Complex number $\omega$ satisfies $\omega^5 = 2$. Find the sum of all possible values of $\omega^4 + \omega^3 + \omega^2 + \omega + 1$.

In a certain city, age is reckoned in terms of real numbers rather than integers. Every two citizens $x$ and $x'$ either know each other or do not know each other. Moreover, if they do not, then there exists a chain of citizens $x = x_0, x_1, \ldots, x_n = x'$ for some integer $n \geq 2$ such that $ x_{i-1}$ and $x_i$ know each other. In a census, all male citizens declare their ages, and there is at least one male citizen. Each female citizen provides only the information that her age is the average of the ages of all the citizens she knows. Prove that this is enough to determine uniquely the ages of all the female citizens.

[asy] size(2.5inch); pair A = (-2,0), B = 2dir(150), D = (2,0), C; draw(A..(0,2)..D--cycle); C = intersectionpoint(A..(0,2)..D,Circle(B,arclength(A--B))); draw(A--B--C--D--cycle); label("$A$",A,W); label("$B$",B,NW); label("$C$",C,N); label("$D$",D,E); label("$4$",A--D,S); label("$1$",A--B,E); label("$1$",B--C,SE); //Credit to chezbgone2 for the diagram[/asy] Quadrilateral $ABCD$ is inscribed in a circle with side $AD$, a diameter of length $4$. If sides $AB$ and $BC$ each have length $1$, then side $CD$ has length $\textbf{(A) }\frac{7}{2}\qquad\textbf{(B) }\frac{5\sqrt{2}}{2}\qquad\textbf{(C) }\sqrt{11}\qquad\textbf{(D) }\sqrt{13}\qquad \textbf{(E) }2\sqrt{3}$

In a triangle $ABC$, right in $A$ and isosceles, let $D$ be a point on the side $AC$ ($A \ne D \ne C$) and $E$ be the point on the extension of $BA$ such that the triangle $ADE$ is isosceles. Let $P$ be the midpoint of segment $BD$, $R$ be the midpoint of the segment $CE$ and $Q$ the intersection point of $ED$ and $BC$. Prove that the quadrilateral $ARQP$ is a square.

We say that a positive integer $k$ is tricubic if there are three positive integers $a, b, c,$ not necessarily different, such that $k=a^3+b^3+c^3.$ a) Prove that there are infinitely many positive integers $n$ that satisfy the following condition: exactly one of the three numbers $n, n+2$ and $n+28$ is tricubic. b) Prove that there are infinitely many positive integers $n$ that satisfy the following condition: exactly two of the three numbers $n, n+2$ and $n+28$ are tricubic. c) Prove that there are infinitely many positive integers $n$ that satisfy the following condition: the three numbers $n, n+2$ and $n+28$ are tricubic.

In $\triangle ABC$, $X, Y$, are points on BC such that $BX=XY=YC$, $M , N$ are points on $AC$ such that $AM=MN=NC$. $BM$ and $BN$ intersect $AY$ at $S$ and $R$ and respectively. If the area of $\triangle ABC$ is $1$, find the area of $SMNR$.

A rectangular sheet of paper is folded so that one corner lies on top of the corner diagonally opposite. The resulting shape is a pentagon whose area is $20\%$ one-sheet thick, and $80\%$ two-sheets-thick. Determine the ratio of the two sides of the original sheet of paper.

Let $p> 3$ be a prime number and let $q = \frac{4^p-1}{3}$. Show that $q$ is a composite integer as well is a divisor of $2^{q-1}- 1$.

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 the smallest positive three-digit integer $n$ such that $3^n+4^n$ is divisible by $5$.

Call $A \in \mathbb{N}^0$ be $number of year$ if all digits of $A$ equals $0$, $1$ or $2$ (in decimal representation). Prove that exist infinity $N \in \mathbb{N}$, such that $N$ can't presented as $A^2+B$ where $A \in \mathbb{N}^0 ; B$- $number of year$.

$f(0) = 0$ $f(1) = 1$ $f(2) = 3$ $f(3) = 5$ $f(4) = 9$ $f(5) = 11$ $f(6) = 29$ $f(11) = 31$ $f(20) = ? $

Consider a sequence of numbers $(a_1, a_2, \ldots , a_{2^n}).$ Define the operation \[S\biggl((a_1, a_2, \ldots , a_{2^n})\biggr) = (a_1a_2, a_2a_3, \ldots , a_{2^{n-1}a_{2^n}, a_{2^n}a_1).}\] Prove that whatever the sequence $(a_1, a_2, \ldots , a_{2^n})$ is, with $a_i \in \{-1, 1\}$ for $i = 1, 2, \ldots , 2^n,$ after finitely many applications of the operation we get the sequence $(1, 1, \ldots, 1).$

Let $A_1A_2 ... A_8$ be a convex octagon such that all of its sides are equal and its opposite sides are parallel. For each $i = 1, ... , 8$, define $B_i$ as the intersection between segments $A_iA_{i+4}$ and $A_{i-1}A_{i+1}$, where $A_{j+8} = A_j$ and $B_{j+8} = B_j$ for all $j$. Show some number $i$, amongst 1, 2, 3, and 4 satisfies \[\frac{A_iA_{i+4}}{B_iB_{i+4}} \leq \frac{3}{2}\]

What is the largest amount of elements that can be taken from the set $\{1, 2, ... , 2012, 2013\}$, such that within them there are no distinct three, say $a$, $b$,and $c$, such that $a$ is a divisor or multiple of $b-c$?

We will say that a rearrangement of the letters of a word has no [i]fixed letters[/i] if, when the rearrangement is placed directly below the word, no column has the same letter repeated. For instance $HBRATA$ is a rearragnement with no fixed letter of $BHARAT$. How many distinguishable rearrangements with no fixed letters does $BHARAT$ have? (The two $A$s are considered identical.)

Let $S(k)$ denote the digit-sum of a positive integer $k$(in base $10$). Determine the smallest positive integer $n$ such that \[S(n^2 ) = S(n) - 7\]

Find all polynomials $p(x)$ with real coefficients that satisfy $$4p(x^2) = 4(p(x))^2 + 4p(x)- 1$$

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

Find all natural numbers less than $105$ that are divisible by $1999$ and whose digits sum (in decimal notation) to $25$.

Let $ABCDE$ be a convex pentagon such that $BC \parallel AE,$ $AB = BC + AE,$ and $\angle ABC = \angle CDE.$ Let $M$ be the midpoint of $CE,$ and let $O$ be the circumcenter of triangle $BCD.$ Given that $\angle DMO = 90^{\circ},$ prove that $2 \angle BDA = \angle CDE.$ [i]Proposed by Nazar Serdyuk, Ukraine[/i]

A family wears clothes of three colors: red,blue and green,with a separate,identical laundry bin for each color. At the beginning of the first week,all bins are empty.Each week,the family generates a total of $10 kg $ of laundry(the proportion of each color is subject to variation).The laundry is sorted by color and placed in the bins.Next,the heaviest bin(only one of them,if there are several that are heaviest)is emptied and its content swashed.What is the minimal possible storing capacity required of the laundry bins in order for them never to overflow?

A positive integer $a$ is selected, and some positive integers are written on a board. Alice and Bob play the following game. On Alice's turn, she must replace some integer $n$ on the board with $n+a$, and on Bob's turn he must replace some even integer $n$ on the board with $n/2$. Alice goes first and they alternate turns. If on his turn Bob has no valid moves, the game ends. After analyzing the integers on the board, Bob realizes that, regardless of what moves Alice makes, he will be able to force the game to end eventually. Show that, in fact, for this value of $a$ and these integers on the board, the game is guaranteed to end regardless of Alice's or Bob's moves.

For any real number $x$, we let $\lfloor x \rfloor$ be the unique integer $n$ such that $n \leq x < n+1$. For example. $\lfloor 31.415 \rfloor = 31$. Compute \[2020^{2021} - \left\lfloor\frac{2020^{2021}}{2021} \right \rfloor (2021).\] [i]2021 CCA Math Bonanza Team Round #3[/i]

[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].