Find all unbounded functions $f:\mathbb Z \rightarrow \mathbb Z$ , such that $f(f(x)-y)|x-f(y)$ holds for any integers $x,y$.

Let $a,b,c,d$ be positive integers satisfying \[\frac{ab}{a+b}+\frac{cd}{c+d}=\frac{(a+b)(c+d)}{a+b+c+d}.\] Determine all possible values of $a+b+c+d$.

Right triangle $ABC$ is inscribed in circle $W$. $\angle{CAB}=65$ degrees, and $\angle{CBA}=25$ degrees. The median from $C$ to $AB$ intersects $W$ and line $D$. Line $l_1$ is drawn tangent to $W$ at $A$. Line $l_2$ is drawn tangent to $W$ at $D$. The lines $l_1$ and $l_2$ intersect at $P$ Determine $\angle{APD}$

Let $\Gamma$ be the circumcircle of $\triangle ABC$. Let $D$ be a point on the side $BC$. The tangent to $\Gamma$ at $A$ intersects the parallel line to $BA$ through $D$ at point $E$. The segment $CE$ intersects $\Gamma$ again at $F$. Suppose $B$, $D$, $F$, $E$ are concyclic. Prove that $AC$, $BF$, $DE$ are concurrent.

Let be a $ 3\times 3 $ real matrix $ A. $ Prove the following statements. [b]a)[/b] $ f(A)\neq O_3, $ for any polynomials $ f\in\mathbb{R} [X] $ whose roots are not real. [b]b)[/b] $ \exists n\in\mathbb{N}\quad \left( A+\text{adj} (A) \right)^{2n} =\left( A \right)^{2n} +\left( \text{adj} (A) \right)^{2n}\iff \text{det} (A)=0 $ [i]Laurențiu Panaitopol[/i]

Let $x$ be a real number such that $3 \sin^4 x -2 \cos^6 x = -\frac{17}{25}$ . Then $3 \cos^4 x - 2 \sin^6 x = \frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $10m + n$.

The plane has a circle $\omega$ and a point $A$ outside it. For any point $C$, the point $B$ on the circle $\omega$ is defined such that $ABC$ is an equilateral triangle with vertices $A, B$ and $C$ listed clockwise. Prove that if point $B$ moves along the circle $\omega$, then point $C$ also moves along a circle.

Find the smallest positive integer $n$ or show no such $n$ exists, with the following property: there are infinitely many distinct $n$-tuples of positive rational numbers $(a_1, a_2, \ldots, a_n)$ such that both $$a_1+a_2+\dots +a_n \quad \text{and} \quad \frac{1}{a_1} + \frac{1}{a_2} + \dots + \frac{1}{a_n}$$ are integers.

Let $ABC$ be an acute-angled triangle with $AC > AB$, let $O$ be its circumcentre, and let $D$ be a point on the segment $BC$. The line through $D$ perpendicular to $BC$ intersects the lines $AO, AC,$ and $AB$ at $W, X,$ and $Y,$ respectively. The circumcircles of triangles $AXY$ and $ABC$ intersect again at $Z \ne A$. Prove that if $W \ne D$ and $OW = OD,$ then $DZ$ is tangent to the circle $AXY.$

A polygonal line of the length $1001$ is given in a unit square. Prove that there exists a line parallel to one of the sides of the square that meets the polygonal line in at least $500$ points.

A right circular cone and a right cylinder with same height $20$ does not have same circular base but the circles are coplanar and their centers are same. If the cone and the cylinder are at the same side of the plane and their base radii are $20$ and $10$, respectively, what is the ratio of the volume of the part of the cone inside the cylinder over the volume of the part of the cone outside the cylinder? $ \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ \frac{5}{3} \qquad\textbf{(D)}\ \frac{4}{3} \qquad\textbf{(E)}\ 1 $

Consider the set of all fractions $\tfrac{x}{y},$ where $x$ and $y$ are relatively prime positive integers. How many of these fractions have the property that if both numerator and denominator are increased by $1$, the value of the fraction is increased by $10\%$? $ \textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad\textbf{(E) }\text{infinitely many} $

Two turtles, the leader and the slave, are crawling along the plane from point $A$ to point $B$. They crawl in turn: first the leader crawls some distance, then the slave crawls some distance in a straight line towards the leading one. Then the leader crawls somewhere again, after which the slave crawls towards the leader, etc. Finally, they both crawl to $B$. Prove that the slave turtle crawled no more than the leading one.

Consider the set $\Omega$ formed by the first twenty natural numbers, $\Omega = \{1, 2, . . . , 20\}$ . A nonempty subset $A$ of $\Omega$ is said to be [i]sumfree [/i ] if for all pair of elements$ x, y \in A$, the sum $(x + y)$ is not in $A$, ( $x$ can be equal to $y$). Prove that $\Omega$ has at least $2018$ sumfree subsets.

Let $ABCD$ be a quadrilateral inscribed in a circle $\Omega.$ Let the tangent to $\Omega$ at $D$ meet rays $BA$ and $BC$ at $E$ and $F,$ respectively. A point $T$ is chosen inside $\triangle ABC$ so that $\overline{TE}\parallel\overline{CD}$ and $\overline{TF}\parallel\overline{AD}.$ Let $K\ne D$ be a point on segment $DF$ satisfying $TD=TK.$ Prove that lines $AC,DT,$ and $BK$ are concurrent.

Let $ABC$ be an acute triangle with orthocenter $H$, and $P$ a point interior to the triangle. Points $D,E,F$ are the reflections of $P$ about $BC,CA,AB$. If $Q$ is the intersection of $HD$ and $EF$, prove that the ratio $HQ/HD$ is independent of the choice of $P$. Luis González

Determine the largest real number $z$ such that \begin{align*} x + y + z = 5 \\ xy + yz + xz = 3 \end{align*} and $x$, $y$ are also real.

Find the largest value of the real number $ \lambda$, such that as long as point $ P$ lies in the acute triangle $ ABC$ satisfying $ \angle{PAB}\equal{}\angle{PBC}\equal{}\angle{PCA}$, and rays $ AP$, $ BP$, $ CP$ intersect the circumcircle of triangles $ PBC$, $ PCA$, $ PAB$ at points $ A_1$, $ B_1$, $ C_1$ respectively, then $ S_{A_1BC}\plus{} S_{B_1CA}\plus{} S_{C_1AB} \geq \lambda S_{ABC}$.

Real numbers $x,y$ are such that $x^2\ge y$ and $y^2\ge x.$ Prove that $\frac{x}{y^2+1}+\frac{y}{x^2+1}\le1.$

Consider a $(2m-1)\times(2n-1)$ rectangular region, where $m$ and $n$ are integers such that $m,n\ge 4.$ The region is to be tiled using tiles of the two types shown: \[ \begin{picture}(140,40) \put(0,0){\line(0,1){40}} \put(0,0){\line(1,0){20}} \put(0,40){\line(1,0){40}} \put(20,0){\line(0,1){20}} \put(20,20){\line(1,0){20}} \put(40,20){\line(0,1){20}} \multiput(0,20)(5,0){4}{\line(1,0){3}} \multiput(20,20)(0,5){4}{\line(0,1){3}} \put(80,0){\line(1,0){40}} \put(120,0){\line(0,1){20}} \put(120,20){\line(1,0){20}} \put(140,20){\line(0,1){20}} \put(80,0){\line(0,1){20}} \put(80,20){\line(1,0){20}} \put(100,20){\line(0,1){20}} \put(100,40){\line(1,0){40}} \multiput(100,0)(0,5){4}{\line(0,1){3}} \multiput(100,20)(5,0){4}{\line(1,0){3}} \multiput(120,20)(0,5){4}{\line(0,1){3}} \end{picture} \] (The dotted lines divide the tiles into $1\times 1$ squares.) The tiles may be rotated and reflected, as long as their sides are parallel to the sides of the rectangular region. They must all fit within the region, and they must cover it completely without overlapping. What is the minimum number of tiles required to tile the region?

Let $P$ and $Q$ be two convex polygons. Let $h$ be the length of the projection of $Q$ onto a line perpendicular to a side of $P$ which is of length $p$. Define $f(P,Q)$ to be the sum of the products $hp$ over all sides of $P$. Prove that $f(P,Q) = f(Q, P)$.

If it is possible to find six elements, whose sum are divisible by $6$, from every set with $n$ elements, what is the least $n$ ? $\textbf{(A)}\ 13 \qquad\textbf{(B)}\ 12 \qquad\textbf{(C)}\ 11 \qquad\textbf{(D)}\ 10 \qquad\textbf{(E)}\ 9$

Let $n>1$ be a positive integer and $d_1<d_2<\dots<d_m$ be its $m$ positive divisors, including $1$ and $n$. Lalo writes the following $2m$ numbers on a board: \[d_1,d_2\dots, d_m,d_1+d_2,d_2+d_3,\dots,d_{m-1}+d_m,N \] where $N$ is a positive integer. Afterwards, Lalo erases any number that is repeated (for example, if a number appears twice, he erases one of them). Finally, Lalo realizes that the numbers left on the board are exactly all the divisors of $N$. Find all possible values that $n$ can take.

The equation $ 3y^2 \plus{} y \plus{} 4 \equal{} 2(6x^2 \plus{} y \plus{} 2)$ where $ y \equal{} 2x$ is satisfied by: $ \textbf{(A)}\ \text{no value of }x \qquad\textbf{(B)}\ \text{all values of }x \qquad\textbf{(C)}\ x \equal{} 0\text{ only}$ $ \textbf{(D)}\ \text{all integral values of }x\text{ only} \qquad\textbf{(E)}\ \text{all rational values of }x\text{ only}$

Monic quadratic polynomials $ P(x)$ and $ Q(x)$ have the property that $ P(Q(x))$ has zeroes at $ x\equal{}\minus{}23,\minus{}21,\minus{}17, \text{and} \minus{}15$, and $ Q(P(x))$ has zeroes at $ x\equal{}\minus{}59, \minus{}57, \minus{}51, \text{and} \minus{}49$. What is the sum of the minimum values of $ P(x)$ and $ Q(x)$? $ \textbf{(A)}\ \text{\minus{}100} \qquad \textbf{(B)}\ \text{\minus{}82} \qquad \textbf{(C)}\ \text{\minus{}73} \qquad \textbf{(D)}\ \text{\minus{}64} \qquad \textbf{(E)}\ 0$