Prove that for any positive integer $n\ge 2$ the following inequality holds: $$\frac{1}{n+1}+\frac{1}{n+2}+...+\frac{1}{2n}>\frac{1}{2}$$

Let $M$ be a nonempty subset of $\mathbb Z^+$ such that for every element $x$ in $M,$ the numbers $4x$ and $\lfloor \sqrt x \rfloor$ also belong to $M.$ Prove that $M = \mathbb Z^+.$

Zara has a collection of $4$ marbles: an Aggie, a Bumblebee, a Steelie, and a Tiger. She wants to display them in a row on a shelf, but does not want to put the Steelie and the Tiger next to one another. In how many ways can she do this? $\textbf{(A) }6 \qquad \textbf{(B) }8 \qquad \textbf{(C) }12 \qquad \textbf{(D) }18 \qquad \textbf{(E) }24$

Given positive integers $k,n$ and a real number $\ell$, where $k,n \geq 1$. Given are also pairwise different positive real numbers $a_1,a_2,\ldots, a_k$. Let $S = \{a_1,a_2,\ldots,a_k, -a_1, -a_2,\ldots, -a_k\}$. Let $A$ be the number of solutions of the equation $$x_1 + x_2 + \ldots + x_{2n} = 0,$$ where $x_1,x_2,\ldots, x_{2n} \in S$. Let $B$ be the number of solutions of the equation $$x_1 + x_2 + \ldots + x_{2n} = \ell,$$ where $x_1,x_2,\ldots,x_{2n} \in S$. Prove that $A \geq B$. Solutions of an equation with only difference in the permutation are different.

A piece of rectangular paper $20 \times 19$, divided into four units, is cut into several square pieces, the cuts being along the sides of the unit squares. Such a square piece is called odd square if the length of its side is an odd number. a) What is the minimum possible number of odd squares? b) What is the smallest value that the sum of the perimeters of the odd squares can take?

A car is being driven so that its wheels, all of radius $a$ feet, have an angular velocity of $\omega$ radians per second. A particle is thrown off from the tire of one of these wheels, where it is supposed that $a \omega^{2} >g$. Neglecting the resistance of the air, show that the maximum height above the roadway which the particle can reach is $$\frac{(a \omega+g \omega^{-1})^{2}}{2g}.$$

Denote by $d(n)$ the number of divisors of the positive integer $n$. A positive integer $n$ is called highly divisible if $d(n) > d(m)$ for all positive integers $m < n$. Two highly divisible integers $m$ and $n$ with $m < n$ are called consecutive if there exists no highly divisible integer $s$ satisfying $m < s < n$. (a) Show that there are only finitely many pairs of consecutive highly divisible integers of the form $(a, b)$ with $a\mid b$. (b) Show that for every prime number $p$ there exist infinitely many positive highly divisible integers $r$ such that $pr$ is also highly divisible.

Prove that every polynomial is a difference of two increasing polynomials.

In the $x$-$y$ plane, for $t>0$, denote by $S(t)$ the area of the part enclosed by the curve $y=e^{t^2x}$, the $x$-axis, $y$-axis and the line $x=\frac{1}{t}.$ Show that $S(t)>\frac 43.$ If necessary, you may use $e^3>20.$

Two cubes are inscribed in a sphere of radius $R$. Calculate the sum of squares of all segments connecting the vertices of one cube with the vertices of the other cube

It is known that the sum of the digits of the natural number $N$ is $100$, and the sum of the digits of the number $5N$ is $50$. Prove that $N$ is even.

Compute the minimum possible value of $(x-1)^2+(x-2)^2+(x-3)^2+(x-4)^2+(x-5)^2$ For real values $x$

For an interior point $D$ of a triangle $ABC,$ let $\Gamma_D$ denote the circle passing through the points $A, \: E, \: D, \: F$ if these points are concyclic where $BD \cap AC=\{E\}$ and $CD \cap AB=\{F\}.$ Show that all circles $\Gamma_D$ pass through a second common point different from $A$ as $D$ varies.

In a trapezoid $ABCD$, the internal bisector of angle $A$ intersects the base $BC$(or its extension) at the point $E$. Inscribed in the triangle $ABE$ is a circle touching the side $AB$ at $M$ and side $BE$ at the point $P$. Find the angle $DAE$ in degrees, if $AB:MP=2$.

Sir Alex plays the following game on a row of 9 cells. Initially, all cells are empty. In each move, Sir Alex is allowed to perform exactly one of the following two operations: [list=1] [*] Choose any number of the form $2^j$, where $j$ is a non-negative integer, and put it into an empty cell. [*] Choose two (not necessarily adjacent) cells with the same number in them; denote that number by $2^j$. Replace the number in one of the cells with $2^{j+1}$ and erase the number in the other cell. [/list] At the end of the game, one cell contains $2^n$, where $n$ is a given positive integer, while the other cells are empty. Determine the maximum number of moves that Sir Alex could have made, in terms of $n$. [i]Proposed by Warut Suksompong, Thailand[/i]

Given positive integers $a,b,$ find the least positive integer $m$ such that among any $m$ distinct integers in the interval $[-a,b]$ there are three pair-wise distinct numbers that their sum is zero. [i]Proposed by Marian Tetiva, Romania[/i]

What is the value of $1234+2341+3412+4123$? $\textbf{(A) } 10,000 \qquad \textbf{(B) }10,010 \qquad \textbf{(C) }10,110 \qquad \textbf{(D) }11,000 \qquad \textbf{(E) }11,110$

Let $A$ be an $n\times n$ nonsingular matrix with complex elements, and let $\overline{A}$ be its complex conjugate. Let $B = A\overline{A}+I$, where $I$ is the $n\times n$ identity matrix. (a) Prove or disprove: $A^{-1}BA = \overline{B}$. (b) Prove or disprove: the determinant of $A\overline{A}+I$ is real.

For which $n \in N$ is it possible to arrange a tennis tournament for doubles with $n$ players such that each player has every other player as an opponent exactly once?

The incircle and the excircle of triangle $ABC$ touch the side $AC$ at points $P$ and $Q$ respectively. The lines $BP$ and $BQ$ meet the circumcircle of triangle $ABC$ for the second time at points $P'$ and $Q'$ respectively. Prove that $$PP' > QQ'$$

Let $m$ be a positive integer and let the lines $13x+11y=700$ and $y=mx-1$ intersect in a point whose coordinates are integers. Then $m$ is: $\text{(A)} \ 4 \qquad \text{(B)} \ 5 \qquad \text{(C)} \ 6 \qquad \text{(D)} \ 7 \qquad \text{(E)} \ \text{one of the integers}~ 4,5,6,7~\text{and one other positive integer}$

Two circles $\omega_1$ and $\omega_2$ touch externally at point $P$.Let $A$ be a point on $\omega_2$ not lying on the line through the centres of the two circles.Let $AB$ and $AC$ be the tangents to $\omega_1$.Lines $BP$ and $CP$ meet $\omega_2$ for the second time at points $E$ and $F$.Prove that the line $EF$,the tangent to $\omega_2$ at $A$ and the common tangent at $P$ concur.

Let $x, y,z$ satisfy the following inequalities $\begin{cases} | x + 2y - 3z| \le 6 \\ | x - 2y + 3z| \le 6 \\ | x - 2y - 3z| \le 6 \\ | x + 2y + 3z| \le 6 \end{cases}$ Determine the greatest value of $M = |x| + |y| + |z|$.

Suppose that $a$ and $ b$ are positive integers such that $$c = a +\frac{b}{a} -\frac{1}{b}$$ is an integer. Prove that $c$ is a perfect square.

Let $m\ge 5$ be an odd integer, and let $D(m)$ denote the number of quadruples $\big(a_1, a_2, a_3, a_4\big)$ of distinct integers with $1\le a_i \le m$ for all $i$ such that $m$ divides $a_1+a_2+a_3+a_4$. There is a polynomial $$q(x) = c_3x^3+c_2x^2+c_1x+c_0$$such that $D(m) = q(m)$ for all odd integers $m\ge 5$. What is $c_1?$ $(\textbf{A})\: {-}6\qquad(\textbf{B}) \: {-}1\qquad(\textbf{C}) \: 4\qquad(\textbf{D}) \: 6\qquad(\textbf{E}) \: 11$