Given $r\in\mathbb{R}$. Alice and Bob plays the following game: An equation with blank is written on the blackboard as below: $$S=|\Box-\Box|+|\Box-\Box|+|\Box-\Box|$$ Every round, Alice choose a real number from $[0,1]$ (not necessary to be different from the numbers chosen before) and Bob fill it in an empty box. After 6 rounds, every blank is filled and $S$ is determined at the same time. If $S\ge r$ then Alice wins, otherwise Bob wins. Find all $r$ such that Alice can guarantee her victory.

Amy, Bill and Celine are friends with different ages. Exactly one of the following statements is true. \begin{align*}\text{I.}&\text{ Bill is the oldest.}\\ \text{II.}&\text{ Amy is not the oldest.}\\ \text{III.}&\text{ Celine is not the youngest.}\end{align*} Rank the friends from the oldest to the youngest. $\textbf{(A)}\ \text{Bill, Amy, Celine}\qquad \textbf{(B)}\ \text{Amy, Bill, Celine}\qquad \textbf{(C)}\ \text{Celine, Amy, Bill}\qquad \\ \textbf{(D)}\ \text{Celine, Bill, Amy}\qquad \textbf{(E)}\ \text{Amy, Celine, Bill}$

You are given a regular hexagon. We say that a square is inscribed in the hexagon if it can be drawn in the interior such that all the four vertices lie on the perimeter of the hexagon. [b](a)[/b] A line segment has its endpoints on opposite edges of the hexagon. Show that, it passes through the centre of the hexagon if and only if it divides the two edges in the same ratio. [b](b)[/b] Suppose, a square $ABCD$ is inscribed in the hexagon such that $A$ and $C$ are on the opposite sides of the hexagon. Prove that, centre of the square is same as that of the hexagon. [b](c)[/b] Suppose, the side of the hexagon is of length $1$. Then find the length of the side of the inscribed square whose one pair of opposite sides is parallel to a pair of opposite sides of the hexagon. [b](d)[/b] Show that, up to rotation, there is a unique way of inscribing a square in a regular hexagon.

Let \(ABCDEF\) be a convex cyclic hexagon such that quadrilateral \(ABDF\) is a square, and the incenter of \(\triangle ACE\) lines on \(\overline{BF}\). Diagonal \(CE\) intersects diagonals \(BD\) and \(DF\) at points \(P\) and \(Q\), respectively. Prove that the circumcircle of \(\triangle DPQ\) is tangent to \(\overline{BF}\). [i]Proposed by Elliott Liu[/i]

Find the smallest positive integer $n$ such that the number $2^n + 2^8 + 2^{11}$ is a perfect square. (A): $8$, (B): $9$, (C): $11$, (D): $12$, (E) None of the above.

(a) Prove that $\{n\sqrt3\} >\frac{1}{n\sqrt3}$ for any positive integer $n$. (b) Is there a constant $c > 1$ such that $\{n\sqrt3\} >\frac{c}{n\sqrt3}$ for all $n \in N$?

Nancy shuffles a deck of $52$ cards and spreads the cards out in a circle face up, leaving one spot empty. Andy, who is in another room and does not see the cards, names a card. If this card is adjacent to the empty spot, Nancy moves the card to the empty spot, without telling Andy; otherwise nothing happens. Then Andy names another card and so on, as many times as he likes, until he says "stop." [list][b](a)[/b] Can Andy guarantee that after he says "stop," no card is in its initial spot? [b](b)[/b] Can Andy guarantee that after he says "stop," the Queen of Spades is not adjacent to the empty spot?[/list]

Consider a graph $G$ with $n$ vertices and at least $n^2/10$ edges. Suppose that each edge is colored in one of $c$ colors such that no two incident edges have the same color. Assume further that no cycles of size $10$ have the same set of colors. Prove that there is a constant $k$ such that $c$ is at least $kn^\frac{8}{5}$ for any $n$. [i]David Yang.[/i]

Define a number to be $boring$ if all the digits of the number are the same. How many positive integers less than $10000$ are both prime and boring?

Let $P(x)$ be a monic polynomial of degree $3$. Suppose that $P(x)$ has remainder $R(x)$ when it is divided by $(x - 1)(x - 4)$ and $2R(x)$ when it is divided by $(x - 2)(x - 3)$. Given that $P(0) = 5$, find $P(5)$.

A [b]palindrome[/b] is a whole number that reads the same forwards and backwards. If one neglects the colon, certain times displayed on a digital watch are palindromes. Three examples are: $ \boxed{1:01} $, $ \boxed{12:21} $. How many times during a 12-hour period will be palindromes? $ \text{(A)}\ 57\qquad\text{(B)}\ 60\qquad\text{(C)}\ 63\qquad\text{(D)}\ 90\qquad\text{(E)}\ 93 $

Let $a_n$ denote the angle opposite to the side of length $4n^2$ units in an integer right angled triangle with lengths of sides of the triangle being $4n^2, 4n^4+1$ and $4n^4-1$ where $n \in N$. Then find the value of $\lim_{p \to \infty} \sum_{n=1}^p a_n$ (A) $\pi/2$ (B) $\pi/4$ (C) $\pi $ (D) $\pi/3$

The polygon ABCDEFG (shown on the right) is a regular octagon. Prove that the area of the rectangle $ADEH$ is one half the area of the whole polygon $ABCDEFGH$. [asy] draw((0,1.414)--(1.414,0)--(3.414,0)--(4.828,1.414)--(4.828,3.414)--(3.414,4.828)--(1.414,4.828)--(0,3.414)--(0,1.414)); fill((0,1.414)--(0,3.414)--(4.828,3.414)--(4.828,1.414)--cycle, mediumgrey); label("$B$",(1.414,0),SW); label("$C$",(3.414,0),SE); label("$D$",(4.828,1.414),SE); label("$E$",(4.828,3.414),NE); label("$F$",(3.414,4.828),NE); label("$G$",(1.414,4.828),NW); label("$H$",(0,3.414),NW); label("$A$",(0,1.414),SW); [/asy]

Let $ABC$ be a triangle. $D$ is the midpoint of $AB$, $E$ is a point on the side $BC$ such that $BE = 2 EC$ and $\angle ADC = \angle BAE$. Find $\angle BAC$.

Mary divides a circle into $12$ sectors. The central angles of these sectors, measured in degrees, are all integers and they form an arithmetic sequence. What is the degree measure of the smallest possible sector angle? $ \textbf{(A)}\ 5\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 12 $

A non-isosceles trapezoid $ABCD$ ($AB // CD$) is given. An arbitrary circle passing through points $A$ and $B$ intersects the sides of the trapezoid at points $P$ and $Q$, and the intersect the diagonals at points $M$ and $N$. Prove that the lines $PQ, MN$ and $CD$ are concurrent.

If $ p$ is a positive integer, then $ \frac {3p \plus{} 25}{2p \minus{} 5}$ can be a positive integer, if and only if $ p$ is: $ \textbf{(A)}\ \text{at least }3 \qquad\textbf{(B)}\ \text{at least }3\text{ and no more than }35 \qquad\textbf{(C)}\ \text{no more than }35$ $ \textbf{(D)}\ \text{equal to }35 \qquad\textbf{(E)}\ \text{equal to }3\text{ or }35$

Prove that the following inequality holds for all positive real numbers $a$, $b$, $c$, $d$, $e$ and $f$ \[\sqrt[3]{\frac{abc}{a+b+d}}+\sqrt[3]{\frac{def}{c+e+f}} < \sqrt[3]{(a+b+d)(c+e+f)} \text{.}\] [i]Proposed by Dimitar Trenevski.[/i]

Suppose $A\subseteq \{0,1,\dots,29\}$. It satisfies that for any integer $k$ and any two members $a,b\in A$($a,b$ is allowed to be same), $a+b+30k$ is always not the product of two consecutive integers. Please find $A$ with largest possible cardinality.

Find the number of solutions in positive integers of the equation $\lfloor \frac{x}{2} \rfloor = \lfloor \frac{x}{11} \rfloor +1$ where $\lfloor A\rfloor$ denotes the integer part of the number $A$, e.g. $\lfloor 2.031\rfloor = 2$, $\lfloor 2\rfloor = 2$, etc.

Find all the triangles such that its side lenghts, area and its angles' measures (in degrees) are rational.

Point L is marked on side BC of triangle ABC so that AL is twice as long as the median CM. Given that angle ALC is equal to 45 degrees prove that AL is perpendicular to CM.

A plane is cut into unit squares, each of which is colored in black or white. It is known that each rectangle 3 x 4 or 4 x 3 contains exactly 8 white squares. In how many ways can this plane be colored?

Let $ S\equal{}\{1,2,\ldots,n\}$ be a set, where $ n\geq 6$ is an integer. Prove that $ S$ is the reunion of 3 pairwise disjoint subsets, with the same number of elements and the same sum of their elements, if and only if $ n$ is a multiple of 3.

Find all functions $f$ that map the set of real numbers into the set of real numbers, satisfying the following conditions: 1) $|f(x)|\ge 1$, 2) $f(x+y)=\frac{f(x)+f(y)}{1+f(x)f(y)}$ of all real values of $x $ and $y$.