Form the infinite graph $A$ by taking the set of primes $p$ congruent to $1\pmod{4}$, and connecting $p$ and $q$ if they are quadratic residues modulo each other. Do the same for a graph $B$ with the primes $1\pmod{8}$. Show $A$ and $B$ are isomorphic to each other. [i]Linus Hamilton.[/i]

If $ n$ runs through all the positive integers, then $ f(n) \equal{} \left[n \plus{} \sqrt {\frac {n}{3}} \plus{} \frac {1}{2} \right]$ runs through all positive integers skipping the terms of the sequence $ a_n \equal{} 3 \cdot n^2 \minus{} 2 \cdot n.$

Let $n$ be a positive integer. Determine the smallest positive integer $k$ with the following property: it is possible to mark $k$ cells on a $2n \times 2n$ board so that there exists a unique partition of the board into $1 \times 2$ and $2 \times 1$ dominoes, none of which contain two marked cells.

Solve: $$\begin{cases} 2x_1 - 5x_2 + 3x_3 \ge 0 \\ 2x_2 - 5x_3 + 3x4 \ge 0 \\ ...\\ 2x_{23} - 5x_{24} + 3x_{25} \ge 0\\ 2x_{24} - 5x_{25} + 3x_1 \ge 0\\ 2x_{25} - 5x_1 + 3x_2 \ge 0 \end{cases}$$

[b]p1.[/b] Find $$2^{\left(0^{\left(2^3\right)}\right)}$$ [b]p2.[/b] Amy likes to spin pencils. She has an $n\%$ probability of dropping the $n$th pencil. If she makes $100$ attempts, the expected number of pencils Amy will drop is $\frac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. Find $p + q$. [b]p3.[/b] Determine the units digit of $3 + 3^2 + 3^3 + 3^4 +....+ 3^{2022} + 3^{2023}$. [b]p4.[/b] Cyclic quadrilateral $ABCD$ is inscribed in circle $\omega$ with center $O$ and radius $20$. Let the intersection of $AC$ and $BD$ be $E$, and let the inradius of $\vartriangle AEB$ and $\vartriangle CED$ both be equal to $7$. Find $AE^2 - BE^2$. [b]p5.[/b] An isosceles right triangle is inscribed in a circle which is inscribed in an isosceles right triangle that is inscribed in another circle. This larger circle is inscribed in another isosceles right triangle. If the ratio of the area of the largest triangle to the area of the smallest triangle can be expressed as $a+b\sqrt{c}$, such that $a, b$ and $c$ are positive integers and no square divides $c$ except $1$, find $a + b + c$. [b]p6.[/b] Jonny has three days to solve as many ISL problems as he can. If the amount of problems he solves is equal to the maximum possible value of $gcd \left(f(x), f(x+1) \right)$ for $f(x) = x^3 +2$ over all positive integer values of $x$, then find the amount of problems Jonny solves. [b]p7.[/b] Three points $X$, $Y$, and $Z$ are randomly placed on the sides of a square such that $X$ and $Y$ are always on the same side of the square. The probability that non-degenerate triangle $\vartriangle XYZ$ contains the center of the square can be written as $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers. Find $a + b$. [b]p8.[/b] Compute the largest integer less than $(\sqrt7 +\sqrt3)^6$. [b]p9.[/b] Find the minimum value of the expression $\frac{(x+y)^2}{x-y}$ given $x > y > 0$ are real numbers and $xy = 2209$. [b]p10.[/b] Find the number of nonnegative integers $n \le 6561$ such that the sum of the digits of $n$ in base $9$ is exactly $4$ greater than the sum of the digits of $n$ in base $3$. [b]p11.[/b] Estimation (Tiebreaker) Estimate the product of the number of people who took the December contest, the sum of all scores in the November contest, and the number of incorrect responses for Problem $1$ and Problem $2$ on the October Contest. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABCDE$ be a convex pentagon with $AB=14,$ $BC=7,$ $CD=24,$ $DE=13,$ $EA=26,$ and $\angle B=\angle E=60^{\circ}.$ For each point $X$ in the plane, define $f(X)=AX+BX+CX+DX+EX.$ The least possible value of $f(X)$ can be expressed as $m+n\sqrt{p},$ where $m$ and $n$ are positive integers and $p$ is not divisible by the square of any prime. Find $m+n+p.$

Let $n$ be a positive integer such that $2+2\sqrt{28n^2 +1}$ is an integer. Show that $2+2\sqrt{28n^2 +1}$ is the square of an integer.

An interior point in a rectangle is connected by line segments to the midpoints of its four sides. Thus four domains (polygons) with the areas $a, b, c$ and $d$ appear (see the figure). Prove that $a + c = b + d$. [img]https://1.bp.blogspot.com/-BipDNHELjJI/XzcCa68P3HI/AAAAAAAAMXY/H2Iqya9VItMLXrRqsdyxHLTXCAZ02nEtgCLcBGAsYHQ/s0/2002%2BMohr%2Bp1.png[/img]

$N> 4$ points move around the circle, each with a constant speed. For Any four of them have a moment in time when they all meet. Prove that is the moment when all the points meet.

Find the positive integer $n{}$ if $$\left(1-\frac{1}{1+2}\right)\cdot\left(1-\frac{1}{1+2+3}\right)\cdot\ldots\cdot\left(1-\frac{1}{1+2+\ldots+n}\right)=\frac{2021}{6057}.$$

If $a$ is a given real number, then the number of subsets of $M=\{x\in\mathbb{R}|x^2-3x-a^2+2=0\}$ is $\text{(A)}1\qquad\text{(B)}2\qquad\text{(C)}4\qquad\text{(D)}$ Not sure

Let $X=\{1,2,3,...,10\}$. Find the number of pairs of $\{A,B\}$ such that $A\subseteq X, B\subseteq X, A\ne B$ and $A\cap B=\{5,7,8\}$.

For each nonnegative integer $n$, we define a set $H_n$ of points in the plane as follows: [list] [*]$H_0$ is the unit square $\{(x,y) \mid 0 \le x, y \le 1\}$. [*]For each $n \ge 1$, we construct $H_n$ from $H_{n-1}$ as follows. Note that $H_{n-1}$ is the union of finitely many square regions $R_1, \ldots, R_k$. For each $i$, divide $R_i$ into four congruent square quadrants. If $n$ is odd, then the upper-right and lower-left quadrants of each $R_i$ make up $H_n$. If $n$ is even, then the upper-left and lower-right quadrants of each $R_i$ make up $H_n$. [/list] The figures $H_0$, $H_1$, $H_2$, and $H_3$ are shown below. [asy] pair[]sq(int n){pair[]a; if(n == 0)a.push((.5,.5)); else for(pair k:sq(n-1)) { pair l=1/2^(n+1)*(1,(-1)^(1+(n%2)));a.push(k+l);a.push(k-l); } return a;} void hh(int n,real k){ pair[] S=sq(n);real r=1/2^(n+1); for(pair p:S)filldraw(shift(p+(k,0))*((r,r)--(r,-r)--(-r,-r)--(-r,r)--cycle)); label("$H_"+string(n)+"$",(k+.5,-.3));} size(7cm); for(int i=0;i<=3;++i)hh(i,1.6*i); [/asy] Suppose that the point $P = (x,y)$ lies in $H_n$ for all $n \ge 0$. The greatest possible value of $xy$ is $\tfrac{m}{n}$, for relatively prime positive integers $m, n$. Compute $100m+n$. [i]Proposed by Michael Tang[/i]

Restore the triangle with a compass and a ruler given the intersection point of altitudes and the feet of the median and angle bisectors drawn to one side. (No research required.)

Find the side lengths of the triangle $ABC$ with area $S$ and $\angle BAC = x$ such that the side $BC$ is as short as possible.

The numbers from $1$ to $2000^2$ were written on a board. Vasya choose $2000$ of them whose sum of them equal to two thousandth of the sum of all numbers. Proof that his friend, Petya, will be able to color each of the remaining numbers by one of other $1999$ colors so that the sum of numbers with each of total $2000$ colors are the same.

Let $ABC$ be a triangle with incenter $I$ and let $X$, $Y$ and $Z$ be the incenters of the triangles $BIC$, $CIA$ and $AIB$, respectively. Let the triangle $XYZ$ be equilateral. Prove that $ABC$ is equilateral too. [i]Proposed by Mirsaleh Bahavarnia, Iran[/i]

Let $\vartriangle ABC$ be a triangle with $\angle BAC = 45^o, \angle BCA = 30^o$, and $AB = 1$. Point $D$ lies on segment $AC$ such that $AB = BD$. Find the square of the length of the common external tangent to the circumcircles of triangles $\vartriangle BDC$ and $\vartriangle ABC$.

Find all $t\in \mathbb R$, such that $\int_{0}^{\frac{\pi}{2}}\mid \sin x+t\cos x\mid dx=1$ .

Let $n$ unit circles be given on a plane. Prove that on one of the circles there is an arc of length at least $\frac{2\pi}n$ not intersecting any other circle.

Consider $2009$ cards, each having one gold side and one black side, lying on parallel on a long table. Initially all cards show their gold sides. Two player, standing by the same long side of the table, play a game with alternating moves. Each move consists of choosing a block of $50$ consecutive cards, the leftmost of which is showing gold, and turning them all over, so those which showed gold now show black and vice versa. The last player who can make a legal move wins. (a) Does the game necessarily end? (b) Does there exist a winning strategy for the starting player? [i]Proposed by Michael Albert, Richard Guy, New Zealand[/i]

The equations are given $$ \begin{array}{c} x^2 + p_1x + q_1 = 0\\ x^2 + p_2x + q_2 = 0\\ x^2 + p_3x + q_3 = 0 \end{array}$$ each two of which have a common root, but all three have no common root. Prove that: 1) $2 (p_1p_2 + p_2p_3 + p_3p_1) - (p_1^2 + p_2^2 + p_3^2) = 4 (q_1 + q_2+ q_3)$ 2) he roots of these equations are rational when the numbers $p_1$, $p_2$ and $p_3$ are rational}.

[b]1.[/b] Let $a_{v} $ and $b_{v} $, ${v= 1,2,\dots,n} $, be real numbers such that $a_{1}\geq a_{2} \geq a_{3}\geq\dots\geq a_{n}> 0 $ and $b_{1}\geq a_{1}, b_{1}b_{2}\geq a_{1}a_{2},\dots,b_{1}b_{2}\dots b_{n}\geq a_{1}a_{2}\dots a_{n} $ Show that $b_{1}+b_{2}+\dots+b_{n}\geq a_{1}+a_{2}+\dots+a_{n} $ [b](S. 4)[/b]

Find the number of eight-digit positive integers that are multiples of $9$ and have all distinct digits.

Find all real numbers $x$ satisfying the equation $x^3 - 8 = 16 \sqrt[3]{x + 1}$.