Given $3n$ cards, each of them will be written with a number from the following sequence: $$2, 3, ..., n, n + 1, n + 3, n + 4, ..., 2n + 1, 2n + 2, 2n + 4, ..., 3n + 3$$ with each number used exactly once. Then every card is arranged from left to right in random order. Determine the probability such that for every $i$ with $1\le i \le 3n$, the number written on the $i$-th card, counted from the left, is greater than or equal to $i$.

Let $ ABCD$ be an isosceles trapezium. Determine the geometric location of all points $ P$ such that \[ |PA| \cdot |PC| \equal{} |PB| \cdot |PD|.\]

Triangle $ABC$ is isosceles with $AB=AC$. The bisectors of angles $ABC$ and $ACB$ meet at $I$. If the measure of angle $CIA$ is $130^\circ$, compute the measure of angle $CAB$. [i]Proposed by Connor Gordon[/i]

Three real numbers $ a,b,c$ satisfy the inequality $ |ax^2\plus{}bx\plus{}c| \le 1$ for all $ x \in [\minus{}1,1]$. Prove that $ |cx^2\plus{}bx\plus{}a| \le 2$ for all $ x \in [\minus{}1,1]$.

What is the maximum value of the difference between the largest real root and the smallest real root of the equation system \[\begin{array}{rcl} ax^2 + bx+ c &=& 0 \\ bx^2 + cx+ a &=& 0 \\ cx^2 + ax+ b &=& 0 \end{array}\], where at least one of the reals $a,b,c$ is non-zero? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ \sqrt 2 \qquad\textbf{(D)}\ 3\sqrt 2 \qquad\textbf{(E)}\ \text{There is no upper-bound} $

Alice and Bob want to play a game. In the beginning of the game, they are teleported to two random position on a train, whose length is $ 1 $ km. This train is closed and dark. So they dont know where they are. Fortunately, both of them have iPhone 133, it displays some information: 1. Your facing direction 2. Your total walking distance 3. whether you are at the front of the train 4. whether you are at the end of the train Moreover, one may see the information of the other one. Once Alice and Bob meet, the game ends. Alice and Bob can only discuss their strategy before the game starts. Find the least value $ x $ so that they are guarantee to end the game with total walking distance $ \le x $ km.

Determine the smallest natural number $a\geq 2$ for which there exists a prime number $p$ and a natural number $b\geq 2$ such that \[\frac{a^p - a}{p}=b^2.\]

How many integral solutions are there of the equation $x^5 -31x+2015=0$ ? [list=1] [*] 2 [*] 4 [*] 1 [*] None of these [/list]

Let $AX,BY,CZ$ be three cevians concurrent at an interior point $D$ of a triangle $ABC$. Prove that if two of the quadrangles $DY AZ,DZBX,DXCY$ are circumscribable, so is the third.

In the sequence $ 2001, 2002, 2003, \ldots$, each term after the third is found by subtracting the previous term from the sum of the two terms that precede that term. For example, the fourth term is $ 2001 \plus{} 2002 \minus{} 2003 \equal{} 2000$. What is the $ 2004^\text{th}$ term in this sequence? $ \textbf{(A)} \minus{} \! 2004 \qquad \textbf{(B)} \minus{} \! 2 \qquad \textbf{(C)}\ 0 \qquad \textbf{(D)}\ 4003 \qquad \textbf{(E)}\ 6007$

Let $S$ be a set of 2006 numbers. We call a subset $T$ of $S$ [i]naughty[/i] if for any two arbitrary numbers $u$, $v$ (not neccesary distinct) in $T$, $u+v$ is [i]not[/i] in $T$. Prove that 1) If $S=\{1,2,\ldots,2006\}$ every naughty subset of $S$ has at most 1003 elements; 2) If $S$ is a set of 2006 arbitrary positive integers, there exists a naughty subset of $S$ which has 669 elements.

Find three consecutive natural numbers, each of which is divisible by the square of the sum of its digits. Prove that there are no five such numbers in a row.

Let $\{a_n \}_n$ be a sequence of real numbers such there there are countably infinite distinct subsequences converging to the same point. We call two subsequences distinct if they do not have a common term. Which of the following statements always holds: (A) $\{a_n \}_n$ is bounded (B) $\{a_n \}_n$ is unbounded (C) The set of convergent subsequence $\{a_n \}_n$ is countable (D) None of these

Suppose that $m, n, k$ are positive integers satisfying $$3mk=(m+3)^n+1.$$ Prove that $k$ is odd.

Each of two boxes contains both black and white marbles, and the total number of marbles in the two boxes is $25.$ One marble is taken out of each box randomly. The probability that both marbles are black is $27/50,$ and the probability that both marbles are white is $m/n,$ where $m$ and $n$ are relatively prime positive integers. What is $m+n?$

Let $ABC$ be a triangle with $\angle{C} = 90^{\circ}$, and let $H$ be the foot of the altitude from $C$. A point $D$ is chosen inside the triangle $CBH$ so that $CH$ bisects $AD$. Let $P$ be the intersection point of the lines $BD$ and $CH$. Let $\omega$ be the semicircle with diameter $BD$ that meets the segment $CB$ at an interior point. A line through $P$ is tangent to $\omega$ at $Q$. Prove that the lines $CQ$ and $AD$ meet on $\omega$.