On a $1\times n$ board there are $n-1$ separating edges between neighbouring cells. Initially, none of the edges contain matches. During a move of size $0 < k < n$ a player chooses a $1\times k$ sub-board which contains no matches inside, and places a matchstick on all of the separating edges bordering the sub-board that don’t already have one. A move is considered legal if at least one matchstick can be placed and if either $k = 1$ or $k{}$ is divisible by 4. Two players take turns making moves, the player in turn must choose one of the available legal moves of the largest size $0 < k < n$ and play it. If someone does not have a legal move, the game ends and that player loses. [i]Beat the organisers twice in a row in this game! First the organisers determine the value of $n{}$, then you get to choose whether you want to play as the first or the second player.[/i]

$2000$ numbers are considered: $11, 101, 1001, . . $. Prove that at least $99\%$ of these numbers are composite.

Let $K$, $L$, $M$, $N$ be the midpoints of sides $BC$, $CD$, $DA$, $AB$ respectively of a convex quadrilateral $ABCD$. The common points of segments $AK$, $BL$, $CM$, $DN$ divide each of them into three parts. It is known that the ratio of the length of the medial part to the length of the whole segment is the same for all segments. Does this yield that $ABCD$ is a parallelogram?

Mrs. Jones is pouring orange juice for her 4 kids into 4 identical glasses. She fills the first 3 full, but only has enough orange juice to fill one third of the last glass. What fraction of a glass of orange juice does she need to pour from the 3 full glasses into the last glass so that all glasses have an equal amount of orange juice? $\textbf{(A) }\frac{1}{12}\qquad\textbf{(B) }\frac{1}{4}\qquad\textbf{(C) }\frac{1}{6}\qquad\textbf{(D) }\frac{1}{8}\qquad\textbf{(E) }\frac{2}{9}$

The sum $\frac{2}{3\cdot 6} +\frac{2\cdot 5}{3\cdot 6\cdot 9} +\ldots +\frac{2\cdot5\cdot \ldots \cdot 2015}{3\cdot 6\cdot 9\cdot \ldots \cdot 2019}$ is written as a decimal number. Find the first digit after the decimal point.

Let $\triangle ABC$ satisfies $\cos A:\cos B:\cos C=1:1:2$, then $\sin A=\sqrt[s]{t}$($s\in\mathbb{N},t\in\mathbb{Q^+}$ and $t$ is an irreducible fraction). Find $s+t$.

What is the value of \[ \log_37\cdot\log_59\cdot\log_711\cdot\log_913\cdots\log_{21}25\cdot\log_{23}27? \] $\textbf{(A) } 3 \qquad \textbf{(B) } 3\log_{7}23 \qquad \textbf{(C) } 6 \qquad \textbf{(D) } 9 \qquad \textbf{(E) } 10 $

The sequence an of non-zero reals satisfies $a_n^2 - a_{n-1}a_{n+1} = 1$ for $n \geq 1$. Prove that there exists a real number $\alpha$ such that $a_{n+1} = \alpha a_n - a_{n-1}$ for $n \geq 1$.

For each positive integer $n$, let $\rho(n)$ be the number of positive divisors of $n$ with exactly the same set of prime divisors as $n$. Show that, for any positive integer $m$, there exists a positive integer $n$ such that $\rho(202^n+1)\geq m.$

A trapezium is given with parallel bases having lengths $1$ and $4$. Split it into two trapeziums by a cut, parallel to the bases, of length $3$. We now want to divide the two new trapeziums, always by means of cuts parallel to the bases, in $m$ and $n$ trapeziums, respectively, so that all the $m + n$ trapezoids obtained have the same area. Determine the minimum possible value for $m + n$ and the lengths of the cuts to be made to achieve this minimum value.

Given a square $ABCD$ of side length $6$, the point $E$ is drawn on the line $AB$ such that the distance $EA$ is less than $EB$ and the triangle $\vartriangle BCE$ has the same area as $ABCD$. Compute the shaded area. [img]https://cdn.artofproblemsolving.com/attachments/a/8/5d945a593aee58af3af94f4e8e967eeaeefa6a.png[/img]

The planes $p$ and $p'$ are parallel. A polygon $P$ on $p$ has $m$ sides and a polygon $P'$ on $p'$ has $n$ sides. Find the largest and smallest distances between a vertex of $P$ and a vertex of $P'$.

Yan and Kirill play a game. At first Kirill says 4 numbers $x_1<x_2<x_3<x_4$, and then Yan says three pairwise different non zero numbers $a_1$, $a_2$ and $a_3$. For all $i$ from $1$ to $3$ they consider the quadratic trinomial $f_i(x)$ which has roots $x_i$ and $x_{i+1}$ and leading coefficient $a_i$, and construct on the plane the graphs of that trinomials. Yan wins if in every pair $(f_1(x),f_2(x))$ and $(f_2(x),f_3(x))$ their graphs intersect at exactly one point, and if in some pair graphs do not intersect or intersect at more than one point Kirill wins. Find which player can guarantee his win regardless of the actions of his opponent. [i]V. Kamianetski[/i]

There are twenty-four 4-digit numbers that use each of the four digits 2, 5, 7, and 4exactly once. Listed in numerical order from smallest to largest, the number in the $17th$ position in the list is $ \text{(A)}\ 4527\qquad\text{(B)}\ 5724\qquad\text{(C)}\ 5742\qquad\text{(D)}\ 7245\qquad\text{(E)}\ 7524 $

Professor Shian Bray is buying CCA Math Bananas$^{\text{TM}}$. He starts with $\$500$. The first CCA Math Bananas$^{\text{TM}}$ he buys costs $\$1$. Each time after he buys a CCA Math Banana$^{\text{TM}}$, the cost of a CCA Math Bananas$^{\text{TM}}$ doubles with probability $\frac{1}{2}$ (otherwise staying the same). Professor Bray buys CCA Math Bananas$^{\text{TM}}$ until he cannot afford any more, ending with $n$ CCA Math Bananas$^{\text{TM}}$. Estimate the expected value of $n$. An estimate of $E$ earns $2^{1-0.25|E-A|}$ points, where $A$ is the actual answer. [i]2020 CCA Math Bonanza Lightning Round #5.1[/i]

Assume that complex numbers $z_1,z_2,...,z_n$ satisfy $|z_i-z_j| \le 1$ for any $1 \le i <j \le n$. Let $$S= \sum_{1 \le i <j \le n} |z_i-z_j|^2.$$ (1) If $n = 6063$, find the maximum value of $S$. (2) If $n= 2021$, find the maximum value of $S$.

What is the greatest integer not exceeding the sum $\sum^{1599}_{n=1} \dfrac{1}{\sqrt{n}}$?

Call a set of positive integers $\textit{good}$ if there is a partition of it into two sets $S$ and $T$, such that there do not exist three elements $a, b, c \in S$ such that $a^b = c$ and such that there do not exist three elements $a, b, c \in T$ such that $a^b = c$ ($a$ and $b$ need not be distinct). Find the smallest positive integer $n$ such that the set $\{2, 3, 4, \dots, n\}$ is \textit{not} good.

[b]p1.[/b] Solve the equation: $\sqrt{x} +\sqrt{x + 1} - \sqrt{x + 2} = 0$. [b]p2.[/b] Solve the inequality: $\ln (x^2 + 3x + 2) \le 0$. [b]p3.[/b] In the trapezoid $ABCD$ ($AD \parallel BC$) $|AD|+|AB| = |BC|+|CD|$. Find the ratio of the length of the sides $AB$ and $CD$ ($|AB|/|CD|$). [b]p4.[/b] Gollum gave Bilbo a new riddle. He put $64$ stones that are either white or black on an $8 \times 8$ chess board (one piece per each of $64$ squares). At every move Bilbo can replace all stones of any horizontal or vertical row by stones of the opposite color (white by black and black by white). Bilbo can make as many moves as he needs. Bilbo needs to get a position when in every horizontal and in every vertical row the number of white stones is greater than or equal to the number of black stones. Can Bilbo solve the riddle and what should be his solution? [b]p5.[/b] Two trolls Tom and Bert caught Bilbo and offered him a game. Each player got a bag with white, yellow, and black stones. The game started with Tom putting some number of stones from his bag on the table, then Bert added some number of stones from his bag, and then Bilbo added some stones from his bag. After that three players started making moves. At each move a player chooses two stones of different colors, takes them away from the table, and puts on the table a stone of the color different from the colors of chosen stones. Game ends when stones of one color only remain on the table. If the remaining stones are white Tom wins and eats Bilbo, if they are yellow, Bert wins and eats Bilbo, if they are black, Bilbo wins and is set free. Can you help Bilbo to save his life by offering him a winning strategy? [b]p6.[/b] There are four roads in Mirkwood that are straight lines. Bilbo, Gandalf, Legolas, and Thorin were travelling along these roads, each along a different road, at a different constant speed. During their trips Bilbo met Gandalf, and both Bilbo and Gandalf met Legolas and Thorin, but neither three of them met at the same time. When meeting they did not stop and did not change the road, the speed, and the direction. Did Legolas meet Thorin? Justify your answer. PS. You should use hide for answers.

Given positive integers $n$ and $k$, say that $n$ is $k$-[i]solvable[/i] if there are positive integers $a_1$, $a_2$, ..., $a_k$ (not necessarily distinct) such that \[ \frac{1}{a_1} + \frac{1}{a_2} + \cdots + \frac{1}{a_k} = 1 \] and \[ a_1 + a_2 + \cdots + a_k = n. \] Prove that if $n$ is $k$-solvable, then $42n + 12$ is $(k + 3)$-solvable.

Let $t$ be an integer. Suppose the equation $$x^2 + (4t - 1) x + 4t^2 = 0$$ has at least one positive integer solution $n$. Show that $n$ is a perfect square.

In a country, there are $2018$ cities, some of which are connected by roads. Each city is connected to at least three other cities. It is possible to travel from any city to any other city using one or more roads. For each pair of cities, consider the shortest route between these two cities. What is the greatest number of roads that can be on such a shortest route?

Sheldon and Bella play a game on an infinite grid of cells. On each of his turns, Sheldon puts one of the following tetrominoes (reflections and rotations aren't permitted) [asy] size(200); draw((0, 0)--(1, 0)--(1, 2)--(0, 2)--cycle); draw((1, 1)--(2, 1)--(2, 3)--(1, 3)--cycle); draw((0,1)--(1,1)); draw((1,2)--(2,2)); draw((5, 0.5)--(6, 0.5)--(6, 1.5)--(5, 1.5)--cycle); draw((6, 0.5)--(7, 0.5)--(7, 1.5)--(6, 1.5)--cycle); draw((6, 1.5)--(7, 1.5)--(7, 2.5)--(6, 2.5)--cycle); draw((7, 1.5)--(8, 1.5)--(8, 2.5)--(7, 2.5)--cycle); [/asy] somewhere on the grid without overlap. Then, Bella colors that tetromino such that it has a different color from any other tetromino that shares a side with it. After $2631$ such moves by each player, the game ends, and Sheldon's score is the number of colors used by Bella. What's the maximum $N$ such that Sheldon can guarantee that his score will be at least $N$?

Let $p$ be an odd prime number and $c$ an integer for which $2c -1$ is divisible by $p$. Prove that $$(-1)^{\frac{p+1}{2}}+\sum_{n=0}^{\frac{p-1}{2}} {2n \choose n}c^n$$ is divisible by $p$.

A parallelepiped is cut by a plane along a 6-gon. Supposed this 6-gon can be put into a certain rectangle $ \pi$ (which means one can put the rectangle $ \pi$ on the parallelepiped's plane such that the 6-gon is completely covered by the rectangle). Show that one also can put one of the parallelepiped' faces into the rectangle $ \pi.$