The vertices of a convex polygon with $n\geqslant 4$ sides are coloured with black and white. A diagonal is called [i]multicoloured[/i] if its vertices have different colours. A colouring of the vertices is [i]good[/i] if the polygon can be partitioned into triangles by using only multicoloured diagonals which do not intersect in the interior of the polygon. Find the number of good colourings. [i]Proposed by S. Berlov[/i]

Let $n$ be a given integer which is greater than $1$ . Find the greatest constant $\lambda(n)$ such that for any non-zero complex $z_1,z_2,\cdots,z_n$ ,have that \[\sum_{k\equal{}1}^n |z_k|^2\geq \lambda(n)\min\limits_{1\le k\le n}\{|z_{k+1}-z_k|^2\},\] where $z_{n+1}=z_1$.

Luffy is playing with some magic boxes and a machine. Each box has a value(number) inside. Opening a box, Luffy sees the value, adds the value to his score(if the box value is negative, Luffy loses points) and destroys the box. Putting a box of value $X$ in the machine, this box vanishes and it is replaced by two new boxes of values $X+1$ and $X-1$(it's [b]not[/b] known which one has the respective value, but he can identify the new boxes). At the beginning, Luffy has $0$ points and has a box whose value is known(it is zero). a) Prove that Luffy can reach at least $1000$ points b) Is it possible that Luffy reaches at least $1000000$ points, [b]without[/b] have less than $-42$ points in any moment?

[b]p1.[/b] A dog on a $10$ meter long leash is tied to a $10$ meter long, infinitely thin section of fence. What is the minimum area over which the dog will be able to roam freely on the leash, given that we can fix the position of the leash anywhere on the fence? [b]p2.[/b] Suppose that the equation $$\begin{tabular}{cccccc} &\underline{C} &\underline{H} &\underline{M}& \underline{M}& \underline{C}\\ +& &\underline{H}& \underline{M}& \underline{M} & \underline{T}\\ \hline &\underline{P} &\underline{U} &\underline{M} &\underline{A} &\underline{C}\\ \end{tabular}$$ holds true, where each letter represents a single nonnegative digit, and distinct letters represent different digits (so that $\underline{C}\, \underline{H}\, \underline{ M}\, \underline{ M}\, \underline{ C}$ and $ \underline{P}\, \underline{U}\, \underline{M}\, \underline{A}\, \underline{C}$ are both five digit positive integers, and the number $\underline{H }\, \underline{M}\, \underline{M}\, \underline{T}$ is a four digit positive integer). What is the largest possible value of the five digit positive integer$\underline{C}\, \underline{H}\, \underline{ M}\, \underline{ M}\, \underline{ C}$ ? [b]p3.[/b] Square $ABCD$ has side length $4$, and $E$ is a point on segment $BC$ such that $CE = 1$. Let $C_1$ be the circle tangent to segments $AB$, $BE$, and $EA$, and $C_2$ be the circle tangent to segments $CD$, $DA$, and $AE$. What is the sum of the radii of circles $C_1$ and $C_2$? [b]p4.[/b] A finite set $S$ of points in the plane is called tri-separable if for every subset $A \subseteq S$ of the points in the given set, we can find a triangle $T$ such that (i) every point of $A$ is inside $T$ , and (ii) every point of $S$ that is not in $A$ is outside$ T$ . What is the smallest positive integer $n$ such that no set of $n$ distinct points is tri-separable? [b]p5.[/b] The unit $100$-dimensional hypercube $H$ is the set of points $(x_1, x_2,..., x_{100})$ in $R^{100}$ such that $x_i \in \{0, 1\}$ for $i = 1$, $2$, $...$, $100$. We say that the center of $H$ is the point $$\left( \frac12,\frac12, ..., \frac12 \right)$$ in $R^{100}$, all of whose coordinates are equal to $1/2$. For any point $P \in R^{100}$ and positive real number $r$, the hypersphere centered at $P$ with radius $r$ is defined to be the set of all points in $R^{100}$ that are a distance $r$ away from $P$. Suppose we place hyperspheres of radius $1/2$ at each of the vertices of the $100$-dimensional unit hypercube $H$. What is the smallest real number $R$, such that a hypersphere of radius $R$ placed at the center of $H$ will intersect the hyperspheres at the corners of $H$? [b]p6.[/b] Greg has a $9\times 9$ grid of unit squares. In each square of the grid, he writes down a single nonzero digit. Let $N$ be the number of ways Greg can write down these digits, so that each of the nine nine-digit numbers formed by the rows of the grid (reading the digits in a row left to right) and each of the nine nine-digit numbers formed by the columns (reading the digits in a column top to bottom) are multiples of $3$. What is the number of positive integer divisors of $N$? [b]p7.[/b] Find the largest positive integer $n$ for which there exists positive integers $x$, $y$, and $z$ satisfying $$n \cdot gcd(x, y, z) = gcd(x + 2y, y + 2z, z + 2x).$$ [b]p8.[/b] Suppose $ABCDEFGH$ is a cube of side length $1$, one of whose faces is the unit square $ABCD$. Point $X$ is the center of square $ABCD$, and $P$ and $Q$ are two other points allowed to range on the surface of cube $ABCDEFHG$. Find the largest possible volume of tetrahedron $AXPQ$. [b]p9.[/b] Deep writes down the numbers $1, 2, 3, ... , 8$ on a blackboard. Each minute after writing down the numbers, he uniformly at random picks some number $m$ written on the blackboard, erases that number from the blackboard, and increases the values of all the other numbers on the blackboard by $m$. After seven minutes, Deep is left with only one number on the black board. What is the expected value of the number Deep ends up with after seven minutes? [b]p10.[/b] Find the number of ordered tuples $(x_1, x_2, x_3, x_4, x_5)$ of positive integers such that $x_k \le 6$ for each index $k = 1$, $2$, $... $,$ 5$, and the sum $$x_1 + x_2 +... + x_5$$ is $1$ more than an integer multiple of $7$. [b]p11.[/b] The equation $$\left( x- \sqrt[3]{13}\right)\left( x- \sqrt[3]{53}\right)\left( x- \sqrt[3]{103}\right)=\frac13$$ has three distinct real solutions $r$, $s$, and $t$ for $x$. Calculate the value of $$r^3 + s^3 + t^3.$$ [b]p12.[/b] Suppose $a$, $b$, and $c$ are real numbers such that $$\frac{ac}{a + b}+\frac{ba}{b + c}+\frac{cb}{c + a}= -9$$ and $$\frac{bc}{a + b}+\frac{ca}{b+c}+\frac{ab}{c + a}= 10.$$ Compute the value of $$\frac{b}{a + b}+\frac{c}{b + c}+\frac{a}{c + a}.$$ [b]p13.[/b] The complex numbers $w$ and $z$ satisfy the equations $|w| = 5$, $|z| = 13$, and $$52w - 20z = 3(4 + 7i).$$ Find the value of the product $wz$. [b]p14.[/b] For $i = 1, 2, 3, 4$, we choose a real number $x_i$ uniformly at random from the closed interval $[0, i]$. What is the probability that $x_1 < x_2 < x_3 < x_4$ ? [b]p15.[/b] The terms of the infinite sequence of rational numbers $a_0$, $a_1$, $a_2$, $...$ satisfy the equation $$a_{n+1} + a_{n-2} = a_na_{n-1}$$ for all integers $n\ge 2$. Moreover, the values of the initial terms of the sequence are $a_0 =\frac52$, $a_1 = 2$ and} $a_2 =\frac52.$ Call a nonnegative integer $m$ lucky if when we write $a_m =\frac{p}{q}$ for some relatively prime positive integers $p$ and $q$, the integer $p + q$ is divisible by $13$. What is the $101^{st}$ smallest lucky number? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In the plane, two intersecting lines $a$ and $b$ are given, along with a circle $\omega$ that has no common points with these lines. For any line $\ell||b$, define $A=\ell\cap a$, and $\{B,C\}=\ell\cap \omega$ such that $B$ is on segment $AC$. Construct the line $\ell$ such that the ratio $\frac{|BC|}{|AB|}$ is maximal.

Let $f(x) = \sqrt{4x^2 - 4x^4}$. Let $A$ be the number of real numbers $x$ that satisfy $$f(f(f(\dots f(x)\dots ))) = x,$$ where the function $f$ is applied to $x$ 2020 times. Compute $A \pmod {1000}$. [i]Proposed by Timothy Qian[/i]

A parallelogram has $3$ of its vertices at $(1, 2)$, $(3,8)$, and $(4, 1)$. Compute the sum of the possible $x$-coordinates for the $4$th vertex.

Carol was given three numbers and was asked to add the largest of the three to the product of the other two. Instead, she multiplied the largest with the sum of the other two, but still got the right answer. What is the sum of the three numbers?

A square is constructed on the side $AB$ of triangle $ABC$ (outside the triangle).$ O$ is the centre of the square. $M$ and $N$ are the midpoints of the sides $BC$ and $AC$. The lengths of these sides are $a$ and $b$ respectively. Find the maximal possible value of the sum $CM + ON$ (when the angle at $C$ changes). (IF Sharygin)

Let $\tau (n)$ denote the number of positive integer divisors of $n$. Find the sum of the six least positive integers $n$ that are solutions to $\tau (n) + \tau (n+1) = 7$.

A bird starts with 300 ml of blood at 100 degrees in its body, 50 ml of blood at 0 degrees in its feet. Every minute, 50 ml of blood flows from the body to the feet, and 50 ml of blood at 40% of the body temperature flows from the feet to the body. The bird feels cold once its internal body temperature (not including the feet) falls below 60 degrees. Compute how many minutes it takes for the bird to feel cold. [i]2022 CCA Math Bonanza Team Round #6[/i]

In a garden, which is organized as a $2024\times 2024$ board, we plant three types of flowers: roses, daisies, and orchids. We want to plant flowers such that the following conditions are satisfied: (i) Each grid is planted with at most one type of flower. Some grids can be left blank and not planted. (ii) For each planted grid $A$, there exist exactly $3$ other planted grids in the same column or row such that those $3$ grids are planted with flowers of different types from $A$'s. (iii) Each flower is planted in at least $1$ grid. What is the maximal number of the grids that can be planted with flowers?

Find all positive integers $n$ with the following property: the $k$ positive divisors of $n$ have a permutation $(d_1,d_2,\ldots,d_k)$ such that for $i=1,2,\ldots,k$, the number $d_1+d_2+\cdots+d_i$ is a perfect square.

Let be given a semicircle with diameter $AB$ and center $O$, and a line intersecting the semicircle at $C$ and $D$ and the line $AB$ at $M$ ($MB < MA$, $MD < MC$). The circumcircles of the triangles $AOC$ and $DOB$ meet again at $L$. Prove that $\angle MKO$ is right. [i]L. Kuptsov[/i]

Let $n$ be a positive integer. If $4^n + 2^n + 1$ is a prime, prove that $n$ is a power of three.

A box contains $5$ chips, numbered $1$, $2$, $3$, $4$, and $5$. Chips are drawn randomly one at a time without replacement until the sum of the values drawn exceeds $4$. What is the probability that $3$ draws are required? $\textbf{(A)} \frac{1}{15} \qquad \textbf{(B)} \frac{1}{10} \qquad \textbf{(C)} \frac{1}{6} \qquad \textbf{(D)} \frac{1}{5} \qquad \textbf{(E)} \frac{1}{4}$

Consider the points $O = (0,0), A = (- 2,0)$ and $B = (0,2)$ in the coordinate plane. Let $E$ and $F$ be the midpoints of $OA$ and $OB$ respectively. We rotate the triangle $OEF$ with a center in $O$ clockwise until we obtain the triangle $OE'F'$ and, for each rotated position, let $P = (x, y)$ be the intersection of the lines $AE'$ and $BF'$. Find the maximum possible value of the $y$-coordinate of $P$.

Find the dimension of the solution space of the problem $\partial u/\partial \overline{z} = \delta(z - i)$ for $\text{Im } z \ge 0$, $\text{Im } u(z) = 0$ for $\text{Im } z = 0$, $u\to 0$ as $z\to\infty$.

Let $\phi(n)$ be the number of positive integers less than $n$ that are relatively prime to $n$, where $n$ is a positive integer. Find all pairs of positive integers $(m,n)$ such that \[2^n + (n-\phi(n)-1)! = n^m+1.\]

Which one of the following points is [u]not[/u] on the graph of $y=\dfrac{x}{x+1}$? $\textbf{(A)}\ (0,0)\qquad \textbf{(B)}\ \left(-\dfrac{1}{2},-1\right) \qquad \textbf{(C)}\ \left(\dfrac{1}{2},\dfrac{1}{3}\right) \qquad \textbf{(D)}\ (-1,1) \qquad \textbf{(E)}\ (-2,2)$

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

At the beginning of a trip, the mileage odometer read $ 56200$ miles. The driver filled the gas tank with $ 6$ gallons of gasoline. During the trip, the driver filled his tank again with $ 12$ gallons of gasoline when the odometer read $ 56560$. At the end of the trip, the driver filled the tank again with $ 20$ gallons of gasoline. The odometer read $ 57060$. To the nearest tenth, what was the car's average miles-per-gallon for the entire trip? \[ \textbf{(A)}\ 22.5 \qquad \textbf{(B)}\ 22.6 \qquad \textbf{(C)}\ 24.0 \qquad \textbf{(D)}\ 26.9 \qquad \textbf{(E)}\ 27.5 \]

Evaluate $\int_2^a \frac{x^a-1-xa^x\ln a}{(x^a-1)^2}dx.$ proposed by kunny

Find the remainder of $$\prod_{n = 2}^{99} (1 - n^2 + n^4)(1 - 2n^2 + n^4)$$ when divided by $101$.

Let $S$ be the set of all polygonal areas in a plane. Prove that there is a function $f : S \to (0,1)$ which satisfies $f(S_1 \cup S_2) = f(S_1)+ f(S_2)$ for any $S_1,S_2 \in S$ which have common points only on their borders