Let $ AD$ is centroid of $ ABC$ triangle. Let $ (d)$ is the perpendicular line with $ AD$. Let $ M$ is a point on $ (d)$. Let $ E, F$ are midpoints of $ MB, MC$ respectively. The line through point $ E$ and perpendicular with $ (d)$ meet $ AB$ at $ P$. The line through point $ F$ and perpendicular with $ (d)$ meet $ AC$ at $ Q$. Let $ (d')$ is a line through point $ M$ and perpendicular with $ PQ$. Prove $ (d')$ always pass a fixed point.

The figure shows a large circle with radius $2$ m and four small circles with radii $1$ m. It is to be painted using the three shown colours. What is the cost of painting the figure? [img]https://1.bp.blogspot.com/-oWnh8uhyTIo/XzP30gZueKI/AAAAAAAAMUY/GlC3puNU_6g6YRf6hPpbQW8IE8IqMP3ugCLcBGAsYHQ/s0/2018%2BMohr%2Bp2.png[/img]

In the beginning, there is a pair of positive integers $(m,n)$ written on the board. Alice and Bob are playing a turn-based game with the following move. At each turn, a player erases one of the numbers written on the board, and writes a different positive number not less than the half of the erased one. If a player cannot write a new number at some turn, he/she loses the game. For how many starting pairs $(m,n)$ from the pairs $(7,79)$, $(17,71)$, $(10,101)$, $(21,251)$, $(50,405)$, can Alice guarantee to win when she makes the first move? $ \textbf{(A)}\ 4 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 1 \qquad\textbf{(E)}\ \text{None of above} $

If $ a\equal{}\minus{}2$, the largest number in the set $ \left \{ \minus{}3a,4a,\frac{24}{a},a^2,1 \right \}$ is \[ \textbf{(A)}\ \minus{}3a \qquad \textbf{(B)}\ 4a \qquad \textbf{(C)}\ \frac{24}{a} \qquad \textbf{(D)}\ a^2 \qquad \textbf{(E)}\ 1 \]

Let $x_n=\binom{2n}{n}$ for all $n\in\mathbb{Z}^+$. Prove there exist infinitely many finite sets $A,B$ of positive integers, satisfying $A \cap B = \emptyset $, and \[\frac{{\prod\limits_{i \in A} {{x_i}} }}{{\prod\limits_{j\in B}{{x_j}} }}=2012.\]

Each side of a triangle $ABC$ is divided into three equal parts, and the middle segment in each of the sides is painted green. In the exterior of $\triangle ABC$ three equilateral triangles are constructed, in such a way that the three green segments are sides of these triangles. Denote by $A',B',C'$ the vertices of these new equilateral triangles that don’t belong to the edges of $\triangle ABC$, respectively. Let $A'',B'',C''$ be the points symmetric to $A',B',C'$ with respect to $BC,CA,AB$. (a) Prove that $\triangle A'B'C'$ and $\triangle A''B''C''$ are equilateral. (b) Prove that $ABC,A'B'C'$, and $A''B''C''$ have a common centroid.

If $p$ and $q$ are natural numbers so that \[ \frac{p}{q}=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+ \ldots -\frac{1}{1318}+\frac{1}{1319}, \] prove that $p$ is divisible with $1979$.

Find all polynomials $f\in \mathbb{Z}[X]$ such that if $p$ is prime then $f(p)$ is also prime.

Mellon Game Lab has come up with a concept for a new game: Square Finder. The premise is as follows. You are given an $n\times n$ grid of squares (for integer $n\geq 2$), each of which is either blank or has an arrow pointing up, down, left, or right. You are also given a $2\times 2$ grid of squares that appears somewhere in this grid, possibly rotated. For example, see if you can find the following $2\times 2$ grid inside the larger $4\times 4$ grid. [asy] size(2cm); defaultpen(fontsize(16pt)); string b = ""; string u = "$\uparrow$"; string d = "$\downarrow$"; string l = "$\leftarrow$"; string r = "$\rightarrow$"; // input should be n x n string[][] input = {{b,u},{r,l}}; int n = input.length; // draw table for (int i=0; i<=n; ++i) { draw((i,0)--(i,n)); draw((0,i)--(n,i)); } // fill table for (int i=1; i<=n; ++i) { for (int j=1; j<=n; ++j) { label(input[i-1][j-1], (j-0.5,n-i+0.5)); } } [/asy] [asy] size(4cm); defaultpen(fontsize(16pt)); string b = ""; string u = "$\uparrow$"; string d = "$\downarrow$"; string l = "$\leftarrow$"; string r = "$\rightarrow$"; // input should be n x n string[][] input = {{u,b,b,r},{b,r,u,d},{d,b,u,b},{u,r,b,l}}; int n = input.length; // draw table for (int i=0; i<=n; ++i) { draw((i,0)--(i,n)); draw((0,i)--(n,i)); } // fill table for (int i=1; i<=n; ++i) { for (int j=1; j<=n; ++j) { label(input[i-1][j-1], (j-0.5,n-i+0.5)); } } [/asy] Did you spot it? It's in the bottom left, rotated by $90^\circ$ clockwise. To make the game as interesting as possible, Mellon Game Lab would like the grid to be as large as possible and for no $2\times 2$ grid to appear more than once in the big grid. The grid above doesn't work, as the following $2\times 2$ grid appears twice, once in the top left corner (rotated $90^\circ$ counterclockwise) and once directly below it (overlapping). [asy] size(2cm); defaultpen(fontsize(16pt)); string b = ""; string u = "$\uparrow$"; string d = "$\downarrow$"; string l = "$\leftarrow$"; string r = "$\rightarrow$"; // input should be n x n string[][] input = {{b,r},{d,b}}; int n = input.length; // draw table for (int i=0; i<=n; ++i) { draw((i,0)--(i,n)); draw((0,i)--(n,i)); } // fill table for (int i=1; i<=n; ++i) { for (int j=1; j<=n; ++j) { label(input[i-1][j-1], (j-0.5,n-i+0.5)); } } [/asy] Let's call a grid that avoids such repeats a [i]repeat-free grid[/i]. We are interested in finding out for which $n$ constructing an $n\times n$ repeat-free grid is possible. Here's what we know so far. [list] [*] Any $2\times 2$ grid is repeat-free, as there is only one subgrid to worry about, and there can't possibly be any repeats. [*] If we can construct an $n\times n$ repeat-free grid, we can also construct a $k\times k$ repeat-free grid for any $k\leq n$ by just taking the top left $k\times k$ of the original one we found. [*] By the previous observation, if it is impossible to construct such an $n\times n$ repeat-free grid, we cannot construct a $k\times k$ repeat-free grid for any $k\geq n$, as otherwise we could take the top left $n\times n$ to get one working for $n$. [/list] These three observations together tell us that either we can construct an $n\times n$ repeat-free grid for all $n\geq 2$, or there exists some upper limit $N\geq 2$ such that we can construct an $n\times n$ repeat-free grid for all $n\leq N$ but cannot construct one for any $n> N$. Your goal is to determine if such an $N$ exists, and if so, place bounds on its value. More precisely, this problem consists of two parts: a lower bound and an upper bound. For the lower bound, to show that $N\geq n$ for some $n$, you need to construct an $n\times n$ repeat-free grid (you do not need to prove your construction works). For the upper bound, to show that $N$ is at most some value $n$, you must prove that it is impossible to construct an $(n+1)\times (n+1)$ repeat-free grid. [i]Proposed by Connor Gordon and Eric Oh[/i]

Find the least positive integer $n$ so that both $n$ and $n+1$ have prime factorizations with exactly four (not necessarily distinct) prime factors.

Prove or disprove the following statement: If $F$ is a finite set with two or more elements, then there exists a binary operation $*$ on $F$ such that for all $x,y,z$ in $F$, $(\text i)$ $x*z=y*z$ implies $x=y$ $(\text{ii})$ $x*(y*z)\ne(x*y)*z$

Let $ABCD$ be a rectangle of centre $O$, such that $\angle DAC=60^{\circ}$. The angle bisector of $\angle DAC$ meets $DC$ at $S$. Lines $OS$ and $AD$ meet at $L$, and lines $BL$ and $AC$ meet at $M$. Prove that lines $SM$ and $CL$ are parallel.

Let $k \ge 0$ and $a, b, c$ be three positive real numbers such that $$\frac{a}{b}+\frac{b}{c}+ \frac{c}{a}= (k + 1)^2 + \frac{2}{k+ 1}.$$ Prove that $$a^2 + b^2 + c^2 \le (k^2 + 1)(ab + bc + ca).$$

(a) If every point of the plane is painted one of three colors, do there necessarily exist two points of the same color exactly one inch apart? (b) What if "three'' is replaced by "nine''?

What is the largest integer that is a divisor of \[ (n\plus{}1)(n\plus{}3)(n\plus{}5)(n\plus{}7)(n\plus{}9) \]for all positive even integers $ n$? $ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 11 \qquad \textbf{(D)}\ 15 \qquad \textbf{(E)}\ 165$

Daniel, Clarence, and Matthew split a \$20.20 dinner bill so that Daniel pays half of what Clarence pays. If Daniel pays \$6.06, what is the ratio of Clarence's pay to Matthew's pay? [i]Proposed by Henry Ren[/i]