\}_{n\ge1}$ is defined by $x_1 = 7$, $x_{n+1} = 2x^2_n - 1$, for all positive integers $n$. Prove that for all positive integers $n$ the number $x_n$ cannot be divisible by $2003$.

There are $10$ people who want to choose a committee of 5 people among them. They do this by first electing a set of $1, 2, 3,$ or $4$ committee leaders, who then choose among the remaining people to complete the 5-person committee. In how many ways can the committee be formed, assuming that people are distinguishable? (Two committees that have the same members but different sets of leaders are considered to be distinct.)

In a triangle $ABC$, $M$ and $N$ are the respective midpoints of the sides $AC$ and $AB$, and $P$ is the point of intersection of $BM$ and $CN$. Prove that, if it is possible to inscribe a circle in the quadrilateral $AMPN$, then the triangle $ABC$ is isosceles.

Find all quadruples $(a,b,c,d)$ of positive real numbers such that $abcd=1,a^{2012}+2012b=2012c+d^{2012}$ and $2012a+b^{2012}=c^{2012}+2012d$.

$2019$ circles split a plane into a number of parts whose boundaries are arcs of those circles. How many colors are needed to color this geographic map if any two neighboring parts must be coloured with different colours?

Prove that every real positive number may be represented as a sum of nine numbers whose decimal representation consists of the digits $0$ and $7$. (E Turkevich)

Find the minimum of $x^2 - 2x$ over all real numbers $x.$

In town $ A,$ there are $ n$ girls and $ n$ boys, and each girl knows each boy. In town $ B,$ there are $ n$ girls $ g_1, g_2, \ldots, g_n$ and $ 2n \minus{} 1$ boys $ b_1, b_2, \ldots, b_{2n\minus{}1}.$ The girl $ g_i,$ $ i \equal{} 1, 2, \ldots, n,$ knows the boys $ b_1, b_2, \ldots, b_{2i\minus{}1},$ and no others. For all $ r \equal{} 1, 2, \ldots, n,$ denote by $ A(r),B(r)$ the number of different ways in which $ r$ girls from town $ A,$ respectively town $ B,$ can dance with $ r$ boys from their own town, forming $ r$ pairs, each girl with a boy she knows. Prove that $ A(r) \equal{} B(r)$ for each $ r \equal{} 1, 2, \ldots, n.$

Let $a > 3$ be an odd integer. Show that for every positive integer $n$ the number $a^{2^n}- 1$ has at least $n + 1$ distinct prime divisors.

The notation $\phi (n)$ is the number of positive integers smaller than $n$ and coprime with $n$, $\pi (n)$ is the number of primes that do not exceed $n$. Prove that for any natural number $n > 1$, we have $$\phi (n) \ge \frac{\pi (n)}{2}$$

Let $a$ and $b$ be real numbers such that $a \ne 0$. Prove that not all the roots of $ax^4 + bx^3 + x^2 + x + 1 = 0$ can be real.

Determine all pairs of natural numbers $(m, r)$ with $2014 \ge m \ge r \ge 1$ that fulfill $\binom{2014}{m}+\binom{m}{r}=\binom{2014}{r}+\binom{2014-r}{m-r} $

Triangle $ABC$ is such that $AB < AC$. The perpendicular bisector of side $BC$ intersects lines $AB$ and $AC$ at points $P$ and $Q$, respectively. Let $H$ be the orthocentre of triangle $ABC$, and let $M$ and $N$ be the midpoints of segments $BC$ and $PQ$, respectively. Prove that lines $HM$ and $AN$ meet on the circumcircle of $ABC$.

Let $\sigma : \{1, 2, . . . , n\} \to \{1, 2, . . . , n\}$ be a bijective mapping. Let $S_n$ be the set of all such mappings and let $d_k(\sigma) = |\sigma(k) - \sigma(k + 1)|$, for all $k \in \{1, 2, ..., n\}$, where $\sigma (n + 1) = \sigma (1)$. Also let $d(\sigma) = \min \{d_k(\sigma) | 1 \le k \le n\}$. Find $\max_{\sigma \in S_n} d(\sigma)$.

A function $g \colon \mathbb{R} \to \mathbb{R}$ is given such that its range is a finite set. Find all functions $f \colon \mathbb{R} \to \mathbb{R}$ that satisfies $$2f(x+g(y))=f(2g(x)+y)+f(x+3g(y))$$ for all $x, y \in \mathbb{R}$.

If $a$ and $b$ are positive integers such that $3\sqrt{2+\sqrt{2+\sqrt{3}}}=a\cos\frac{\pi}{b}$, find $a+b$.

Let $n \geq 3$ and $a_1,a_2,...,a_n$ be complex numbers different from $0$ with $|a_i| < 1$ for all $i \in \{1,2,...,n-1 \}.$ If the coefficients of $f = \prod_{i=1}^n (X-a_i)$ are integers, prove that $\textbf{a)}$ The numbers $a_1,a_2,...,a_n$ are distinct. $\textbf{b)}$ If $a_j^2 = a_ia_k,$ then $i=j=k.$

An infinite sequence of integers, $a_0,a_1,a_2,\dots,$ with $a_0>0$, has the property that for $n\ge 0$, $a_{n+1}=a_n-b_n$, where $b_n$ is the number having the same sign as $a_n$, but having the digits written in the reverse order. For example if $a_0=1210,a_1=1089$ and $a_2=-8712$, etc. Find the smallest value of $a_0$ so that $a_n\neq 0$ for all $n\ge 1$.

Real numbers $ a_{1}$, $ a_{2}$, $ \ldots$, $ a_{n}$ are given. For each $ i$, $ (1 \leq i \leq n )$, define \[ d_{i} \equal{} \max \{ a_{j}\mid 1 \leq j \leq i \} \minus{} \min \{ a_{j}\mid i \leq j \leq n \} \] and let $ d \equal{} \max \{d_{i}\mid 1 \leq i \leq n \}$. (a) Prove that, for any real numbers $ x_{1}\leq x_{2}\leq \cdots \leq x_{n}$, \[ \max \{ |x_{i} \minus{} a_{i}| \mid 1 \leq i \leq n \}\geq \frac {d}{2}. \quad \quad (*) \] (b) Show that there are real numbers $ x_{1}\leq x_{2}\leq \cdots \leq x_{n}$ such that the equality holds in (*). [i]Author: Michael Albert, New Zealand[/i]

Some manors of Lipshire county are connected by roads. The inhabitants of manors connected by a road are called neighbours. Is it always possible to settle in each manor a knight (who always tells truth) or a liar (who always lies) so that every inhabitant can say ”The number of liars among my neighbours is at least twice the number of knights”?

Let $f : \mathbb{N} \to \mathbb{N}$ be a function such that the following conditions hold: $\qquad\ (1) \; f(1) = 1.$ $\qquad\ (2) \; \dfrac{(x + y)}{2} < f(x + y) \le f(x) + f(y) \; \forall \; x, y \in \mathbb{N}.$ $\qquad\ (3) \; f(4n + 1) < 2f(2n + 1) \; \forall \; n \ge 0.$ $\qquad\ (4) \; f(4n + 3) \le 2f(2n + 1) \; \forall \; n \ge 0.$ Find the sum of all possible values of $f(2023)$.

Let $n$ be a fixed positive integer. Ben is playing a computer game. The computer picks a tree $T$ such that no vertex of $T$ has degree $2$ and such that $T$ has exactly $n$ leaves, labeled $v_1,\ldots, v_n$. The computer then puts an integer weight on each edge of $T$, and shows Ben neither the tree $T$ nor the weights. Ben can ask queries by specifying two integers $1\leq i < j \leq n$, and the computer will return the sum of the weights on the path from $v_i$ to $v_j$. At any point, Ben can guess whether the tree's weights are all zero. He wins the game if he is correct, and loses if he is incorrect. (a) Show that if Ben asks all $\binom n2$ possible queries, then he can guarantee victory. (b) Does Ben have a strategy to guarantee victory in less than $\binom n2$ queries? [i]Brandon Wang[/i]

Suppose $ f(x)$ is defined for all real numbers $ x$; $ f(x)>0$ for all $ x$, and $ f(a)f(b)\equal{}f(a\plus{}b)$ for all $ a$ and $ b$. Which of the following statements is true? $ \text{I. } f(0)\equal{}1$ $ \text{II. } f(\minus{}a)\equal{}1/f(a) \text{ for all } a$ $ \text{III. } f(a) \equal{} \sqrt[3]{f(3a)} \text{ for all } a$ $ \text{IV. } f(b)>f(a) \text{ if } b>a$ $ \textbf{(A)}\ \text{III and IV only} \qquad$ $ \textbf{(B)}\ \text{I, III, and IV only} \qquad$ $ \textbf{(C)}\ \text{I, II, and IV only} \qquad$ $ \textbf{(D)}\ \text{I, II, and III only} \qquad$ $ \textbf{(E)}\ \text{All are true.}$

All of the numbers $1, 2,3,...,1000000$ are initially colored black. On each move it is possible to choose the number $x$ (among the colored numbers) and change the color of $x$ and of all of the numbers that are not co-prime with $x$ (black into white, white into black). Is it possible to color all of the numbers white?