Prove that for any natural number $ n $ there are $ n $ consecutive numbers, each one of these numbers having the following property: the sum of the positive divisors of a number $ x $ is greater than $ 2x. $

Arrange the following four numbers from smallest to largest $ a \equal{} (10^{100})^{10}$, $ b \equal{} 10^{(10^{10})}$, $ c \equal{} 1000000!$, $ d \equal{} (100!)^{10}$

Let $M$ be a set of points in a plane with at least two elements. Prove that if $M$ has two axes of symmetry $g_1$ and $g_2$ intersecting at an angle $\alpha = q\pi$, where $q$ is irrational, then $M$ must be infinite.

Prove that for any triangle $ABC$ there exists a point P in the plane of the triangle and three points $A' , B'$ , and $C'$ on the lines $BC, AC$, and $AB$ respectively such that \[AB \cdot PC'= AC \cdot PB'= BC \cdot PA'= 0.3M^2,\] where $M = max\{AB,AC,BC\}$.

Each side of the convex quadrilateral was continued into both sides and on all eight extensions set aside equal segments. It turned out that the resulting $8$ points are the outer ends of the construction the given segments are different and lie on the same circle. Prove that the original quadrilateral is a square.

consider a $2008 \times 2008$ chess board. let $M$ be the smallest no of rectangles that can be drawn on the chess board so that sides of every cell of the board is contained in the sides of one of the rectangles. find the value of $M$. (eg for $2\times 3$ chessboard, the value of $M$ is 3.)

Find all integers $x,y$ such that $x^2(y-1)+y^2(x-1) = 1$.

Let $\alpha$ be the unique real root of the polynomial $x^3-2x^2+x-1$. It is known that $1<\alpha<2$. We define the sequence of polynomials $\left\{{p_n(x)}\right\}_{n\ge0}$ by taking $p_0(x)=x$ and setting \begin{align*} p_{n+1}(x)=(p_n(x))^2-\alpha \end{align*} How many distinct real roots does $p_{10}(x)$ have?

Find all positive integers $n$ for which $2^n + 2021n$ is a perfect square.

[i]Nocycleland[/i] is a country with $500$ cities and $2013$ two-way roads, each one of them connecting two cities. A city $A$ [i]neighbors[/i] $B$ if there is one road that connects them, and a city $A$ [i]quasi-neighbors[/i] $B$ if there is a city $C$ such that $A$ neighbors $C$ and $C$ neighbors $B$. It is known that in Nocycleland, there are no pair of cities connected directly with more than one road, and there are no four cities $A$, $B$, $C$ and $D$ such that $A$ neighbors $B$, $B$ neighbors $C$, $C$ neighbors $D$, and $D$ neighbors $A$. Show that there is at least one city that quasi-neighbors at least $57$ other cities.

Nick is a runner, and his goal is to complete four laps around a circuit at an average speed of $10$ mph. If he completes the first three laps at a constant speed of only $9$ mph, what speed does he need to maintain in miles per hour on the fourth lap to achieve his goal?

A line through the point $(-a,0)$ cuts from the second quadrant a triangular region with area $T$. The equation of the line is: $ \textbf{(A)}\ 2Tx+a^2y+2aT=0 \qquad\textbf{(B)}\ 2Tx-a^2y+2aT=0 \qquad$ ${{\textbf{(C)}\ 2Tx+a^2y-2aT=0 \qquad\textbf{(D)}\ 2Tx-a^2y-2aT=0 }\qquad\textbf{(E)}\ \text{none of these} } $

Given positive numbers $a_1$ and $b_1$, consider the sequences defined by \[a_{n+1}=a_n+\frac{1}{b_n},\quad b_{n+1}=b_n+\frac{1}{a_n}\quad (n \ge 1)\] Prove that $a_{25}+b_{25} \geq 10\sqrt{2}$.

In the figure, BC  is a diameter of the circle, where $BC=\sqrt{901}, BD=1$, and $DA=16$. If $EC=x$, what is the value of x? [asy]size(2inch); pair O,A,B,C,D,E; B=(0,0); O=(2,0); C=(4,0); D=(.333,1.333); A=(.75,2.67); E=(1.8,2); draw(Arc(O,2,0,360)); draw(B--C--A--B); label("$A$",A,N); label("$B$",B,W); label("$C$",C,E); label("$D$",D,W); label("$E$",E,N); label("Figure not drawn to scale",(2,-2.5),S); [/asy]

Find all pairs of integers $(a,b)$ such that $a^2+ab+1\mid b^2+ab+a+b-1$.

For positive real numbers $b_1,b_2,\ldots,b_n$ define $$a_1=\frac{b_1}{b_1+b_2+\ldots+b_n}\enspace\text{ and }\enspace a_k=\frac{b_1+\ldots+b_k}{b_1+\ldots+b_{k-1}}\text{ for }k>1.$$Prove that $a_1+a_2+\ldots+a_n\le\frac1{a_1}+\frac1{a_2}+\ldots+\frac1{a_n}$

Let $n\in \mathbb{N}^*$ and a matrix $A\in \mathcal{M}_n(\mathbb{C}),\ A=(a_{ij})_{1\le i, j\le n}$ such that: \[a_{ij}+a_{jk}+a_{ki}=0,\ (\forall)i,j,k\in \{1,2,\ldots,n\}\] Prove that $\text{rank}\ A\le 2$.

Let $\mathcal{S}$ be a finite set of at least two points in the plane. Assume that no three points of $\mathcal S$ are collinear. A [i]windmill[/i] is a process that starts with a line $\ell$ going through a single point $P \in \mathcal S$. The line rotates clockwise about the [i]pivot[/i] $P$ until the first time that the line meets some other point belonging to $\mathcal S$. This point, $Q$, takes over as the new pivot, and the line now rotates clockwise about $Q$, until it next meets a point of $\mathcal S$. This process continues indefinitely. Show that we can choose a point $P$ in $\mathcal S$ and a line $\ell$ going through $P$ such that the resulting windmill uses each point of $\mathcal S$ as a pivot infinitely many times. [i]Proposed by Geoffrey Smith, United Kingdom[/i]

What is the maximum area of an isosceles trapezoid that has legs of length $1$ and one base twice as long as the other? $ \textbf{(A) }\frac 54 \qquad \textbf{(B) } \frac 87 \qquad \textbf{(C)} \frac{5\sqrt2}4 \qquad \textbf{(D) } \frac 32 \qquad \textbf{(E) } \frac{3\sqrt3}4 $

When Walter drove up to the gasoline pump, he noticed that his gasoline tank was $\frac{1}{8}$ full. He purchased $7.5$ gallons of gasoline for $ \$10$. With this additional gasoline, his gasoline tank was then $\frac{5}{8}$ full. The number of gallons of gasoline his tank holds when it is full is $\text{(A)}\ 8.75 \qquad \text{(B)}\ 10 \qquad \text{(C)}\ 11.5 \qquad \text{(D)}\ 15 \qquad \text{(E)}\ 22.5$

Let $a, b, c$ be integers not all the same with $a, b, c\ge 4$ that satisfy $$4abc = (a + 3) (b + 3) (c + 3).$$ Find the numerical value of $a + b + c$.

Circles $\omega_1$ and $\omega_2$ intersect at points $A$ and $B$. A straight line is drawn through point $B$, which again intersects circles $\omega_1$ and $\omega_2$ at points $C$ and $D$, respectively. Point $E$, located on circle $\omega_1$ , satisfies the relation $CE = CB$ , and point $F$, located on circle $\omega_2$, satisfies the relation $DB = DF$. The line $BF$ intersects again the circle $\omega_1$ at the point $P$, and the line $BE$ intersects again the circle $\omega_2$ at the point $Q$. Prove that the points $A, P$, and $Q$ are collinear.

Considering all numbers of the form $n = \lfloor \frac{k^3}{2012} \rfloor$, where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$, and $k$ ranges from $1$ to $2012$, how many of these $n$’s are distinct?

Let $x$, $y$, and $z$ be positive real numbers satisfying $x^2+y^2+z^2=1$. Prove that \[x^2yz+xy^2z+xyz^2\le\frac{1}{3}.\]

A point $A_0$ and two lines $d_1$ and $d_2$ are given in the space. For each nonnegative integer $n$ we denote by $B_n$ the projection of $A_n$ on $d_2,$ and by $A_{n+1}$ the projection of $B_n$ on $d_1.$ Prove that there exist two segments $[A'A''] \subset d_1$ and $[B'B''] \subset d_2$ of length $0.001$ and a nonnegative integer $N$ such that $A_n \in [A'A'']$ and $B_n \in [B'B'']$ for any $n \ge N.$