Positive integer $n$ cannot be divided by $2$ and $3$, there are no nonnegative integers $a$ and $b$ such that $|2^a-3^b|=n$. Find the minimum value of $n$.

What is $2021+20+21+2+0+2+1$? [i]Proposed by Nathan Xiong[/i]

$AB$ is a chord of length $6$ in a circle of radius $5$ and centre $O$. A square is inscribed in the sector $OAB$ with two vertices on the circumference and two sides parallel to $ AB$. Find the area of the square.

Let $a, b, c$ be positive real numbers such that $a + b + c = 1$. Show that $$\frac{1 - a^2}{a + bc} + \frac{1 - b^2}{b + ca} + \frac{1 - c^2}{c + ab} \ge 6$$

Find all triples of real numbers $(a, b, c)$ that satisfy the system of equations $$\begin{cases} b^2 = 4a(\sqrt{c} - 1) \\ c^2 = 4b (\sqrt{a} - 1) \\ a^2 = 4c(\sqrt{b} - 1) \end{cases}$$

Positive integers are written into squares of a infinite chessboard such that a number $n$ is written $n$ times. If the absolute differences of numbers written into any two squares having a common side is not greater than $k$, what is the least possible value of $k$? $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \text{None of the preceding} $

If $ 8\cdot2^x \equal{} 5^{y \plus{} 8}$, then when $ y \equal{} \minus{} 8,x \equal{}$ $ \textbf{(A)}\ \minus{} 4 \qquad\textbf{(B)}\ \minus{} 3 \qquad\textbf{(C)}\ 0 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 8$

In an acute triangle $ABC$, let $D$, $E$, $F$ be the feet of the perpendiculars from the points $A$, $B$, $C$ to the lines $BC$, $CA$, $AB$, respectively, and let $P$, $Q$, $R$ be the feet of the perpendiculars from the points $A$, $B$, $C$ to the lines $EF$, $FD$, $DE$, respectively. Prove that $p\left(ABC\right)p\left(PQR\right) \ge \left(p\left(DEF\right)\right)^{2}$, where $p\left(T\right)$ denotes the perimeter of triangle $T$ . [i]Proposed by Hojoo Lee, Korea[/i]

Prove that for all natural $n$ there exists a polynomial $P(x)$ divisible by $(x-1)^n$ such that its degree is not greater than $2^n$ and each of its coefficients is equal to $1$, $0$ or $-1$. (D. Fomin, Leningrad)

A mass $m$ is resting at equilibrium suspended from a vertical spring of natural length $L$ and spring constant $k$ inside a box as shown: [asy] //The Spring import graph; size(10cm); guide coil(path g, real width=0.1, real margin = 1*width) { real L = arclength(g); real r = width / 2; pair startpoint = arcpoint(g, margin); real[][] isectiontimes = intersections(g, circle(c=startpoint,r=r)); real initialcirclecentertime = (isectiontimes.length == 1 ? isectiontimes[0][0] : isectiontimes[1][0]); pair startdir = dir(startpoint - point(g,initialcirclecentertime)); real startangle = atan2(startdir.y, startdir.x); real startarctime = arclength(subpath(g, 0, initialcirclecentertime)); write(startarctime); pair endpoint = arcpoint(g, L - margin); real finalcirclecentertime = intersections(g, circle(c=endpoint,r=r))[0][0]; pair enddir = dir(endpoint - point(g,finalcirclecentertime)); real endangle = atan2(enddir.y, enddir.x); real endarctime = arclength(subpath(g, 0, finalcirclecentertime)); write(endarctime); real coillength = 2r; real lengthalongcoils = L - 2*margin; int numcoils = ceil(lengthalongcoils / coillength); real anglesubtended = 2pi * numcoils - startangle + endangle; real angleat(real arctime) { return (arctime - startarctime) * (anglesubtended / (endarctime - startarctime)) + startangle; } pair f(real t) { return arcpoint(g,t) + r * expi(angleat(t)); } return subpath(g, 0, arctime(g, margin)) & graph(f, startarctime, endarctime, n=max(length(g), 20*numcoils+2), operator..) & subpath(g, arctime(g, L-margin), length(g)); } draw(coil((0,0.25)--(0,1))); //Outer Box draw((-1,1)--(1,1),linewidth(2)); draw((-1,1)--(-1,-1.2),linewidth(2)); draw((-1,-1.2)--(1,-1.2),linewidth(2)); draw((1,1)--(1,-1.2),linewidth(2)); //Inner Box draw((-0.2,0.25)--(0.2,0.25),linewidth(2)); path arc1=arc((-0.2,0.15),(-0.2,0.25),(-0.3,0.15)); path arc2=arc((0.2,0.15),(0.3,0.15),(0.2,0.25)); draw(arc1,linewidth(2)); draw(arc2,linewidth(2)); draw((-0.3,0.15)--(-0.3,-0.3),linewidth(2)); draw((0.3,0.15)--(0.3,-0.3),linewidth(2)); path arc3=arc((-0.2,-0.3),(-0.3,-0.3),(-0.2,-0.4)); draw(arc3,linewidth(2)); path arc4=arc((0.2,-0.3),(0.2,-0.4),(0.3,-0.3)); draw((-0.2,-0.4)--(0.2,-0.4),linewidth(2)); draw(arc4,linewidth(2)); [/asy] The box begins accelerating upward with acceleration $a$. How much closer does the equilibrium position of the mass move to the bottom of the box? (a) $(a/g)L$ (b) $(g/a)L$ (c) $m(g + a)/k$ (d) $m(g - a)/k$ (e) $ma/k$

Let $ S$ be the set of all real numbers strictly greater than −1. Find all functions $ f: S \to S$ satisfying the two conditions: (a) $ f(x \plus{} f(y) \plus{} xf(y)) \equal{} y \plus{} f(x) \plus{} yf(x)$ for all $ x, y$ in $ S$; (b) $ \frac {f(x)}{x}$ is strictly increasing on each of the two intervals $ \minus{} 1 < x < 0$ and $ 0 < x$.

Find all integers $m, n$ such that $2n^3 - m^3 = mn^2 + 11$.

A polygon is [b]gridded[/b] if the internal angles of the polygon are either $90$ or $270$, it has integer side lengths and its sides don't intersect with each other. Prove that for all $n \ge 8$, it exist a gridded polygon with area $2n$ and perimeter $2n$.

The positive real numbers $a,b,c,$ satisfy: $$\frac{a}{2b+1} + \frac{2b}{3c+1} + \frac{3c}{a+1} = 1$$ $$\frac{1}{a+1} + \frac{1}{2b+1} + \frac{1}{3c+1} = 2$$ What is the value of $\frac{1}{a} + \frac{1}{b} + \frac{1}{c}$

Three friends have a total of $6$ identical pencils, and each one has at least one pencil. In how many ways can this happen? $\textbf{(A)}\ 1\qquad \textbf{(B)}\ 3\qquad \textbf{(C)}\ 6\qquad \textbf{(D)}\ 10\qquad \textbf{(E)}\ 12$

Let $ n, m$ be positive integers of different parity, and $ n > m$. Find all integers $ x$ such that $ \frac {x^{2^n} \minus{} 1}{x^{2^m} \minus{} 1}$ is a perfect square.