Factor the polynomial $x^3+x^2z+xyz+y^2z-y^3$.

George walks $1$ mile to school. He leaves home at the same time each day, walks at a steady speed of $3$ miles per hour, and arrives just as school begins. Today he was distracted by the pleasant weather and walked the first $\frac{1}{2}$ mile at a speed of only $2$ miles per hour. At how many miles per hour must George run the last $\frac{1}{2}$ mile in order to arrive just as school begins today? $\textbf{(A) }4\qquad\textbf{(B) }6\qquad\textbf{(C) }8\qquad\textbf{(D) }10\qquad \textbf{(E) }12$

In the spring round of the Tournament of Towns this year, $6$ problems were posed in the Senior A-Level paper. In a certain country, each problem was solved by exactly $1000$ participants, but no two participants solved all $6$ problems between them. What is the smallest possible number of participants from this country in the spring round Senior A-Level paper? (R Zhenodarov)

[b]p1.[/b] Two robots race on the plane from $(0, 0)$ to $(a, b)$, where $a$ and $b$ are positive real numbers with $a < b$. The robots move at the same constant speed. However, the first robot can only travel in directions parallel to the lines $x = 0$ or $y = 0$, while the second robot can only travel in directions parallel to the lines $y = x$ or $y = -x$. Both robots take the shortest possible path to $(a, b)$ and arrive at the same time. Find the ratio $\frac{a}{b}$ . [b]p2.[/b] Suppose $x + \frac{1}{x} + y + \frac{1}{y} = 12$ and $x^2 + \frac{1}{x^2} + y^2 + \frac{1}{y^2} = 70$. Compute $x^3 + \frac{1}{x^3} + y^3 + \frac{1}{y^3}$. [b]p3.[/b] Find the largest non-negative integer $a$ such that $2^a$ divides $$3^{2^{2018}}+ 3.$$ [b]p4.[/b] Suppose $z$ and $w$ are complex numbers, and $|z| = |w| = z \overline{w}+\overline{z}w = 1$. Find the largest possible value of $Re(z + w)$, the real part of $z + w$. [b]p5.[/b] Two people, $A$ and $B$, are playing a game with three piles of matches. In this game, a move consists of a player taking a positive number of matches from one of the three piles such that the number remaining in the pile is equal to the nonnegative difference of the numbers of matches in the other two piles. $A$ and $B$ each take turns making moves, with $A$ making the first move. The last player able to make a move wins. Suppose that the three piles have $10$, $x$, and $30$ matches. Find the largest value of $x$ for which $A$ does not have a winning strategy. [b]p6.[/b] Let $A_1A_2A_3A_4A_5A_6$ be a regular hexagon with side length $1$. For $n = 1$,$...$, $6$, let $B_n$ be a point on the segment $A_nA_{n+1}$ chosen at random (where indices are taken mod $6$, so $A_7 = A_1$). Find the expected area of the hexagon $B_1B_2B_3B_4B_5B_6$. [b]p7.[/b] A termite sits at the point $(0, 0, 0)$, at the center of the octahedron $|x| + |y| + |z| \le 5$. The termite can only move a unit distance in either direction parallel to one of the $x$, $y$, or $z$ axes: each step it takes moves it to an adjacent lattice point. How many distinct paths, consisting of $5$ steps, can the termite use to reach the surface of the octahedron? [b]p8.[/b] Let $$P(x) = x^{4037} - 3 - 8 \cdot \sum^{2018}_{n=1}3^{n-1}x^n$$ Find the number of roots $z$ of $P(x)$ with $|z| > 1$, counting multiplicity. [b]p9.[/b] How many times does $01101$ appear as a not necessarily contiguous substring of $0101010101010101$? (Stated another way, how many ways can we choose digits from the second string, such that when read in order, these digits read $01101$?) [b]p10.[/b] A perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself. For example, $28$ is a perfect number because $1 + 2 + 4 + 7 + 14 = 28$. Let $n_i$ denote the ith smallest perfect number. Define $$f(x) =\sum_{i|n_x}\sum_{j|n_i}\frac{1}{j}$$ (where $\sum_{i|n_x}$ means we sum over all positive integers $i$ that are divisors of $n_x$). Compute $f(2)$, given there are at least $50 $perfect numbers. [b]p11.[/b] Let $O$ be a circle with chord $AB$. The perpendicular bisector to $AB$ is drawn, intersecting $O$ at points $C$ and $D$, and intersecting $AB$ at the midpoint $E$. Finally, a circle $O'$ with diameter $ED$ is drawn, and intersects the chord $AD$ at the point $F$. Given $EC = 12$, and $EF = 7$, compute the radius of $O$. [b]p12.[/b] Suppose $r$, $s$, $t$ are the roots of the polynomial $x^3 - 2x + 3$. Find $$\frac{1}{r^3 - 2}+\frac{1}{s^3 - 2}+\frac{1}{t^3 - 2}.$$ [b]p13.[/b] Let $a_1$, $a_2$,..., $a_{14}$ be points chosen independently at random from the interval $[0, 1]$. For $k = 1$, $2$,$...$, $7$, let $I_k$ be the closed interval lying between $a_{2k-1}$ and $a_{2k}$ (from the smaller to the larger). What is the probability that the intersection of $I_1$, $I_2$,$...$, $I_7$ is nonempty? [b]p14.[/b] Consider all triangles $\vartriangle ABC$ with area $144\sqrt3$ such that $$\frac{\sin A \sin B \sin C}{ \sin A + \sin B + \sin C}=\frac14.$$ Over all such triangles $ABC$, what is the smallest possible perimeter? [b]p15.[/b] Let $N$ be the number of sequences $(x_1,x_2,..., x_{2018})$ of elements of $\{1, 2,..., 2019\}$, not necessarily distinct, such that $x_1 + x_2 + ...+ x_{2018}$ is divisible by $2018$. Find the last three digits of $N$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

$\vartriangle ABC$ is right with $\angle C = 90^o$. The internal angle bisectors of $\angle A$ and $\angle B$ meet at point $D$, while the external angle bisectors of $\angle A$ and $\angle B$ meet at point $E$. Suppose that $AD = 1$ and $BD = 2$. The value of $DE^2$ can be expressed as $x+y \sqrt{z}$ for integers $x$, $y$, and $z$, where $z$ is greater than $1$ and not divisible by the square of any prime. Compute $100x + 10y + z$. Note: For a generic triangle $\vartriangle PQR$, if we let $Q'$ be the reflection of $Q$ over $P$, then the external angle bisector of $\angle P$ is the line that contains the internal angle bisector of $\angle Q'PR$.

Let $m,n\in\mathbb{Z_{+}}$ be such numbers that set $\{1,2,\ldots,n\}$ contains exactly $m$ different prime numbers. Prove that if we choose any $m+1$ different numbers from $\{1,2,\ldots,n\}$ then we can find number from $m+1$ choosen numbers, which divide product of other $m$ numbers.

Given an acute triangle $ABC$, let $AD$ be an altitude and $H$ the orthocenter. Let $E$ denote the reflection of $H$ with respect to $A$. Point $X$ is chosen on the circumcircle of triangle $BDE$ such that $AC\| DX$ and point $Y$ is chosen on the circumcircle of triangle $CDE$ such that $DY\| AB$. Prove that the circumcircle of triangle $AXY$ is tangent to that of $ABC$.

Let $ABC$ be a triangle such that $|AC|>|AB|$. Let $X$ be on line $AB$ (closer to $A$) such that $|BX|=|AC|$ and let $Y$ be on the segment $AC$ such that $|CY|=|AB|$. Intersection of lines $XY$ and bisector of $BC$ is point $P$. Prove that $\angle BPC+\angle BAC = 180^\circ$.

For any natural number $ n\ge 2, $ define $ m(n) $ to be the minimum number of elements of a set $ S $ that simultaneously satisfy: $ \text{(i)}\quad \{ 1,n\} \subset S\subset \{ 1,2,\ldots ,n\} $ $ \text{(ii)}\quad $ any element of $ S, $ distinct from $ 1, $ is equal to the sum of two (not necessarily distinct) elements from $ S. $ [b]a)[/b] Prove that $ m(n)\ge 1+\left\lfloor \log_2 n \right\rfloor ,\quad\forall n\in\mathbb{N}_{\ge 2} . $ [b]b)[/b] Prove that there are infinitely many natural numbers $ n\ge 2 $ such that $ m(n)=m(n+1). $ $ \lfloor\rfloor $ denotes the usual integer part.

For what polynomials $P(n)$ with integer coefficients can a positive integer be assigned to every lattice point in $\mathbb{R}^3$ so that for every integer $n \ge 1$, the sum of the $n^3$ integers assigned to any $n \times n \times n$ grid of lattice points is divisible by $P(n)$? [i]Proposed by Andre Arslan[/i]

Let $ABC$ be a triangle with incenter $I$. The line $AI$ intersects $BC$ and the circumcircle of $ABC$ at the points $T$ and $S$, respectively. Let $K$ and $L$ be the incenters of $SBT$ and $SCT$, respectively, $M$ be the midpoint of $BC$ and $P$ be the reflection of $I$ with respect to $KL$. a) Prove that $M$, $T$, $K$ and $L$ are concyclic. b) Determine the measure of $\angle BPC$.

The sequence of real numbers $a_0,a_1,a_2,\ldots$ is defined recursively by \[a_0=-1,\qquad\sum_{k=0}^n\dfrac{a_{n-k}}{k+1}=0\quad\text{for}\quad n\geq 1.\]Show that $ a_{n} > 0$ for all $ n\geq 1$. [i]Proposed by Mariusz Skalba, Poland[/i]

If the expression $ \begin{pmatrix}a & c \\ d & b \end{pmatrix}$ has the value $ ab\minus{}cd$ for all values of $a, b, c$ and $d$, then the equation $ \begin{pmatrix}2x & 1 \\ x & x \end{pmatrix} = 3$: $\textbf{(A)}\ \text{Is satisfied for only 1 value of }x \qquad\\ \textbf{(B)}\ \text{Is satisified for only 2 values of }x \qquad\\ \textbf{(C)}\ \text{Is satisified for no values of }x \qquad\\ \textbf{(D)}\ \text{Is satisfied for an infinite number of values of }x \qquad\\ \textbf{(E)}\ \text{None of these.}$

Consider two uniform spherical planets of equal density but unequal radius. Which of the following quantities is the same for both planets? (a) The escape velocity from the planet’s surface. (b) The acceleration due to gravity at the planet’s surface. (c) The orbital period of a satellite in a circular orbit just above the planet’s surface. (d) The orbital period of a satellite in a circular orbit at a given distance from the planet’s center. (e) None of the above.

How many solutions does the system have: $ \{\begin{matrix}&(3x+2y) *(\frac{3}{x}+\frac{1}{y})=2\\ & x^2+y^2\leq 2012\\ \end{matrix} $ where $ x,y $ are non-zero integers