Consider a $3\times7$ grid of squares. Each square may be coloured green or white. [list] (a) Is it possible to find a colouring so that no subrectangle has all four corner squares of the same colour? (b) Is it possible for a $4\times 6$ grid? [/list] [i]Subrectangles must have their corners at grid-points of the original diagram. The corner squares of a subrectangle must be different. The original diagram is a subrectangle of itself.[/i]

Let $m$ be the product of the first 100 primes, and let $S$ denote the set of divisors of $m$ greater than 1 (hence $S$ has exactly $2^{100} - 1$ elements). We wish to color each element of $S$ with one of $k$ colors such that $\ \bullet \ $ every color is used at least once; and $\ \bullet \ $ any three elements of $S$ whose product is a perfect square have exactly two different colors used among them. Find, with proof, all values of $k$ for which this coloring is possible.

There are $256$ players in a tennis tournament who are ranked from $1$ to $256$, with $1$ corresponding to the highest rank and $256$ corresponding to the lowest rank. When two players play a match in the tournament, the player whose rank is higher wins the match with probability $\frac{3}{5}$. In each round of the tournament, the player with the highest rank plays against the player with the second highest rank, the player with the third highest rank plays against the player with the fourth highest rank, and so on. At the end of the round, the players who win proceed to the next round and the players who lose exit the tournament. After eight rounds, there is one player remaining and they are declared the winner. Determine the expected value of the rank of the winner.

In the $xy$-coordinate plane, the horizontal line $y = k$ intersects the graph of the cubic $2x^3 + 6x^2 - 4x + 5$ in three points $P$, $Q$, and $R$. Given that $Q$ is the midpoint of $P$ and $R$, what is $k$?

A mixture is prepared by adding $50.0$ mL of $0.200$ M $\ce{NaOH}$ to $75.0$ mL of $0.100$ M $\ce{NaOH}$. What is the $\[[OH^-]$ in the mixture? $ \textbf{(A) }\text{0.0600 M}\qquad\textbf{(B) }\text{0.0800 M}\qquad\textbf{(C) }\text{0.140 M}\qquad\textbf{(D) }\text{0.233 M}\qquad$

We want to construct an isosceles triangle using a compass and a straightedge. We are given two of the following four data: the length of the base of the triangle $(a),$ the length of the leg of the triangle $(b),$ the radius of the inscribed circle $(r),$ and the radius of the circumscribed circle $(R).$ In which of the six possible cases will we definitely be able to construct the triangle? [i]Proposed by György Rubóczky, Budapest[/i]

If $0<a,b<1$ and $p,q\geq 0 ,\ p+q=1$ are real numbers , then prove that: \[a^pb^q+(1-a)^p(1-b)^q\le 1\]

Show that if $f: [0,1]\rightarrow [0,1]$ is a continous function and it has topological transitivity then periodic points of $f$ are dense in $[0,1]$. Topological transitivity means there for every open sets $U$ and $V$ there is $n>0$ such that $f^n(U)\cap V\neq \emptyset$.

For each integer from 1 through 2019, Tala calculated the product of its digits. Compute the sum of all 2019 of Tala's products.

In the rectangle in the figure, we are going to write $12$ numbers from $1$ to $9$, so that the sum of the four numbers written in each line is the same and the sum of the three is also equal numbers in each column. Six numbers have already been written. Determine the sum of the numbers of each row and every column. [img]https://cdn.artofproblemsolving.com/attachments/7/f/3db9ded1e703c5392f258e1608a1800760d78c.png[/img]

A simple polygon in the plane is a figure bounded by a closed nonself-intersecting broken line. (a) Do there exist two congruent simple $7$-gons in the plane such that all the seven vertices of one of the $7$-gons are the vertices of the other one and yet these two $7$-gons have no common sides? (b) Do there exist three such $7$-gons? (V Proizvolov)

There is a board numbered $-10$ to $10$. Each square is coloured either red or white, and the sum of the numbers on the red squares is $n$. Maureen starts with a token on the square labeled $0$. She then tosses a fair coin ten times. Every time she flips heads, she moves the token one square to the right. Every time she flips tails, she moves the token one square to the left. At the end of the ten flips, the probability that the token finishes on a red square is a rational number of the form $\frac a b$. Given that $a + b = 2001$, determine the largest possible value for $n$.

Let $ f: \mathbb R \rightarrow \mathbb R$ be a function such that $ f(tx_1\plus{}(1\minus{}t)x_2)\leq tf(x_1)\plus{}(1\minus{}t)f(x_2)$ for all $ x_1 , x_2 \in \mathbb R$ and $ t\in (0,1)$. Show that $ \sum_{k\equal{}1}^{2003}f(a_{k\plus{}1})a_k \geq \sum_{k\equal{}1}^{2003}f(a_k)a_{k\plus{}1}$ for all real numbers $ a_1,a_2,...,a_{2004}$ such that $ a_1\geq a_2\geq ... \geq a_{2003}$ and $ a_{2004}\equal{}a_1$

Compute \[\sum_{a_1=0}^\infty\sum_{a_2=0}^\infty\cdots\sum_{a_7=0}^\infty\dfrac{a_1+a_2+\cdots+a_7}{3^{a_1+a_2+\cdots+a_7}}.\]

Which regular polygon can be obtained (and how) by cutting a cube with a plane ?

How many positive integers a less than $100$ such that $4a^2 + 3a + 5$ is divisible by $6$.

Let $ABCD$ be a square. The points $N$ and $P$ are chosen on the sides $AB$ and $AD$ respectively, such that $NP=NC$. The point $Q$ on the segment $AN$ is such that that $\angle QPN=\angle NCB$. Prove that $\angle BCQ=\frac{1}{2}\angle AQP$.

If $ \{a_k\}$ is a sequence of real numbers, call the sequence $ \{a'_k\}$ defined by $ a_k' \equal{} \frac {a_k \plus{} a_{k \plus{} 1}}2$ the [i]average sequence[/i] of $ \{a_k\}$. Consider the sequences $ \{a_k\}$; $ \{a_k'\}$ - [i]average sequence[/i] of $ \{a_k\}$; $ \{a_k''\}$ - average sequence of $ \{a_k'\}$ and so on. If all these sequences consist only of integers, then $ \{a_k\}$ is called [i]Good[/i]. Prove that if $ \{x_k\}$ is a [i]good[/i] sequence, then $ \{x_k^2\}$ is also [i]good[/i].

The six-pointed star in the figure is regular: all interior angles of the small triangles are equal. To each of the thirteen points marked are assigned a color: green or red. Prove that there will always be three points of the same color that are vertices of an equilateral triangle. [img]https://cdn.artofproblemsolving.com/attachments/b/f/c50a1f8cb81ea861f16a6a47c3b758c5993213.png[/img]

Each of the $24$ students in Mr. Friedman's class cut up a $7\times 7$ grid of squares while he read them short stories by Mark Twain. While not all of the students cut their squares up in the same way, each of them cut their $7\times 7$ square into at most the three following types (shapes) of pieces. [asy] size(350); defaultpen(linewidth(0.8)); real r = 4.5, s = 9; filldraw(origin--(2,0)--(2,1)--(1,1)--(1,2)--(0,2)--cycle,blue); draw((0,1)--(1,1)--(1,0)); filldraw((r,0)--(r+2,0)--(r+2,2)--(r,2)--cycle,green); draw((r+1,0)--(r+1,2)^^(r,1)--(r+2,1)); filldraw((s,0)--(s+2,0)--(s+2,1)--(s+3,1)--(s+3,2)--(s+1,2)--(s+1,1)--(s,1)--cycle,red); draw((s+1,0)--(s+1,1)--(s+2,1)--(s+2,2)); [/asy] Let $a$, $b$, and $c$ be the number of total pieces of each type from left to right respectively after all $24$ $7\times 7$ squares are cut up. How many ordered triples $(a,b,c)$ are possible?

Let $a_1=1$ and $a_n=n(a_{n-1}+1)$ for $n\ge2$. Define $$p_n=\prod_{i=1}^n\left(1+\frac1{a_i}\right)$$Then $\lim_{n\to\infty}p_n$ is $\textbf{(A)}~1+e$ $\textbf{(B)}~e$ $\textbf{(C)}~1$ $\textbf{(D)}~\infty$

Find all natural numbers $ k\ge 1 $ and $ n\ge 2, $ which have the property that there exist two matrices $ A,B\in M_n\left(\mathbb{Z}\right) $ such that $ A^3=O_n $ and $ A^kB +BA=I_n. $

Find the necessary and sufficient condition that a trapezoid can be formed out of a given four-bar linkage.

We consider the set \begin{align*} \mathbb{C}^{\nu} = \{ (z_1,z_2, \ldots , z_{\nu}) \in \mathbb{C} \},\qquad \nu \geq 2 \end{align*} and the function $\phi : \mathbb{C}^{\nu} \longrightarrow \mathbb{C}^{\nu}$ mapping every element $(z_1,z_2, \ldots , z_{\nu}) \in \mathbb{C}^{\nu}$ to \begin{align*}\phi ( z_1,z_2, \ldots , z_{\nu})= \left( z_1-z_2, z_2-z_3, \ldots, z_{\nu}-z_1 \right) \end{align*} We also consider the $\nu-$tuple $(\omega_0, \omega_1, \ldots , \omega_{\nu-1} )$ $\in \mathbb{C}^{\nu}$ of the $n-$th roots of $-1$, where \begin{align*} \omega_{\mu} = \cos \left( \frac{\pi + 2\mu \pi }{\nu} \right) + \iota \sin \left( \frac{\pi + 2\mu \pi}{\nu} \right) \qquad \mu =0,1, \ldots , \nu -1 \end{align*} Let after $\kappa$ (where $\kappa$ $\in$ $\mathbb{N}$ ), successive applications of $\phi$ to the element $(\omega_0, \omega_1, \ldots , \omega_{\nu-1} )$, we obtain the element \begin{align*} \phi ^{(\kappa)} \left( \omega_0, \omega_1, \ldots , \omega_{\nu-1} \right) =\left( Z_{\kappa 1}, Z_{\kappa 2}, \ldots , Z_{\kappa \nu } \right) \end{align*} Determine [list=i] [*] the values of $\nu$ for which all coordinates of $\phi ^{(\kappa)} \left( \omega_0, \omega_1, \ldots , \omega_{\nu-1} \right) $ have measures less than or equal to $1$ [*] for $\nu =4$, the minimal value of $\kappa \in \mathbb{N}$, for which \begin{align*} \mid Z_{\kappa i} \mid \geq 2^{100} \qquad \qquad 1 \le i \le 4 \end{align*}

Consider a partition of $\{1,2,\ldots,900\}$ into $30$ subsets $S_1,S_2,\ldots,S_{30}$ each with $30$ elements. In each $S_k$, we paint the fifth largest number blue. Is it possible that, for $k=1,2,\ldots,30$, the sum of the elements of $S_k$ exceeds the sum of the blue numbers?