We call polynomial $S(x)\in\mathbb{R}[x]$ sadeh whenever it's divisible by $x$ but not divisible by $x^2$. For the polynomial $P(x)\in\mathbb{R}[x]$ we know that there exists a sadeh polynomial $Q(x)$ such that $P(Q(x))-Q(2x)$ is divisible by $x^2$. Prove that there exists sadeh polynomial $R(x)$ such that $P(R(x))-R(2x)$ is divisible by $x^{1401}$.

Maria has a $6 \times 5$ board with some shaded squares, as in the figure. She writes, in some order, the digits $1, 2, 3, 4$ and $5$ in the first row and then completes the board as follows: look at the number written in the shaded box and write the number that occupies the position indicated by the box shaded as the last number in the next row, and repeat the other numbers in the first four squares, following the same order as in the previous row. For example, if you wrote $2, 3, 4, 1, 5$ in the first row, then since $4$ is in the shaded box, the number that occupies the fourth place $(1)$ is written in the last box of the second row and completes it with the remaining numbers in the order in which. They were. She remains: $2, 3, 4, 5, 1$. Then, to complete the third row, as in the shaded box is $3$, the number located in the third place $(4)$ writes it in the last box and gets $2, 3, 5, 1, 4$. Following in the same way, he gets the board of the figure. Show a way to locate the numbers in the first row to get the numbers $2, 4, 5, 1, 3$ in the last row.

We consider a square $ABCD$ and a point $E$ on the segment $CD$. The bisector of $\angle EAB$ cuts the segment $BC$ in $F$. Prove that $BF + DE = AE$.

When $4444^{4444}$ is written in decimal notation, the sum of its digits is $ A.$ Let $B$ be the sum of the digits of $A.$ Find the sum of the digits of $ B.$ ($A$ and $B$ are written in decimal notation.)

Let $ABCD$ be a parallelogram. An interior circle of the $ABCD$ is tangent to the lines $AB$ and $AD$ and intersects the diagonal $BD$ at $E$ and $F$. Prove that exists a circle that passes through $E$ and $F$ and is tangent to the lines $CB$ and $CD$.

For any positive integer $n$, let $S_n$ denote the set of positive integers which cannot be written in the form $an+2017b$ for nonnegative integers $a$ and $b$. Let $A_n$ denote the average of the elements of $S_n$ if the cardinality of $S_n$ is positive and finite, and $0$ otherwise. Compute \[\left\lfloor\displaystyle\sum_{n=1}^{\infty}\frac{A_n}{2^n}\right\rfloor.\] [i]Proposed by Tristan Shin[/i]

[b]p1.[/b] Find all real solutions of the system $$x^5 + y^5 = 1$$ $$x^6 + y^6 = 1$$ [b]p2.[/b] The centers of three circles of the radius $R$ are located in the vertexes of equilateral triangle. The length of the sides of the triangle is $a$ and $\frac{a}{2}< R < a$. Find the distances between the intersection points of the circles, which are outside of the triangle. [b]p3.[/b] Prove that no positive integer power of $2$ ends with four equal digits. [b]p4.[/b] A circle is divided in $10$ sectors. $90$ coins are located in these sectors, $9$ coins in each sector. At every move you can move a coin from a sector to one of two neighbor sectors. (Two sectors are called neighbor if they are adjoined along a segment.) Is it possible to move all coins into one sector in exactly$ 2004$ moves? [b]p5.[/b] Inside a convex polygon several points are arbitrary chosen. Is it possible to divide the polygon into smaller convex polygons such that every one contains exactly one given point? Justify your answer. [b]p6.[/b] A troll tried to spoil a white and red $8\times 8$ chessboard. The area of every square of the chessboard is one square foot. He randomly painted $1.5\%$ of the area of every square with black ink. A grasshopper jumped on the spoiled chessboard. The length of the jump of the grasshopper is exactly one foot and at every jump only one point of the chessboard is touched. Is it possible for the grasshopper to visit every square of the chessboard without touching any black point? Justify your answer. PS. You should use hide for answers.

Let $M$ be the greatest integer such that both $M + 1213$ and $M + 3773$ are perfect squares. What is the units digit of $M$? $ \textbf{(A) }1 \qquad \textbf{(B) }2 \qquad \textbf{(C) }3 \qquad \textbf{(D) }6 \qquad \textbf{(E) }8 \qquad $

The fence consists of $25$ vertical bars. The heights of the bars are pairwise distinct positive integers from $1$ to $25$. The width of every bar is $1$. Find the maximum $S$ for which regardless of the order of the bars one can find a rectangle of area $S$ formed by the fence.

$ABCD$ is a convex quadrilateral with $AB + BD = AC + CD$. Prove that $AB < AC$.

Find all functions $f:\mathbb{R}^+\to\mathbb{R}^+$ such that for all $x,y,z\in\mathbb{R}^+$ $$f(x+f(y)+f(f(z)))=z+f(y+f(x))$$

Transform by inversion two concentric and coplanar circles into two equal.

A set $ S$ of points from the space will be called [b]completely symmetric[/b] if it has at least three elements and fulfills the condition that for every two distinct points $ A$ and $ B$ from $ S$, the perpendicular bisector plane of the segment $ AB$ is a plane of symmetry for $ S$. Prove that if a completely symmetric set is finite, then it consists of the vertices of either a regular polygon, or a regular tetrahedron or a regular octahedron.

Let be the sequence $ \left( I_n \right)_{n\ge 1} $ defined as $ I_n=\int_0^{\pi } \frac{dx}{x+\sin^n x +\cos^n x} . $ [b]a)[/b] Study the monotony of $ \left( I_n \right)_{n\ge 1} . $ [b]b)[/b] Calculate the limit of $ \left( I_n \right)_{n\ge 1} . $

Given that $2x + 5 - 3x + 7 = 8$, what is the value of $x$? $\textbf{(A) }{-}4\qquad\textbf{(B) }{-}2\qquad\textbf{(C) }0\qquad\textbf{(D) }2\qquad\textbf{(E) }4$

Find the smallest constant $ k$ such that $ \frac {x}{\sqrt {x \plus{} y}} \plus{} \frac {y}{\sqrt {y \plus{} z}} \plus{} \frac {z}{\sqrt {z \plus{} x}}\leq k\sqrt {x \plus{} y \plus{} z}$ for all positive $ x$, $ y$, $ z$.

A function $h : \mathbb{R} \rightarrow \mathbb{R}$ is differentiable and satisfies $h(ax) = bh(x)$ for all $x$, where $a$ and $b$ are given positive numbers and $0 \not = |a| \not = 1$. Suppose that $h'(0) \not = 0$ and the function $h'$ is continuous at $x = 0$. Prove that $a = b$ and that there is a real number $c$ such that $h(x) = cx$ for all $x$.

(a) Let $m$ and $n$ be positive integers and $p$ a positive rational number, with $m > n$, such that $\sqrt{m} -\sqrt{n}= p$. Prove that $m$ and $n$ are perfect squares. (b) Find all four-digit numbers $\overline{abcd}$, where each letter $a, b, c$ and $d$ represents a digit, such that $\sqrt{\overline{abcd}} -\sqrt{\overline{acd}}= \overline{bb}$.

$ 2.46\times 8.163\times (5.17+4.829) $ is closest to: $ \text{(A)}\ 100\qquad\text{(B)}\ 200\qquad\text{(C)}\ 300\qquad\text{(D)}\ 400\qquad\text{(E)}\ 500 $

$ABCD$ is a convex quadrilateral, both $\angle ABC$ and $\angle BCD$ acute. $E$ is a point inside $ABCD$ satisfying $AE=DE$, and $X, Y$ are the intersection of $AD$ and $CE, BE$ respectively, but not $X=A$ or $Y=D$. If $ABEX$ and $CDEY$ are both inscribed quadrilaterals, prove that the distance between $E$ and the lines $AB, BC, CD$ are all equal.

Let segments $KN,KL$ be tangent to circle $C$ at points $N,L$, respectively. $M$ is a point on the extension of the segment $KN$ and $P$ is the other meet point of the circle $C$ and the circumcircle of $\triangle KLM$. $Q$ is on $ML$ such that $NQ$ is perpendicular to $ML$. Prove that \[ \angle MPQ=2\angle KML. \]

We are given $1998$ normal coins, $1$ heavy coin and $1$ light coin, which all look the same. We wish to determine whether the average weight of the two abnormal coins is less than, equal to, or greater than the weight of a normal coin. Show how to do this using a balance $4$ times or less.

In the $xyz$ space with the origin $O$, Let $K_1$ be the surface and inner part of the sphere centered on the point $(1,\ 0,\ 0)$ with radius 2 and let $K_2$ be the surface and inner part of the sphere centered on the point $(-1,\ 0,\ 0)$ with radius 2. For three points $P,\ Q,\ R$ in the space, consider points $X,\ Y$ defined by \[\overrightarrow{OX}=\overrightarrow{OP}+\overrightarrow{OQ},\ \overrightarrow{OY}=\frac 13(\overrightarrow{OP}+\overrightarrow{OQ}+\overrightarrow{OR}).\] (1) When $P,\ Q$ move every cranny in $K_1,\ K_2$ respectively, find the volume of the solid generated by the whole points of the point $X$. (2) Find the volume of the solid generated by the whole points of the point $R$ for which for any $P$ belonging to $K_1$ and any $Q$ belonging to $K_2$, $Y$ belongs to $K_1$. (3) Find the volume of the solid generated by the whole points of the point $R$ for which for any $P$ belonging to $K_1$ and any $Q$ belonging to $K_2$, $Y$ belongs to $K_1\cup K_2$.

Prove that any group of people can be divided into two disjoint groups $A$ and $B$ such that any member from $A$ has at least half of his acquaintances in $B$ and any member from $B$ has at least half of his acquaintances in $A$ (acquaintance is reciprocal).

MO Space City plans to construct $n$ space stations, with a unidirectional pipeline connecting every pair of stations. A station directly reachable from station P without passing through any other station is called a directly reachable station of P. The number of stations jointly directly reachable by the station pair $\{P, Q\}$ is to be examined. The plan requires that all station pairs have the same number of jointly directly reachable stations. (1) Calculate the number of unidirectional cyclic triangles in the space city constructed according to this requirement. (If there are unidirectional pipelines among three space stations A, B, C forming $A \rightarrow B \rightarrow C \rightarrow A$, then triangle ABC is called a unidirectional cyclic triangle.) (2) Can a space city with $n$ stations meeting the above planning requirements be constructed for infinitely many integers $n \geq 3$?