Let $f(x) = x^3 - 3x + b$ and $g(x) = x^2 + bx -3$, where $b$ is a real number. What is the sum of all possible values of $b$ for which the equations $f(x)$ = 0 and $g(x) = 0$ have a common root?

Find all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ which satisfy \[ f(m + f(n)) = f(m) + n \] for all $m,n \in \mathbb{Z}$.

[b]p1.[/b] Four witches are riding their brooms around a circle with circumference $10$ m. They are standing at the same spot, and then they all start to ride clockwise with the speed of $1$, $2$, $3$, and $4$ m/s, respectively. Assume that they stop at the time when every pair of witches has met for at least two times (the first position before they start counts as one time). What is the total distance all the four witches have travelled? [b]p2.[/b] Suppose $A$ is an equilateral triangle, $O$ is its inscribed circle, and $B$ is another equilateral triangle inscribed in $O$. Denote the area of triangle $T$ as $[T]$. Evaluate $\frac{[A]}{[B]}$. [b]p3. [/b]Tim has bought a lot of candies for Halloween, but unfortunately, he forgot the exact number of candies he has. He only remembers that it's an even number less than $2020$. As Tim tries to put the candies into his unlimited supply of boxes, he finds that there will be $1$ candy left if he puts seven in each box, $6$ left if he puts eleven in each box, and $3$ left if he puts thirteen in each box. Given the above information, find the total number of candies Tim has bought. [b]p4.[/b] Let $f(n)$ be a function defined on positive integers n such that $f(1) = 0$, and $f(p) = 1$ for all prime numbers $p$, and $$f(mn) = nf(m) + mf(n)$$ for all positive integers $m$ and $n$. Let $$n = 277945762500 = 2^23^35^57^7$$ Compute the value of $\frac{f(n)}{n}$ . [b]p5.[/b] Compute the only positive integer value of $\frac{404}{r^2-4}$ , where $r$ is a rational number. [b]p6.[/b] Let $a = 3 +\sqrt{10}$ . If $$\prod^{\infty}_{k=1} \left( 1 + \frac{5a + 1}{a^k + a} \right)= m +\sqrt{n},$$ where $m$ and $n$ are integers, find $10m + n$. [b]p7.[/b] Charlie is watching a spider in the center of a hexagonal web of side length $4$. The web also consists of threads that form equilateral triangles of side length $1$ that perfectly tile the hexagon. Each minute, the spider moves unit distance along one thread. If $\frac{m}{n}$ is the probability, in lowest terms, that after four minutes the spider is either at the edge of her web or in the center, find the value of $m + n$. [b]p8.[/b] Let $ABC$ be a triangle with $AB = 10$; $AC = 12$, and $\omega$ its circumcircle. Let $F$ and $G$ be points on $\overline{AC}$ such that $AF = 2$, $FG = 6$, and $GC = 4$, and let $\overrightarrow{BF}$ and $\overrightarrow{BG}$ intersect $\omega$ at $D$ and $E$, respectively. Given that $AC$ and $DE$ are parallel, what is the square of the length of $BC$? [b]p9.[/b] Two blue devils and $4$ angels go trick-or-treating. They randomly split up into $3$ non-empty groups. Let $p$ be the probability that in at least one of these groups, the number of angels is nonzero and no more than the number of devils in that group. If $p = \frac{m}{n}$ in lowest terms, compute $m + n$. [b]p10.[/b] We know that$$2^{22000} = \underbrace{4569878...229376}_{6623\,\,\, digits}.$$ For how many positive integers $n < 22000$ is it also true that the first digit of $2^n$ is $4$? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A positive integer is called [i]downhill[/i] if the digits in its decimal representation form a nonstrictly decreasing sequence from left to right. Suppose that a polynomial $P(x)$ with rational coefficients takes on an integer value for each downhill positive integer $x$. Is it necessarily true that $P(x)$ takes on an integer value for each integer $x$?

Let $r_1$, $r_2$, $r_3$ be the distinct real roots of $x^3-2019x^2-2020x+2021=0$. Prove that $r_1^3+r_2^3+r_3^3$ is an integer multiple of $3$.

Find the smallest value that the expression takes $x^4 + y^4 - x^2y - xy^2$, for positive numbers $x$ and $y$ satisfying $x + y \le 1$.

A sequence $\{a_n\}$ is defined by: $a_1 = 1, a_{n+1} = a_n + \dfrac{1}{\sqrt{a_n}}$ for $n = 1, 2, 3, \ldots$. Find all real numbers $q$ such that the sequence $\{u_n\}$ defined by $u_n = a_n^q$, $n = 1, 2, 3, \ldots$ has nonzero finite limit when $n$ goes to infinity. THERE MIGHT BE A TYPO!

$$\lim_{n \to \infty} nr\sqrt[2]{1-\cos \frac{2\pi}{n}}=?$$

Define the sequences $(a_n),(b_n)$ by \begin{align*} & a_n, b_n > 0, \forall n\in\mathbb{N_+} \\ & a_{n+1} = a_n - \frac{1}{1+\sum_{i=1}^n\frac{1}{a_i}} \\ & b_{n+1} = b_n + \frac{1}{1+\sum_{i=1}^n\frac{1}{b_i}} \end{align*} 1) If $a_{100}b_{100} = a_{101}b_{101}$, find the value of $a_1-b_1$; 2) If $a_{100} = b_{99}$, determine which is larger between $a_{100}+b_{100}$ and $a_{101}+b_{101}$.

[b]p1.[/b] How many real solutions does the following system of equations have? Justify your answer. $$x + y = 3$$ $$3xy -z^2 = 9$$ [b]p2.[/b] After the first year the bank account of Mr. Money decreased by $25\%$, during the second year it increased by $20\%$, during the third year it decreased by $10\%$, and during the fourth year it increased by $20\%$. Does the account of Mr. Money increase or decrease during these four years and how much? [b]p3.[/b] Two circles are internally tangent. A line passing through the center of the larger circle intersects it at the points $A$ and $D$. The same line intersects the smaller circle at the points $B$ and $C$. Given that $|AB| : |BC| : |CD| = 3 : 7 : 2$, find the ratio of the radiuses of the circles. [b]p4.[/b] Find all integer solutions of the equation $\frac{1}{x}+\frac{1}{y}=\frac{1}{19}$ [b]p5.[/b] Is it possible to arrange the numbers $1, 2, . . . , 12$ along the circle so that the absolute value of the difference between any two numbers standing next to each other would be either $3$, or $4$, or $5$? Prove your answer. [b]p6.[/b] Nine rectangles of the area $1$ sq. mile are located inside the large rectangle of the area $5$ sq. miles. Prove that at least two of the rectangles (internal rectangles of area $1$ sq. mile) overlap with an overlapping area greater than or equal to $\frac19$ sq. mile PS. You should use hide for answers.

Show that the inequality $(x^2 + y^2z^2) (y^2 + x^2z^2) (z^2 + x^2y^2) \ge 8xy^2z^3$ is valid for all integers $x, y$ and $z$.When does equality apply?

Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$, such that $$f(f(x)+y)(x-f(y)) = f(x)^2-f(y^2).$$

Find all complex solutions to the system \begin{align*} (a+ic)^3+(ia+b)^3+(-b+ic)^3&=-6,\\ (a+ic)^2+(ia+b)^2+(-b+ic)^2&=6,\\ (1+i)a+2ic&=0.\end{align*}

Let $ a,b,c$ be complex numbers. Prove that if all the roots of the equation $ x^3\plus{}ax^2\plus{}bx\plus{}c\equal{}0$ are of module $ 1$, then so are the roots of the equation $ x^3\plus{}|a|x^2\plus{}|b|x\plus{}|c|\equal{}0$.

Determine all infinite sequences of nonnegative integers $a_1,a_2,\ldots$ such that: 1. Every positive integer appears in the sequence at least once, and; 2. $a_i$ is the smallest integer $j$ such that $a_{j+2}=i$, for all $i\ge 1$. [i](Proposed by Ho Janson)[/i]

Let $a_0, a_1, a_2,\ldots $ be a sequence of real numbers satisfying $a_0=1$ and $a_n=a_{\lfloor 7n/9\rfloor}+a_{\lfloor n/9\rfloor}$ for $n=1, 2,\ldots $ Prove that there exists a positive integer $k$ with $a_k<\frac{k}{2001!}$.

Prove that for every natural number $ n $ the inequality holds $$ 1 + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \ldots + \frac{1}{\sqrt{n}} > \sqrt{n-1}.$$

Show that it is possible to color the points of $\mathbb Q\times\mathbb Q$ in two colors in such a way that any two points having distance $1$ have distinct colors.

Find all functions $ f: \mathbb{Z} \minus{} \{ 0 \} \rightarrow \mathbb{Q}$ satisfying: $ f \left( \frac{x\plus{}y}{3} \right)\equal{}\frac {f(x)\plus{}f(y)}{2},$ whenever $ x,y,\frac{x\plus{}y}{3} \in \mathbb{Z} \minus{} \{ 0 \}.$

Find all he positive integers $x, y, z, t, w$, such as: $x+\frac{1}{y+\frac{1}{z+\frac{1}{t+\frac{1}{w}}}}=\frac{1998}{115}$

Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that, for all real numbers $x$ and $y$ satisfy, $$f\left(x+yf(x+y)\right)=y^2+f(x)f(y)$$ [i]Proposed by Dorlir Ahmeti, Kosovo[/i]

Let $n$ be a positive integer. Given is a subset $A$ of $\{0,1,...,5^n\}$ with $4n+2$ elements. Prove that there exist three elements $a<b<c$ from $A$ such that $c+2a>3b$. [i]Proposed by Dominik Burek and Tomasz Ciesla, Poland[/i]

Given are the non zero natural numbers $a,b,c$ such that the number $\frac{a\sqrt2+b\sqrt3}{b\sqrt2+c\sqrt3}$ is rational. Prove that the number $\frac{a^2+b^2+c^2}{a+b+c}$ is an integer .

Determine the set of all real numbers $p$ for which the polynomial $Q(x) = x^3 + px^2 - px - 1$ has three distinct real roots.

For any set $A=\{a_1,a_2,\cdots,a_m\}$, let $P(A)=a_1a_2\cdots a_m$. Let $n={2010\choose99}$, and let $A_1, A_2,\cdots,A_n$ be all $99$-element subsets of $\{1,2,\cdots,2010\}$. Prove that $2010|\sum^{n}_{i=1}P(A_i)$.