Let $a_0,a_1,a_2,\dots$ be a periodic sequence of real numbers(that is, there is a fixed positive integer $k$ such that $a_n=a_{n+k}$ for every integer $n\geq 0$). The following equality is true, for all $n\geq 0$: $a_{n+2}=\frac{1}{n+2} (a_n - \frac{n+1}{a_{n+1}})$ if $a_0=2020$, determine the value of $a_1$.

Define a $100 \times 100$ square grid $G$. Initially color all cells of $G$ white. A move consists of selecting a $1 \times 7$ or $7 \times 1$ subgrid of $G$ and flipping the colors of all cells in this subgrid from white to black or vice versa. Is it possible that after a series of moves, all cells are colored black? [i]Proposed by Evan Chang (squareman), USA[/i]

There were $n> 1$ aborigines living on an island, each of them telling only the truth or only lying, and each having at least one friend among the others. The new governor asked each aborigine whether there are more truthful aborigines or liars among his friends, or an equal number of both. Each aborigine answered that there are more liars than truthful aborigines among his friends. The governor then ordered one of the aborigines to be executed for being a liar and asked each of the remaining $n- 1$ aborigines the same question again. This time each aborigine answered that there are more truthful aborigines than liars among his friends. Determine whether the executed aborigine was truthful or a liar, and whether there are more truthful aborigines or liars remaining on the island.

Let $x,y,z$ be positive integers such that $\frac{x+1}{y}+\frac{y+1}{z}+\frac{z+1}{x}$ is an integer. Let $d$ be the greatest common divisor of $x,y$ and $z$. Prove that $d\le \sqrt[3]{xy+yz+zx}$.

Let $a_1 ,a_2 ,\ldots, a_n$ be a permutation of the integers $1,2,\ldots, n.$ Call $a_i$ a [i]big[/i] integer if $a_i >a_j$ for all $i<j.$ Find the mean number of big integers over all permutations on the first $n$ postive integers.

A square board is dissected into $n^2$ rectangular cells by $n-1$ horizontal and $n-1$ vertical lines. The cells are painted alternately black and white in a chessboard pattern. One diagonal consists of $n$ black cells which are squares. Prove that the total area of all black cells is not less than the total area of all white cells.

How many integers of the form $n^{2023-n}$ are perfect squares, where $n$ is a positive integer between $1$ and $2023$ inclusive?

In a triangle $ABC$, the altitude from $A$ meets the circumcircle again at $T$ . Let $O$ be the circumcenter. The lines $OA$ and $OT$ intersect the side $BC$ at $Q$ and $M$, respectively. Prove that \[\frac{S_{AQC}}{S_{CMT}} = \biggl( \frac{ \sin B}{\cos C} \biggr)^2 .\]

Find the maximal count of shapes that can be placed over a chessboard with size $8\times8$ in such a way that no three shapes are not on two squares, lying next to each other by diagonal parallel $A1-H8$ ($A1$ is the lowest-bottom left corner of the chessboard, $H8$ is the highest-upper right corner of the chessboard). [i]V. Chukanov[/i]

There was a rook at some square of a $10 \times 10{}$ chessboard. At each turn it moved to a square adjacent by side. It visited each square exactly once. Prove that for each main diagonal (the diagonal between the corners of the board) the following statement is true: in the rook’s path there were two consecutive steps at which the rook first stepped away from the diagonal and then returned back to the diagonal. [i]Alexandr Gribalko[/i]

For each permutation $\sigma$ of the set $\{1, 2, \ldots , N\}$ we define its [i]correctness[/i] as the number of triples $1 \leqslant i < j < k \leqslant N$ such that the number $\sigma(j)$ lies between the numbers $\sigma(i)$ and $\sigma(k)$. Find the difference between the number of permutations with even correctness and the number of permutations with odd correctness if a) $N = 2018$ and b) $N = 2019$.

A list of integers has mode $ 32$ and mean $ 22$. The smallest number in the list is $ 10$. The median $ m$ of the list is a member of the list. If the list member $ m$ were replaced by $ m \plus{} 10$, the mean and median of the new list would be $ 24$ and $ m \plus{} 10$, respectively. If $ m$ were instead replaced by $ m \minus{} 8$, the median of the new list would be $ m \minus{} 4$. What is $ m$? $ \textbf{(A)}\ 16\qquad \textbf{(B)}\ 17\qquad \textbf{(C)}\ 18\qquad \textbf{(D)}\ 19\qquad \textbf{(E)}\ 20$

Consider equilateral triangle $ABC$ with side length $1$. Suppose that a point $P$ in the plane of the triangle satisfies \[2AP=3BP=3CP=\kappa\] for some constant $\kappa$. Compute the sum of all possible values of $\kappa$. [i]2018 CCA Math Bonanza Lightning Round #3.4[/i]

Find the largest constant $c>0$ such that for every positive integer $n\ge 2$, there always exist a positive divisor $d$ of $n$ such that $$d\le \sqrt{n}\hspace{0.5cm} \text{and} \hspace{0.5cm} \tau(d)\ge c\sqrt{\tau(n)}$$ where $\tau(n)$ is the number of divisors of $n$. [i]Proposed by Mohd. Suhaimi Ramly[/i]

Determine the values of the positive integer $n$ for which the following system of equations has a solution in positive integers $x_1, x_2,...,, x_n$. Find all solutions for each such $n$. $$\begin{cases} x_1 + x_2 +...+ x_n = 16 \\ \\ \dfrac{1}{x_1} + \dfrac{1}{x_2} +...+ \dfrac{1}{x_n} = 1\end{cases}$$

There are $k$ heaps on the table, each containing a different positive number of stones. Juri and Mari make moves alternatingly, Juri starts. On each move, the player making the move has to pick a heap and remove one or more stones in it from the table; in addition, the player is allowed to distribute any number of remaining stones from that heap in any way between other non-empty heaps. The player to remove the last stone from the table wins. For which positive integers $k$ does Juri have a winning strategy for any initial state that satisfies the conditions?

Let $m$ and $n$ be natural numbers. Every one of the $m*n$ squares of the $m*n$ board is colored either black or white, so that no 2 neighbouring squares are the same color(the board is colored like in chess").In one step we can pick 2 neighbouring squares and change their colors like this: [b]- [/b]a white square becomes black; [b]-[/b]a black square becomes blue; [b]-[/b]a blue square becomes white. For which $m$ and $n$ can we ,in a finite sequence of these steps, switch the starting colors from white to black and vice versa.

Let $a_1 < a_2 < ... < a_n$ be a sequence of natural numbers such that for $i < j$ the decimal representation of $a_i$ does not occur as the leftmost digits of the decimal representation of $a_j$ . (For example, $137$ and $13729$ cannot both occur in the sequence.) Prove that $\sum_{i=1}^n \frac{1}{a_i} \le 1+\frac12 +\frac13 +...+\frac19$ .

All perfect squares, and all perfect squares multiplied by two, are written in a row in increasing order. let $f(n)$ be the $n$-th number in this sequence. (For instance, $f(1)=1,f(2)=2,f(3)=4,f(4)=8$.) Is there an integer $n$ such that all the numbers \[f(n),f(2n),f(3n),\dots,f(10n^2)\] are perfect squares?

Each cell of a $29 \times 29$ table contains one of the integers $1, 2, 3, \ldots , 29$, and each of these integers appears $29$ times. The sum of all the numbers above the main diagonal is equal to three times the sum of all the numbers below this diagonal. Determine the number in the central cell of the table.

A football game was played between two teams, the Cougars and the Panthers. The two teams scored a total of 34 points, and the Cougars won by a margin of 14 points. How many points did the Panthers score? $ \textbf{(A) } 10 \qquad \textbf{(B) } 14 \qquad \textbf{(C) } 17 \qquad \textbf{(D) } 20 \qquad \textbf{(E) } 24$

$A, B, C$, and $D$ are all on a circle, and $ABCD$ is a convex quadrilateral. If $AB = 13$, $BC = 13$, $CD = 37$, and $AD = 47$, what is the area of $ABCD$?

Let $a\in (0,1)$, $f(z)=z^2-z+a, z\in \mathbb{C}$. Prove the following statement holds: For any complex number z with $|z| \geq 1$, there exists a complex number $z_0$ with $|z_0|=1$, such that $|f(z_0)| \leq |f(z)|$.

10. Michael is playing basketball. He makes $10\%$ of his shots, and gets the ball back after $90\%$ of his missed shots. If he does not get the ball back he stops playing. What is the probability that Michael eventually makes a shot? 11. How many subsets $S$ of the set $\{1, 2, \ldots , 10\}$ satisfy the property that, for all $i \in [1, 9]$, either $i$ or $i + 1$ (or both) is in S? 12. A positive integer $\overline{ABC}$, where $A, B, C$ are digits, satisfies $$\overline{ABC} = B^C - A$$ Find $\overline{ABC}$.

Prove that the equation $$1320x^3=(y_1+y_2+y_3+y_4)(z_1+z_2+z_3+z_4)(t_1+t_2+t_3+t_4+t_5)$$ has infinitely many solutions in the set of Fibonacci numbers. [i]Proposed by Mihály Bencze[/i]