There are given two positive real number $a<b$. Show that there exist positive integers $p, \ q, \ r, \ s$ satisfying following conditions: $1$. $a< \frac{p}{q} < \frac{r}{s} < b$. $2.$ $p^2+q^2=r^2+s^2$.

The points $A, B, C$ and $D$ lie in this order on the circle $k$. Let $t$ be the tangent at $k$ through $C$ and $s$ the reflection of $AB$ at $AC$. Let $G$ be the intersection of the straight line $AC$ and $BD$ and $H$ the intersection of the straight lines $s$ and $CD$. Show that $GH$ is parallel to $t$.

For each positive number $x$, let \[f(x)=\displaystyle\frac{\left(x+\dfrac{1}{x}\right)^6-\left(x^6+\dfrac{1}{x^6}\right)-2}{\left(x+\frac{1}{x}\right)^3+\left(x^3+\frac{1}{x^3}\right)}.\] The minimum value of $f(x)$ is $\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad\textbf{(E) }6$

Determine all pairs $(a, b)$ of positive integers for which the equation $x^3 - 17x^2 + ax - b^2 = 0$ has three integer roots (not necessarily different).

Let $R$ and $S$ be different points on a circle $\Omega$ such that $RS$ is not a diameter. Let $\ell$ be the tangent line to $\Omega$ at $R$. Point $T$ is such that $S$ is the midpoint of the line segment $RT$. Point $J$ is chosen on the shorter arc $RS$ of $\Omega$ so that the circumcircle $\Gamma$ of triangle $JST$ intersects $\ell$ at two distinct points. Let $A$ be the common point of $\Gamma$ and $\ell$ that is closer to $R$. Line $AJ$ meets $\Omega$ again at $K$. Prove that the line $KT$ is tangent to $\Gamma$. [i]Proposed by Charles Leytem, Luxembourg[/i]

Circles with centers $A ,~ B,$ and $C$ each have radius $r$, where $1 < r < 2$. The distance between each pair of centers is $2$. If $B'$ is the point of intersection of circle $A$ and circle $C$ which is outside circle $B$, and if $C'$ is the point of intersection of circle $A$ and circle $B$ which is outside circle $C$, then length $B'C'$ equals $\textbf{(A) }3r-2\qquad\textbf{(B) }r^2\qquad\textbf{(C) }r+\sqrt{3(r-1)}\qquad$ $\textbf{(D) }1+\sqrt{3(r^2-1)}\qquad\textbf{(E) }\text{none of these}$ [asy] //Holy crap, CSE5 is freaking amazing! import cse5; pathpen=black; pointpen=black; dotfactor=3; size(200); pair A=(1,2),B=(2,0),C=(0,0); D(CR(A,1.5)); D(CR(B,1.5)); D(CR(C,1.5)); D(MP("$A$",A)); D(MP("$B$",B)); D(MP("$C$",C)); pair[] BB,CC; CC=IPs(CR(A,1.5),CR(B,1.5)); BB=IPs(CR(A,1.5),CR(C,1.5)); D(BB[0]--CC[1]); MP("$B'$",BB[0],NW);MP("$C'$",CC[1],NE); //Credit to TheMaskedMagician for the diagram [/asy]

Consider the equation $ y^2\minus{}k\equal{}x^3$, where $ k$ is an integer. Prove that the equation cannot have five integer solutions of the form $ (x_1,y_1),(x_2,y_1\minus{}1),(x_3,y_1\minus{}2),(x_4,y_1\minus{}3),(x_5,y_1\minus{}4)$. Also show that if it has the first four of these pairs as solutions, then $ 63|k\minus{}17$.

Find $f: \mathbb{Z}_+ \rightarrow \mathbb{Z}_+$, such that for any $x,y \in \mathbb{Z}_+$, $$f(f(x)+y)\mid x+f(y).$$

Let $ a\ge 2 $ be a natural number. Show that the following relations are equivalent: $ \text{(i)} \ a $ is the hypothenuse of a right triangle whose sides are natural numbers. $ \text{(ii)}\quad $ there exists a natural number $ d $ for which the polynoms $ X^2-aX\pm d $ have integer roots.

Evaluate $\int_{\sqrt{2}}^{\sqrt{3}} (x^2+\sqrt{x^4-1})(\frac{1}{\sqrt{x^2+1}}+{\frac{1}{\sqrt{x^2-1}})dx.}$ [i]Proposed by kunny[/i]

$a,b$ are real numbers, satisfying that $\frac{1}{a}+\frac{1}{b}=1$. Prove that for any $n\in\mathbb{Z}_+$, $(a+b)^{2n}-a^n-b^n\geq2^{2n}-2^{n+1}$.

9.7 Is there an infinite arithmetic sequence $\{a_n\}\subset \mathbb N$ s.t. $a_n+...+a_{n+9}\mid a_n...a_{n+9}$ for all $n$? ([i]V. Senderov[/i])

In a square $ABCD$ of diagonals $AC$ and $BD$, we call $O$ at the center of the square. A square $PQRS$ is constructed with sides parallel to those of $ABCD$ with $P$ in segment $AO, Q$ in segment $BO, R$ in segment $CO, S$ in segment $DO$. If area of $ABCD$ equals two times the area of $PQRS$, and $M$ is the midpoint of the $AB$ side, calculate the measure of the angle $\angle AMP$.

Arman, starting from a number, calculates the sum of the cubes of the digits of that number, and again calculates the sum of the cubes of the digits of the resulting number and continues the same process. Arman calls a number $Good$ if it reaches $1$ after performing a number of steps. Prove that there is an arithmetic progression of length $1402$ of good numbers. [i]Proposed by Navid Safaei [/i]

a) Let $f\in C[0,b]$, $g\in C(\mathbb R)$ and let $g$ be periodic with period $b$. Prove that $\int_0^b f(x) g(nx)\,\mathrm dx$ has a limit as $n\to\infty$ and $$\lim_{n\to\infty}\int_0^b f(x)g(nx)\,\mathrm dx=\frac 1b \int_0^b f(x)\,\mathrm dx\cdot\int_0^b g(x)\,\mathrm dx$$ b) Find $$\lim_{n\to\infty}\int_0^\pi \frac{\sin x}{1+3\cos^2nx}\,\mathrm dx$$

We have a 4x4 board.All 1x1 squares are white.A move is changing colours of all squares of a 1x3 rectangle from black to white and from white to black.It is possible to make all the 1x1 squares black after several moves?

Let $ABCD$ be a parallelogram. Suppose a line passing through $C$ and lying outside the parallelogram meets $AB$ and $AD$ produced at $E$ and $F$ respectively. Show that \[ AC^2 + CE \cdot CF = AB \cdot AE + AD \cdot AF . \]

i) If $b=\dfrac{a^2+ \dfrac{1}{a^2}}{a^2-\dfrac{1}{a^2}}$ , express $c=\dfrac{a^4+\dfrac{1}{a^4}}{a^4-\dfrac{1}{a^4}}$ , in terms of $b$. ii) If $k= \frac{x^{n}+\dfrac{1}{x^{n}}}{x^{n}-\dfrac{1}{x^{n}}}$, express $m= \frac{x^{2n}+\dfrac{1}{x^{2n}}}{x^{2n}-\dfrac{1}{x^{2n}}}$ in terms of $k$.

Find the least positive integer which is a multiple of $13$ and all its digits are the same. [i](Adapted from Gazeta Matematică 1/1982, Florin Nicolăită)[/i]

Anton is playing a game with shapes. He starts with a circle $\omega_1$ of radius $1$, and to get a new circle $\omega_2$, he circumscribes a square about $\omega_1$ and then circumscribes circle $\omega_2$ about that square. To get another new circle $\omega_3$, he circumscribes a regular octagon about circle $\omega_2$ and then circumscribes circle $\omega_3$ about that octagon. He continues like this, circumscribing a $2n$-gon about $\omega_{n-1}$ and then circumscribing a new circle $\omega_n$ about the $2n$-gon. As $n$ increases, the area of $\omega_n$ approaches a constant $A$. Compute $A$.

Let $n\in\mathbb{Z}$, $n\geq 3$. A matrix $A\in\mathcal{M}_n(\mathbb{C})$ is said to have property $(\mathcal{P})$ if $\det(A+X_{ij})=\det(A+X_{ji})$, for all $i,j\in\{1,2,\dots ,n\}$, where $X_{ij}\in\mathcal{M}_n(\mathbb{C})$ is the matrix with $1$ on position $(i,j)$ and $0$ otherwise. [list=a] [*] Show that if $A\in\mathcal{M}_n(\mathbb{C})$ has property $(\mathcal{P})$ and $\det(A)\neq 0$, then $A=A^T$. [*] Give an example of a matrix $A\in\mathcal{M}_n(\mathbb{C})$ with property $(\mathcal{P})$ such that $A\neq A^T$.

The sequences $(x_n), (y_n), (z_n)$ are given by $x_{n+1}=y_n +\frac{1}{x_n}$,$ y_{n+1}=z_n +\frac{1}{y_n}$,$z_{n+1}=x_n +\frac{1}{z_n} $ for $n \ge 0$ where $x_0,y_0, z_0$ are given positive numbers. Prove that these sequences are unbounded.

Let $a, b, c $ be positive real numbers such that $abc = \frac {2} {3}. $ Prove that: $$\frac {ab}{a + b} + \frac {bc} {b + c} + \frac {ca} {c + a} \geqslant \frac {a+b+c} {a^3+b ^ 3 + c ^ 3}.$$

In a sequence of numbers, a term is called [i]golden [/i] if it is divisible by the term immediately before it. What is the maximum possible number of golden terms in a permutation of $1, 2, 3, . . . , 2021$?

Find the number of functions $f(x)$ from $\{1,2,3,4,5\}$ to $\{1,2,3,4,5\}$ that satisfy $f(f(x)) = f(f(f(x)))$ for all $x$ in $\{1,2,3,4,5\}$.