Find all positive integers $n$ for which $$3x^n + n(x + 2) - 3 \geq nx^2$$ holds for all real numbers $x.$

Let $P_1P_2\ldots P_n$ be an $n$-gon such that for all $i,j \in \{1,2,\ldots,n\}$ where $i \neq j$, there exists $k \neq i,j$ such that $\angle P_iP_kP_j = 60^\circ$. Prove that $n=3$.

Find the smallest positive integer $k$ for which there exists a colouring of the positive integers $\mathbb{Z}_{>0}$ with $k$ colours and a function $f:\mathbb{Z}_{>0}\to \mathbb{Z}_{>0}$ with the following two properties: $(i)$ For all positive integers $m,n$ of the same colour, $f(m+n)=f(m)+f(n).$ $(ii)$ There are positive integers $m,n$ such that $f(m+n)\ne f(m)+f(n).$ [i]In a colouring of $\mathbb{Z}_{>0}$ with $k$ colours, every integer is coloured in exactly one of the $k$ colours. In both $(i)$ and $(ii)$ the positive integers $m,n$ are not necessarily distinct.[/i]

Let $BC$ be a fixed chord of a circle $\omega$. Let $A$ be a variable point on the major arc $BC$ of $\omega$. Let $H$ be the orthocenter of $ABC$. The points $D, E$ lie on $AB, AC$ such that $H$ is the midpoint of $DE$. $O_A$ is the circumcenter of $ADE$. Prove that as $A$ varies, $O_A$ lies on a fixed circle.

A [i]short-sighted[/i] rook is a rook that beats all squares in the same column and in the same row for which he can not go more than $60$-steps. What is the maximal amount of short-sighted rooks that don't beat each other that can be put on a $100\times 100$ chessboard.

Let $ABC$ be an acute, scalene triangle, and let $P$ be an arbitrary point in its interior. Let $\mathcal{P}_A$ be the parabola with focus $P$ and directrix $BC$, and define $\mathcal{P}_B$ and $\mathcal{P}_C$ similarly. (a) Show that if $Q$ is an intersection point of $\mathcal{P}_B$ and $\mathcal{P}_C$, then $P$ and $Q$ are on the same side of $AB$, and $P$ and $Q$ are on the same side of $AC$. (b) You are given that $\mathcal{P}_B$ and $\mathcal{P}_C$ intersect at exactly two points. Let $\ell_A$ be the line between these points, and define $\ell_B$ and $\ell_C$ similarly. Show that $\ell_A$, $\ell_B$, and $\ell_C$ concur. [i]Note: A parabola with focus point $X$ and directrix line $\ell$ is the set of all points $Z$ that are the same distance from $X$ and $\ell$.[/i]

Let $\triangle ABC$ be a triangle with $\angle BAC>90^{\circ}$, $AB=5$ and $AC=7$. Points $D$ and $E$ lie on segment $BC$ such that $BD=DE=EC$. If $\angle BAC+\angle DAE=180^{\circ}$, compute $BC$.

There are $n\ge 2$ lamps, each with two states: $\textbf{on}$ or $\textbf{off}$. For each non-empty subset $A$ of the set of these lamps, there is a $\textit{soft-button}$ which operates on the lamps in $A$; that is, upon $\textit{operating}$ this button each of the lamps in $A$ changes its state(on to off and off to on). The buttons are identical and it is not known which button corresponds to which subset of lamps. Suppose all the lamps are off initially. Show that one can always switch all the lamps on by performing at most $2^{n-1}+1$ operations.

Suppose $A\in{M_2(\mathbb{C})}$ is not a scalar matrix. Let $S=\{B\in{M_2(\mathbb{C})}|\ AB=BA\}$. If $X,\ Y\in{S}$, then prove that $XY=YX$.

Let $p$ be an odd prime. Show that $\frac{1}{\pi}\cdot\cos^{-1}\left(\frac{1}{p}\right)$ is irrational. (Note: $\cos^{-1}\left(x\right)$ is defined to be the unique $y$ with $0\leq y\leq\pi$ such that $\cos\left(y\right)=x$.)

Find the least positive integer $n$ such that \[ \frac 1{\sin 45^\circ\sin 46^\circ}+\frac 1{\sin 47^\circ\sin 48^\circ}+\cdots+\frac 1{\sin 133^\circ\sin 134^\circ}=\frac 1{\sin n^\circ}. \]

Let $n$ be a positive integer and consider an arrangement of $2n$ blocks in a straight line, where $n$ of them are red and the rest blue. A swap refers to choosing two consecutive blocks and then swapping their positions. Let $A$ be the minimum number of swaps needed to make the first $n$ blocks all red and $B$ be the minimum number of swaps needed to make the first $n$ blocks all blue. Show that $A+B$ is independent of the starting arrangement and determine its value.

Let $f :\mathbb R \to\mathbb R$ a function $ n \geq 2$ times differentiable so that: $ \lim_{x \to \infty} f(x) = l \in \mathbb R$ and $ \lim_{x \to \infty} f^{(n)}(x) = 0$. Prove that: $ \lim_{x \to \infty} f^{(k)}(x) = 0 $ for all $ k \in \{1, 2, \dots, n - 1\} $, where $f^{(k)}$ is the $ k $ - th derivative of $f$.

Prove that if a lattice parallellogram contains an odd number of lattice points, then its centroid.

We call S $(n)$ the sum of the digits of the integer $n$. For example, $S (327)=3+2+7=12$. Find the value of $$A=S(1)-S(2)+S(3)-S(4)+...+S(2011)-S(2012).$$ ($A$ has $2012$ terms).

A point $(a,b)$ in the plane is called [i]sparkling[/i] if it also lies on the line $ax+by=1$. Find the maximum possible distance between two sparkling points. [i]Proposed by Evan Chen[/i]

Let $n \geq 3$ be a positive integer. Find the maximum number of diagonals in a regular $n$-gon one can select, so that any two of them do not intersect in the interior or they are perpendicular to each other.

Let $A_{10}$ denote the answer to problem $10$. Two circles lie in the plane; denote the lengths of the internal and external tangents between these two circles by $x$ and $y$, respectively. Given that the product of the radii of these two circles is $15/2$, and that the distance between their centers is $A_{10}$, determine $y^2-x^2$.

Find the least positive integer $ n$ for which $ \frac{n\minus{}13}{5n\plus{}6}$ is non-zero reducible fraction. $ \textbf{(A)}\ 45 \qquad \textbf{(B)}\ 68 \qquad \textbf{(C)}\ 155 \qquad \textbf{(D)}\ 226 \qquad \textbf{(E)}\ \text{none of these}$

Determine the smallest possible value of the expression $$\frac{ab+1}{a+b}+\frac{bc+1}{b+c}+\frac{ca+1}{c+a}$$ where $a,b,c \in \mathbb{R}$ satisfy $a+b+c = -1$ and $abc \leqslant -3$

Find the unique integer $\overline{CA7DB}$ with nonzero digits so that $\overline{ABCD} \cdot 3 = \overline{CA7DB}.$

[i]Reduced name[/i] of a natural number $A$ with $n$ digits ($n \ge 2$) a number of $n-1$ digits obtained by deleting one of the digits of $A$: For example, the [i]reduced names[/i] of $1024$ is $124$, $104$ and $120$. Determine how many seven-digit numbers cannot be written as the sum of one natural numbers $A$ and a [i]reduced name[/i] of $A$.

Prove that if the function $ ax^2 + bx + c $ takes an integer value for every integer value of the variable $ x $, then $ 2a $, $ a + b $, $ c $ are integers and vice versa.

Suppose that $S\subseteq \mathbb Z$ has the following property: if $a,b\in S$, then $a+b\in S$. Further, we know that $S$ has at least one negative element and one positive element. Is the following statement true? There exists an integer $d$ such that for every $x\in \mathbb Z$, $x\in S$ if and only if $d|x$. [i]proposed by Mahyar Sefidgaran[/i]

[asy] draw(unitsquare);draw((0,0)--(.25,sqrt(3)/4)--(.5,0)); label("Z",(0,1),NW);label("Y",(1,1),NE);label("A",(0,0),SW);label("X",(1,0),SE);label("B",(.5,0),S);label("P",(.25,sqrt(3)/4),N); //Credit to Zimbalono for the diagram[/asy] Equilateral triangle $ABP$ (see figure) with side $AB$ of length $2$ inches is placed inside square $AXYZ$ with side of length $4$ inches so that $B$ is on side $AX$. The triangle is rotated clockwise about $B$, then $P$, and so on along the sides of the square until $P$ returns to its original position. The length of the path in inches traversed by vertex $P$ is equal to $\textbf{(A) }20\pi/3\qquad\textbf{(B) }32\pi/3\qquad\textbf{(C) }12\pi\qquad\textbf{(D) }40\pi/3\qquad \textbf{(E) }15\pi$