A shepherd left, as an inheritance, to his children a flock of $k$ sheep, distributed as follows: the oldest received $\left\lfloor\frac{k}{2}\right\rfloor$ sheep, the middle one $\left\lfloor\frac{k}{3}\right\rfloor$ sheep and the youngest $\left\lfloor\frac{k}{5}\right\rfloor$ sheep. Knowing that there are no sheep left, determine all possible values for $k$.

Consider the factorials of the first $100$ positive integers, namely, $1!, 2!$, $...$, $100!$. Is it possible to delete one of them so that the product of the remaining ones is a perfect square? (S Tokarev)

Find all pairs of integers $a, b$ such that the following system of equations has a unique integral solution $(x , y , z )$ : $\begin{cases}x + y = a - 1 \\ x(y + 1) - z^2 = b \end{cases}$

Let $f :\mathbb{N} \rightarrow\mathbb{N}$ be an injective function such that $f(1) = 2,~ f(2) = 4$ and \[f(f(m) + f(n)) = f(f(m)) + f(n)\] for all $m, n \in \mathbb{N}$. Prove that $f(n) = n + 2$ for all $n \ge 2$.

A line in the $xy$-plane has positive slope, passes through the point $(x, y) = (0, 29)$, and lies tangent to the ellipse defined by $\frac{x^2}{100} +\frac{y^2}{400} = 1$. What is the slope of the line?

Let triangle $ABC$ have altitudes $BE$ and $CF$ which meet at $H$. The reflection of $A$ over $BC$ is $A'$. Let $(ABC)$ meet $(AA'E)$ at $P$ and $(AA'F)$ at $Q$. Let $BC$ meet $PQ$ at $R$. Prove that $EF \parallel HR$. [i]Proposed by Daniel Hu[/i]

[asy] size(5cm); defaultpen(fontsize(6pt)); draw((0,0)--(4,0)--(4,4)--(0,4)--cycle); draw((0,0)--(-4,0)--(-4,-4)--(0,-4)--cycle); draw((1,-1)--(1,3)--(-3,3)--(-3,-1)--cycle); draw((-1,1)--(-1,-3)--(3,-3)--(3,1)--cycle); draw((-4,-4)--(0,-4)--(0,-3)--(3,-3)--(3,0)--(4,0)--(4,4)--(0,4)--(0,3)--(-3,3)--(-3,0)--(-4,0)--cycle, red+1.2); label("1", (-3.5,0), S); label("2", (-2,0), S); label("1", (-0.5,0), S); label("1", (3.5,0), S); label("2", (2,0), S); label("1", (0.5,0), S); label("1", (0,3.5), E); label("2", (0,2), E); label("1", (0,0.5), E); label("1", (0,-3.5), E); label("2", (0,-2), E); label("1", (0,-0.5), E); [/asy] Compute the area of the resulting shape, drawn in red above. [i]Proposed by Nathan Xiong[/i]

Suppose $x$, $y$, $z$ are positive numbers such that $x+y+z=1$. Prove that \[ \frac{(1+xy+yz+zx)(1+3x^3 + 3y^3 + 3z^3)}{9(x+y)(y+z)(z+x)} \ge \left( \frac{x \sqrt{1+x} }{\sqrt[4]{3+9x^2}} + \frac{y \sqrt{1+y} }{\sqrt[4]{3+9y^2}} + \frac{z \sqrt{1+z}}{\sqrt[4]{3+9z^2}} \right)^2. \]

$ABC$ is a non-isosceles triangle. $T_A$ is the tangency point of incircle of $ABC$ in the side $BC$ (define $T_B$,$T_C$ analogously). $I_A$ is the ex-center relative to the side BC (define $I_B$,$I_C$ analogously). $X_A$ is the mid-point of $I_BI_C$ (define $X_B$,$X_C$ analogously). Show that $X_AT_A$,$X_BT_B$,$X_CT_C$ meet in a common point, colinear with the incenter and circumcenter of $ABC$.

Let $n \ge 1$ and $x_1, \ldots, x_n \ge 0$. Prove that $$ (x_1 + \frac{x_2}{2} + \ldots + \frac{x_n}{n}) (x_1 + 2x_2 + \ldots + nx_n) \le \frac{(n+1)^2}{4n} (x_1 + x_2 + \ldots + x_n)^2 .$$

Prove that the following inequality holds for all positive $\{a_i\}$: $$\frac{1}{a_1} + \frac{2}{a_1+a_2} + ... +\frac{ n}{a_1+...+a_n} < 4\left(\frac{1}{a_1} + ... + \frac{1}{a_n}\right)$$

Find the number of distinct non-empty subsequences of the binary string \[01001111010110.\] Note: A subsequence of a string $S$ is any string which can be formed by deleting some characters from $S$ while keeping the order of the remaining characters. For example, ``ab'' and ``ccm'' are a subsequences of ``ccamb'', but ``abc'' is not. [i]Team #8[/i]

Suppose $f$ is a polynomial in $\mathbb{Z}[X]$ and m is integer .Consider the sequence $a_i$ like this $a_1=m$ and $a_{i+1}=f(a_i)$ find all polynomials $f$ and alll integers $m$ that for each $i$: \[ a_i | a_{i+1}\]

There is a room having a form of right-angled parallelepiped. Four maps of the same scale are hung (generally, on different levels over the floor) on four walls of the room, so that sides of the maps are parallel to sides of the wall. It is known that the four points corresponding to each of Stockholm, Moscow, and Istanbul are coplanar. Prove that the four points coresponding to Hong Kong are coplanar as well.

The vertices of a regular $45$-gon are painted into three colors so that the number of vertices of each color is the same. Prove that three vertices of each color can be selected so that three triangles formed by the chosen vertices of the same color are all equal.

In a convex quadrilateral $ABCD$, the diagonal $AC$ intersects the diagonal $BD$ at its midpoint $S$. The radii of incircles of triangles $ABS,BCS,CDS,DAS$ are $r_1,r_2,r_3,r_4$, respectively. Prove that $$|r_1-r_2+r_3-r_4|\le\frac18|AB-BC+CD-DA|.$$

Let $K$ be a point on the angle bisector, such that $\angle BKL=\angle KBL=30^\circ$. The lines $AB$ and $CK$ intersect in point $M$ and lines $AC$ and $BK$ intersect in point $N$. Determine $\angle AMN$.

In a tetrahedron $ ABCD $ the edges $AD$, $ BD $, $ CD $ are equal. $ ABC $ Non-collinear points are chosen in the plane. $ A_1$, $B_1$, $C_1 $ The lines $DA_1$, $DB_1$, $DC_1 $ intersect the surface of the sphere circumscribed about the tetrahedron at points $ A_2$, $B_2$, $C_2 $, different from the point $ D $. Prove that the points $A_1$, $B_1$, $C_1$, $A_2$, $B_2$, $C_2$ lie on the surface of a certain sphere.

For each positive integer $n$, denote by $O(n)$ its greatest odd divisor. Given any positive integers $x_1 = a$ and $x_2 = b$, construct an innite sequence of positive integers as follows: $x_n = O(x_{n-1} + x_{n-2})$, where $n = 3,4,...$ (a) Prove that starting from some place, all terms of the sequence are equal to the same integer. (b) Express this integer in terms of $a$ and $b$.

In elections to the City Duma, each voter, if he goes to the polls, casts a vote for himself (if he is a candidate) and for those candidates who are his friends. The forecast of the sociological service of the mayor's office is considered good if it correctly predicts the number of votes cast for at least one of the candidates, and bad otherwise. Prove that for any forecast, voters can turn out to vote in such a way that this forecast turns out to be bad.

Find all possible values of the number $$\frac{a + b}{c}+\frac{a + c}{b}+\frac{b + c}{a},$$ where $a, b, c$ are positive integers, and $\frac{a + b}{c},\frac{a + c}{b},\frac{b + c}{a}$ are also positive integers.

For a set with $n$ elements, how many subsets are there whose cardinality is respectively $\equiv 0$ (mod $3$), $\equiv 1$ (mod $3$), $ \equiv 2$ (mod $3$)? In other words, calculate $$s_{i,n}= \sum_{k\equiv i \;(\text{mod} \;3)} \binom{n}{k}$$ for $i=0,1,2$. Your result should be strong enough to permit direct evaluation of the numbers $s_{i,n}$ and to show clearly the relationship of $s_{0,n}, s_{1,n}$ and $s_{2,n}$ to each other for all positive integers $n$. In particular, show the relationships among these three sums for $n = 1000$.

[b]a)[/b] Give example of two irrational numbers $ a,b $ having the property that $ a^3,b^3,a+b $ are all rational. [b]b)[/b] Prove that if $ x,y $ are two nonnegative real numbers having the property that $ x^3,y^3,x+y $ are rational, then $ x $ and $ y $ are both rational. [i]Mihai Piticari[/i] and [i]Vladimir Cerbu[/i]

Let $ABC$ be a triangle. Let $BE$ and $CF$ be internal angle bisectors of $\angle B$ and $\angle C$ respectively with $E$ on $AC$ and $F$ on $AB$. Suppose $X$ is a point on the segment $CF$ such that $AX$ perpendicular $CF$; and $Y$ is a point on the segment $BE$ such that $AY$ perpendicular $BE$. Prove that $XY = (b + c-a)/2$ where $BC = a, CA = b $and $AB = c$.

Consider the sequence of real numbers $a_n$ satisfying the recurrence $$a_na_{n+2}-a_{n+1}^2-(n+1)a_na_{n+1}=0.$$ Given that $a_1=1$ and $a_2=2018$, compute $$\frac{a_{2018}\cdot a_{2016}}{a_{2017}^2}.$$