Consider the numbers $x_1, x_2,...,x_n$ that satisfy: $\bullet$ $x_i \in \{-1,1\}$, with $i = 1, 2,...,n$ $\bullet$ $x_1x_2x_3x_4 + x_2x_3x_4x_5 +...+ x_nx_1x_2x_3 = 0$ Prove that $n$ is a multiple of $4$.

P6. Determine all integer pairs $(x, y)$ satisfying the following system of equations. \[ \begin{cases} x + y - 6 &= \sqrt{2x + y + 1} \\ x^2 - x &= 3y + 5 \end{cases} \] P7. Determine the sum of all (positive) integers $n \leq 2019$ such that $1^2 + 2^2 + 3^2 + \cdots + n^2$ is an odd number and $1^1 + 2^2 + 3^3 + \cdots + n^n$ is also an odd number. P8. Two quadrilateral-based pyramids where the length of all its edges are the same, have their bases coincide, forming a new 3D figure called "8-plane" (octahedron). If the volume of such "8-plane" (octahedron) is $a^3\sqrt{2}$ cm$^3$, determine the volume of the largest sphere that can be fit inside such "8-plane" (octahedron). P9. Six-digit numbers $\overline{ABCDEF}$ with distinct digits are arranged from the digits 1, 2, 3, 4, 5, 6, 7, 8 with the rule that the sum of the first three numbers and the sum of the last three numbers are the same. Determine the probability that such arranged number has the property that either the first or last three digits (might be both) form an arithmetic sequence or a geometric sequence. [hide=Remarks (Answer spoiled)]It's a bit ambiguous whether the first or last three digits mentioned should be in that order, or not. If it should be in that order, the answer to this problem would be $\frac{1}{9}$, whereas if not, it would be $\frac{1}{3}$. Some of us agree that the correct interpretation should be the latter (which means that it's not in order) and the answer should be $\frac{1}{3}$. However since this is an essay problem, your interpretation can be written in your solution as well and it's left to the judges' discretion to accept your interpretation, or not. This problem is very bashy.[/hide] P10. $X_n$ denotes the number which is arranged by the digit $X$ written (concatenated) $n$ times. As an example, $2_{(3)} = 222$ and $5_{(2)} = 55$. For $A, B, C \in \{1, 2, \ldots, 9\}$ and $1 \leq n \leq 2019$, determine the number of ordered quadruples $(A, B, C, n)$ satisfying: \[ A_{(2n)} = 2 \left ( B_{(n)} \right ) + \left ( C_{(n)} \right )^2. \]

Let $ b_1, b_2, \ldots, b_{1989}$ be positive real numbers such that the equations \[ x_{r\minus{}1} \minus{} 2x_r \plus{} x_{r\plus{}1} \plus{} b_rx_r \equal{} 0 \quad (1 \leq r \leq 1989)\] have a solution with $ x_0 \equal{} x_{1989} \equal{} 0$ but not all of $ x_1, \ldots, x_{1989}$ are equal to zero. Prove that \[ \sum^{1989}_{k\equal{}1} b_k \geq \frac{2}{995}.\]

Triangle $ABC$ has $BC = 7, CA = 8, AB = 9$. Let $D, E, F$ be the midpoints of $BC, CA, AB$ respectively, and let $G$ be the intersection of $AD$ and $BE$. $G'$ is the reflection of $G$ across $D$. Let $G'E$ meet $CG$ at $P$, and let $G'F$ meet $BG$ at $Q$. Determine the area of $APG'Q$.

The game of backgammon has a "doubling" cube, which is like a standard 6-faced die except that its faces are inscribed with the numbers 2, 4, 8, 16, 32, and 64, respectively. After rolling the doubling cube four times at random, we let $a$ be the value of the first roll, $b$ be the value of the second roll, $c$ be the value of the third roll, and $d$ be the value of the fourth roll. What is the probability that $\frac{a + b}{c + d}$ is the average of $\frac{a}{c}$ and $\frac{b}{d}$ ?

Let $A=20132013...2013$ be formed by joining $2013$, $165$ times. Prove that $2013^2 \mid A$.

[hide=C stands for Cantor, G stands for Gauss]they had two problem sets under those two names[/hide] [u]Set 4[/u] [b]G.16[/b] A number $k$ is the product of exactly three distinct primes (in other words, it is of the form $pqr$, where $p, q, r$ are distinct primes). If the average of its factors is $66$, find $k$. [b]G.17[/b] Find the number of lattice points contained on or within the graph of $\frac{x^2}{3} +\frac{y^2}{2}= 12$. Lattice points are coordinate points $(x, y)$ where $x$ and $y$ are integers. [b]G.18 / C.23[/b] How many triangles can be made from the vertices and center of a regular hexagon? Two congruent triangles with different orientations are considered distinct. [b]G.19[/b] Cindy has a cone with height $15$ inches and diameter $16$ inches. She paints one-inch thick bands of paint in circles around the cone, alternating between red and blue bands, until the whole cone is covered with paint. If she starts from the bottom of the cone with a blue strip, what is the ratio of the area of the cone covered by red paint to the area of the cone covered by blue paint? [b]G.20 / C.25[/b] An even positive integer $n$ has an odd factorization if the largest odd divisor of $n$ is also the smallest odd divisor of n greater than 1. Compute the number of even integers $n$ less than $50$ with an odd factorization. [u] Set 5[/u] [b]G.21[/b] In the magical tree of numbers, $n$ is directly connected to $2n$ and $2n + 1$ for all nonnegative integers n. A frog on the magical tree of numbers can move from a number $n$ to a number connected to it in $1$ hop. What is the least number of hops that the frog can take to move from $1000$ to $2018$? [b]G.22[/b] Stan makes a deal with Jeff. Stan is given 1 dollar, and every day for $10$ days he must either double his money or burn a perfect square amount of money. At first Stan thinks he has made an easy $1024$ dollars, but then he learns the catch - after $10$ days, the amount of money he has must be a multiple of $11$ or he loses all his money. What is the largest amount of money Stan can have after the $10$ days are up? [b]G.23[/b] Let $\Gamma_1$ be a circle with diameter $2$ and center $O_1$ and let $\Gamma_2$ be a congruent circle centered at a point $O_2 \in \Gamma_1$. Suppose $\Gamma_1$ and $\Gamma_2$ intersect at $A$ and $B$. Let $\Omega$ be a circle centered at $A$ passing through $B$. Let $P$ be the intersection of $\Omega$ and $\Gamma_1$ other than $B$ and let $Q$ be the intersection of $\Omega$ and ray $\overrightarrow{AO_1}$. Define $R$ to be the intersection of $PQ$ with $\Gamma_1$. Compute the length of $O_2R$. [b]G.24[/b] $8$ people are at a party. Each person gives one present to one other person such that everybody gets a present and no two people exchange presents with each other. How many ways is this possible? [b]G.25[/b] Let $S$ be the set of points $(x, y)$ such that $y = x^3 - 5x$ and $x = y^3 - 5y$. There exist four points in $S$ that are the vertices of a rectangle. Find the area of this rectangle. PS. You should use hide for answers. C1-15/ G1-10 have been posted [url=https://artofproblemsolving.com/community/c3h2790674p24540132]here [/url] and C16-30/G10-15, G25-30 [url=https://artofproblemsolving.com/community/c3h2790676p24540145]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url]

We are given a bag of sugar, a two-pan balance, and a weight of $1$ gram. How do we obtain $1$ kilogram of sugar in the smallest possible number of weighings?

Given a $4 \times 4$ grid board with each square painted a different color, you want to cut it into two pieces of equal area by making a single cut along the grid lines. In how many ways can it be done?

Let $ABCD$ be a parallelogram. A variable line $g$ through the vertex $A$ intersects the rays $BC$ and $DC$ at the points $X$ and $Y$, respectively. Let $K$ and $L$ be the $A$-excenters of the triangles $ABX$ and $ADY$. Show that the angle $\measuredangle KCL$ is independent of the line $g$. [i]Proposed by Vyacheslev Yasinskiy, Ukraine[/i]

Let $ABC$ be an equilateral triangle, and let $P$ be some point in its circumcircle. Determine all positive integers $n$, for which the value of the sum $S_n (P) = |PA|^n + |PB|^n + |PC|^n$ is independent of the choice of point $P$.

For each positive integer $ n$, let $ f(n)$ denote the greatest common divisor of $ n!\plus{}1$ and $ (n\plus{}1)!$. Find, without proof, a formula for $ f(n)$.

Let $P_1$ and $P_2$ be two non-parallel planes in space, and $A$ a point that does not It is in none of them. For each point $X$, let $X_1$ denote its reflection with respect to $P_1$, and $X_2$ its reflection with respect to $P_2$. Determine the locus of points $X$ for the which $X_1, X_2$ and $A$ are collinear.

Let $ M,M'$ be two conjugates point in triangle $ ABC$ (in the sense that $ \angle MAB\equal{}\angle M'AC,\dots$). Let $ P,Q,R,P',Q',R'$ be foots of perpendiculars from $ M$ and $ M'$ to $ BC,CA,AB$. Let $ E\equal{}QR\cap Q'R'$, $ F\equal{}RP\cap R'P'$ and $ G\equal{}PQ\cap P'Q'$. Prove that the lines $ AG, BF, CE$ are parallel.

Let $\alpha,\beta,\gamma$ be the three real roots of the polynomial $x^3-x^2-2x+1=0$. Find all possible values of $\tfrac{\alpha}{\beta}+\tfrac{\beta}{\gamma}+\tfrac{\gamma}{\alpha}$.

For a rational point (x,y), if xy is an integer that divided by 2 but not 3, color (x,y) red, if xy is an integer that divided by 3 but not 2, color (x,y) blue. Determine whether there is a line segment in the plane such that it contains exactly 2017 blue points and 58 red points.