If $\frac{x}{x-1}=\frac{y^2+2y-1}{y^2-2y-2}$, then $x$ equals $\text{(A)}\ y^2+2y-1 \qquad \text{(B)}\ y^2+2y-2 \qquad \text{(C)}\ y^2+2y+2$ $\text{(D)}\ y^2+2y+1 \qquad \text{(E)}\ -y^2-2y+1$

Let $f(x)=x^n + a_{n-1}x^{n-1} + ...+ a_0 $ be a polynomial of degree $ n\ge 1 $ with $ n$ (not necessarily distinct) integer roots. Assume that there exist distinct primes $p_0,p_1,..,p_{n-1}$ such that $a_i > 1$ is a power of $p_i$, for all $ i=0,1,..,n-1$. Find all possible values of $ n$.

Let $ABC$ be a triangle, $I_a$ the center of the excircle at side $BC$, and $M$ its reflection across $BC$. Prove that $AM$ is parallel to the Euler line of the triangle $BCI_a$.

The polynomial $x^3 + px^2 + qx + r$ has three roots in the interval $(0,2)$. Prove that $-2 <p + q + r < 0$.

The ellipsoid $E$ is contained in the simplex $S$, which is located in the unit ball B space $R^n$. Prove that the sum of the principal semi-axes of the ellipsoid $E$ is no more than units.

Find all the numbers of $5$ non-zero digits such that deleting consecutively the digit of the left, in each step, we obtain a divisor of the previous number.

Two boundary points of a ball of radius 1 are joined by a curve contained in the ball and having length less than 2. Prove that the curve is contained entirely within some hemisphere of the given ball.

A house has an even number of lamps distributed among its rooms in such a way that there are at least three lamps in every room. Each lamp shares a switch with exactly one other lamp, not necessarily from the same room. Each change in the switch shared by two lamps changes their states simultaneously. Prove that for every initial state of the lamps there exists a sequence of changes in some of the switches at the end of which each room contains lamps which are on as well as lamps which are off. [i]Proposed by Australia[/i]

Find all positive integers $m$ and $n$ such that $(2^{2^{n}}+1)(2^{2^{m}}+1) $ is divisible by $m\cdot n $ .

We consider two sequences of real numbers $x_{1} \geq x_{2} \geq \ldots \geq x_{n}$ and $\ y_{1} \geq y_{2} \geq \ldots \geq y_{n}.$ Let $z_{1}, z_{2}, .\ldots, z_{n}$ be a permutation of the numbers $y_{1}, y_{2}, \ldots, y_{n}.$ Prove that $\sum \limits_{i=1}^{n} ( x_{i} -\ y_{i} )^{2} \leq \sum \limits_{i=1}^{n}$ $( x_{i} - z_{i})^{2}.$

The complex numbers $z$ and $w$ satisfy the system \begin{align*}z+\frac{20i}{w}&=5+i,\\w+\frac{12i}{z}&=-4+10i.\end{align*} Find the smallest possible value of $|zw|^2$.

Prove the inequality \[ \frac{a_1+ a_3}{a_1 + a_2} + \frac{a_2 + a_4}{a_2 + a_3} + \frac{a_3 + a_1}{a_3 + a_4} + \frac{a_4 + a_2}{a_4 + a_1} \geq 4, \] where $a_i > 0, i = 1, 2, 3, 4.$

Consider a trapezium $ ABCD $ in which $ AB\parallel CD. $ Show that $$ (AC^2+AB^2-BC^2)(BD^2-BC^2+CD^2) =(AC^2-AD^2+CD^2)(BD^2+AB^2-AD^2) . $$

The diagonals of a cyclic quadrilateral $ABCD$ meet at $P$. Let $K$ and $L$ be points on the segments $CP$ and $DP$ such that the circumcircle of $PKL$ is tangent to $CD$ at $M$. Let $X$ and $Y$ be points on the segments $AP$ and $BP$ such that $AX=CK$ and $BY=DL$. Points $Z$ and $W$ are the midpoints of $PK$ and $PL$. Prove that if $C,D,X$ and $Y$ are concyclic, then $\angle MZP = \angle MWP$.

[b]p1[/b]. Consider the following four statements referring to themselves: 1. At least one of these statements is true. 2. At least two of these statements are false. 3. At least three of these statements are true. 4. All four of these statements are false. Determine which statements are true and which are false. Justify your answer. [b]p2.[/b] Let $f(x) = a_{2017}x^{2017} + a_{2016}x^{2016} + ... + a_1x + a_0$ where the coefficients $a_0, a_1, ... , a_{2017}$ have not yet been determined. Alice and Bob play the following game: $\bullet$ Alice and Bob alternate choosing nonzero integer values for the coefficients, with Alice going first. (For example, Alice’s first move could be to set $a_{18}$ to $-3$.) $\bullet$ After all of the coefficients have been chosen: - If f(x) has an integer root then Alice wins. - If f(x) does not have an integer root then Bob wins. Determine which player has a winning strategy and what the strategy is. Make sure to justify your answer. [b]p3.[/b] Suppose that a circle can be inscribed in a polygon $P$ with $2017$ equal sides. Prove that $P$ is a regular polygon; that is, all angles of $P$ are also equal. [b]p4.[/b] A $3 \times 3 \times 3$ cube of cheese is sliced into twenty-seven $ 1 \times 1 \times 1$ blocks. A mouse starts anywhere on the outside and eats one of the $1\times 1\times 1$ cubes. He then moves to an adjacent cube (in any direction), eats that cube, and continues until he has eaten all $27$ cubes. (Two cubes are considered adjacent if they share a face.) Prove that no matter what strategy the mouse uses, he cannot eat the middle cube last. [Note: One should neglect gravity – intermediate configurations don’t collapse.] p5. Suppose that a constant $c > 0$ and an infinite sequence of real numbers $x_0, x_1, x_2, ...$ satisfy $x_{k+1} =\frac{x_k + 1}{1 - cx_k}$ for all $k \ge 0$. Prove that the sequence $x_0, x_1, x_2, ....$ contains infinitely many positive terms and also contains infinitely many negative terms. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

For a vertex $X$ of a quadrilateral, let $h(X)$ be the sum of the distances from $X$ to the two sides not containing $X$. Show that if a convex quadrilateral $ABCD$ satisfies $h(A) = h(B) = h(C) = h(D)$, then it must be a parallelogram.

Find all integers $n$ for which $9n + 16$ and $16n + 9$ are both perfect squares.

Find all real non-zero polynomials satisfying $P(x)^3+3P(x)^2=P(x^{3})-3P(-x)$ for all $x\in\mathbb{R}$.

Find all integers $n \leq 3$ such that there is a set $S_n$ formed by $n$ points of the plane that satisfy the following two conditions: Any three points are not collinear. No point is found inside the circle whose diameter has ends at any two points of $S_n$. [b]NOTE: [/b] The points on the circumference are not considered to be inside the circle.

Find the number of ordered pairs $(a, b)$ of positive integers that are solutions of the following equation: \[a^2 + b^2 = ab(a+b).\]

Let $M$ be a positive integer, and let $a,b,c$ be integers in the interval $[M,M+\sqrt{\frac{M}{2}})$ such that $a^3b+b^3c+c^3a$ is divisible by $abc$. Prove that $a=b=c$. Proposed by Shantanu Nene

$ n>2$ is an integer such that $ n^2$ can be represented as a difference of cubes of 2 consecutive positive integers. Prove that $ n$ is a sum of 2 squares of positive integers, and that such $ n$ does exist.

Given that \[ \sum_{x=1}^{70} \sum_{y=1}^{70} \frac{x^{y}}{y} = \frac{m}{67!} \] for some positive integer $m$, find $m \pmod{71}$.

Find the smallest positive $m$ such that if $a,b,c$ are three side lengths of a triangle with $a^2 +b^2 > mc^2$, then $c$ must be the length of shortest side.

Line segment $AB$ has length $10$. A circle centered at $A$ has radius $5$, and a circle centered at $B$ has radius $5\sqrt{3}$. What is the area of the intersection of the two circles?