Consider the polynomial \[P(x)=x^3+3x^2+6x+10.\] Let its three roots be $a$, $b$, $c$. Define $Q(x)$ to be the monic cubic polynomial with roots $ab$, $bc$, $ca$. Compute $|Q(1)|$. [i]Proposed by Nathan Xiong[/i]

A quadrilateral whose perimeter is equal to $P$ is inscribed in a circle of radius $R$ and is circumscribed around a circle of radius $r$. Check whether the inequality $P\le \frac{r+\sqrt{r^2+4R^2}}{2}$ holds. Try to find the corresponding inequalities for the $n$-gon ($n \ge 5$) inscribed in a circle of radius $R$ and circumscribed around a circle of radius $r$.

A grasshopper is jumping along the number line. Initially it is situated at zero. In $k$-th step, the length of his jump is $k$. a) If the jump length is even, then it jumps to the left, otherwise it jumps to the right (for example, firstly it jumps one step to the right, then two steps to the left, then three steps to the right, then four steps to the left...). Will it visit on every integer at least once? b) If the jump length is divisible by three, then it jumps to the left, otherwise it jumps to the right (for example, firstly it jumps one step to the right, then two steps to the right, then three steps to the left, then four steps to the right...). Will it visit every integer at least once? Proposed by Matko Ljulj

A three-digit number $N$ is equal to $36$ times the sum of its digits. Find the sum of all possible values of $N$. $\textbf{(A) }576\qquad\textbf{(B) }648\qquad\textbf{(C) }972\qquad\textbf{(D) }1152\qquad\textbf{(E) }1620$

$k$ vertices of a regular $n$-gon $P$ are coloured. A colouring is called almost uniform if for every positive integer $m$ the following condition is satisfied: If $M_1$ is a set of m consecutive vertices of $P$ and $M_2$ is another such set then the number of coloured vertices of $M_1$ differs from the number of coloured vertices of $M_2$ at most by $1$. Prove that for all positive integers $k$ and $n$ ($k \le n$) an almost uniform colouring exists and that it is unique within a rotation. (M Kontsevich, Moscow)

Prove that if you choose $10^{100}$ points on a circle and arrange numbers from $1$ to $10^{100}$ on them in some order, then you can choose $100$ pairwise disjoint chords with ends at the selected points such that the sums of the numbers at the ends of all of them are equal to each other.

A $ABC$ triangle divide by a $d$ line such that, new two pieces' areas are equal. $d$ line intersects with $[AB]$ at $D$, $[AC]$ at $E$. Prove that $$\frac{AD+AE}{BD+DE+EC+CB} > \frac{1}{4}$$

Andy chooses not necessarily distinct digits $G$, $U$, $N$, and $A$ such that the $5$ digit number $GUNGA$ is divisible by $44$. Find the least possible value of $G+U+N+G+A$. [i]Proposed by Andy Xu[/i]

A game is played on an $m \times n$ chessboard. At the beginning, there is a coin on one of the squares. Two players take turns to move the coin to an adjacent square (horizontally or vertically). The coin may never be moved to a square that has been occupied before. If a player cannot move any more, he loses. Prove: [list] [*] If the size (number of squares) of the board is even, then the player to move first has a winning strategy, regardless of the initial position. [*] If the size of the board is odd, then the player to move first has a winning strategy if and only if the coin is initially placed on a square whose colour is not the same as the colour of the corners. [/list]

Let $ABCDEF$ be a hexagon inscribed in a circle in which $AB = BC, CD = DE$ and $EF = FA$. Prove that the lines $AD, BE$ and $CF$ intersect at one point.

Let $n$ be a positive integer. We have $n$ boxes where each box contains a non-negative number of pebbles. In each move we are allowed to take two pebbles from a box we choose, throw away one of the pebbles and put the other pebble in another box we choose. An initial configuration of pebbles is called [i]solvable[/i] if it is possible to reach a configuration with no empty box, in a finite (possibly zero) number of moves. Determine all initial configurations of pebbles which are not solvable, but become solvable when an additional pebble is added to a box, no matter which box is chosen.

Given two conditions: A: $\sqrt{1+\sin\theta}=a$ B: $\sin\frac{\theta}{2}+\cos\frac{\theta}{2}=a$ Then, which one of the followings are true? $(\text{A})$A is sufficient and necessary condition of B. $(\text{B})$A is necessary but insufficient condition of B. $(\text{C})$A is sufficient but unnecessary condition of B. $(\text{D})$A is insufficient and unnecessary condition of B.

Is there a convex pentagon in which each diagonal is equal to some side?

In the plane, there are $n \geqslant 6$ pairwise disjoint disks $D_{1}, D_{2}, \ldots, D_{n}$ with radii $R_{1} \geqslant R_{2} \geqslant \ldots \geqslant R_{n}$. For every $i=1,2, \ldots, n$, a point $P_{i}$ is chosen in disk $D_{i}$. Let $O$ be an arbitrary point in the plane. Prove that \[O P_{1}+O P_{2}+\ldots+O P_{n} \geqslant R_{6}+R_{7}+\ldots+R_{n}.\] (A disk is assumed to contain its boundary.)

Find the area of the figure on the coordinate plane bounded by the straight lines $x = 0$, $x = 2$ and the graphs of the functions $y =\sqrt{x^3+ 1}$ and $y = - \sqrt[3]{x^2+ 2x}$.

Find the shortest distance between the graphs of $y=x^2+5$ and $x=y^2+5$. [i]Proposed by Nathan Ramesh

A soldier needs to check if there are any mines in the interior or on the sides of an equilateral triangle $ABC.$ His detector can detect a mine at a maximum distance equal to half the height of the triangle. The soldier leaves from one of the vertices of the triangle. Which is the minimum distance that he needs to traverse so that at the end of it he is sure that he completed successfully his mission?

Find all pairs $(x,y)$ of positive integers such that $x(x+y)=y^2+1.$

[u]Round 9[/u] [b]p25.[/b] A positive integer is called spicy if it is divisible by the sum if its digits. Find the number of spicy integers between $100$ and $200$ inclusive. [b]p26.[/b] Rectangle $ABCD$ has points $E$ and $F$ on sides $AB$ and $BC$, respectively. Given that $\frac{AE}{BE} = \frac{BF}{FC} =\frac12$, $\angle ADE = 30^o$, and $[DEF] = 25$, find the area of rectangle $ABCD$. [b]p27.[/b] Find the largest value of $n$ for which $3^n$ divides ${100 \choose 33}$. [u]Round 10[/u] [b]p28.[/b] Isosceles trapezoid $ABCD$ is inscribed in a circle such that $AB \parallel CD$, $AB = 2$, $CD = 4$, and $AC = 9$. What is the radius of the circle? [b]p29.[/b] Find the product of all possible positive integers $n$ less than $11$ such that in a group of $n$ people, it is possible for every person to be friends with exactly $3$ other people within the group. Assume that friendship is amutual relationship. [b]p30.[/b] Compute the infinite product $$\left( 1+ \frac{1}{2^1} \right) \left( 1+ \frac{1}{2^2} \right) \left( 1+ \frac{1}{2^4} \right) \left( 1+ \frac{1}{2^8} \right) \left( 1+ \frac{1}{2^{16}} \right) ...$$ [u]Round 11[/u] [b]p31.[/b] Find the sum of all possible values of $x y$ if $x +\frac{1}{y}= 12$ and $\frac{1}{x}+ y = 8$. [b]p32.[/b] Find the number of ordered pairs $(a,b)$, where $0 < a,b < 1999$, that satisfy $a^2 +b^2 \equiv ab$ (mod $1999$) [b]p33.[/b] Let $f :N\to Q$ be a function such that $f(1) =0$, $f (2) = 1$ and $f (n) = \frac{f(n-1)+f (n-2)}{2}$ . Evaluate $$\lim_{n\to \infty} f (n).$$ [u]Round 12[/u] [b]p34.[/b] Estimate the sumof the digits of $2018^{2018}$. The number of points you will receive is calculated using the formula $\max \,(0,15-\log_{10}(A-E))$, where $A$ is the true value and $E$ is your estimate. [b]p35.[/b] Let $C(m,n)$ denote the number of ways to tile an $m$ by $n$ rectangle with $1\times 2$ tiles. Estimate $\log_{10}(C(100, 2))$. The number of points you will recieve is calculated using the formula $\max \,(0,15- \log_{10}(A-E))$, where $A$ is the true value and $E$ is your estimate. [b]p36.[/b] Estimate $\log_2 {1000 \choose 500}$. The number of points you earn is equal to $\max \,(0,15-|A-E|)$, where $A$ is the true value and $E$ is your estimate. PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3165983p28809209]here [/url] and 5-8 [url=https://artofproblemsolving.com/community/c3h3165992p28809294]here[/url].. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let the three roots of $x^3+ax+1=0$ be $\alpha,\beta,\gamma$ where $a$ is a positive real number. Let the three roots of $x^3+bx^2+cx-1=0$ be $\frac{\alpha}{\beta},\frac{\beta}{\gamma},\frac{\gamma}{\alpha}$. Find the minimum value of $\dfrac{|b|+|c|}{a}$.

The figure below shows a smaller square within a larger square. Both squares have integer side lengths. The region inside the larger square but outside the smaller square has area $52$. Find the area of the larger square. [img]https://cdn.artofproblemsolving.com/attachments/a/f/2cb8c70109196bf30f88aef0c53bbac07d6cc3.png[/img]

Find values of the parameter $u$ for which the expression \[y = \frac{ \tan(x-u) + \tan(x) + \tan(x+u)}{ \tan(x-u)\tan(x)\tan(x+u)}\] does not depend on $x.$

A convex polygon with $2n$ sides is called [i]rhombic [/i] if its sides are equal and all pairs of opposite sides are parallel. A rhombic polygon can be partitioned into rhombic quadrilaterals. For what value of$ n$, a $2n$-sided rhombic polygon splits into $666$ rhombic quadrilaterals?

Let $X =\{x_0, x_1,\ldots , x_n\}$ be the basis set of a finite metric space, where the points are inductively listed such that $x_k$ maximizes the product of the distances from the points $\{x_0, x_1,\ldots , x_{k-1}\}$ for each $1\leqslant k\leqslant n.$ Prove that if for each $x\in X$ we let $\Pi_x$ be the product of the distances from $x{}$ to every other point, then $\Pi_{x_n}\leqslant 2^{n-1}\Pi_x$ for any $x\in X.$

Sarah and Darah play the following game. Sarah puts $ n$ coins numbered with $ 1,\dots,n$ on a table (Each coin is in HEAD or TAIL position.) At each step Darah gives a coin to Sarah and she (Sarah) let him (Dara) to change the position of all coins with number multiple of a desired number $ k$. At the end, all of the coins that are in TAIL position will be given to Sarah and all of the coins with HEAD position will be given to Darah. Prove that Sarah can put the coins in a position at the beginning of the game such that she gains at least $ \Omega(n)$ coins. [hide="Hint:"]Chernov inequality![/hide]