In the dog dictionary the words are any sequence of letters $A$ and $U$ for example $AA$, $UAU$ and $AUAU$. For each word, your "profundity" will be the quantity of subwords we can obtain by the removal of some letters. For each positive integer $n$, determine the largest "profundity" of word, in dog dictionary, can have with $n$ letters. Note: The word $AAUUA$ has "profundity" $14$ because your subwords are $A, U, AU, AA, UU, UA, AUU, UUA, AAU, AUA, AAA, AAUU, AAUA, AUUA$.

There is a test for the dangerous bifurcation virus that is $ 99\%$ accurate. In other words, if someone has the virus, there is a $ 99\%$ chance that the test will be positive, and if someone does not have it, then there is a $ 99\%$ chance the test will be negative. Assume that exactly $ 1\%$ of the general population has the virus. Given an individual that has tested positive from this test, what is the probability that he or she actually has the disease? Express your answer as a percentage.

How many even integers between 4000 and 7000 have four different digits?

$\boxed{A3}$The sequence $a_1,a_2,a_3,...$ is defined by $a_1=a_2=1,a_{2n+1}=2a_{2n}-a_n$ and $a_{2n+2}=2a_{2n+1}$ for $n\in{N}.$Prove that if $n>3$ and $n-3$ is divisible by $8$ then $a_n$ is divisible by $5$

Let $f(0, 0) = 5^{2003}, f(0, n) = 0$ for every integer $n \neq 0$ and \[\begin{array}{c}\ f(m, n) = f(m-1, n) - 2 \cdot \Bigg\lfloor \frac{f(m-1, n)}{2}\Bigg\rfloor + \Bigg\lfloor\frac{f(m-1, n-1)}{2}\Bigg\rfloor + \Bigg\lfloor\frac{f(m-1, n+1)}{2}\Bigg\rfloor \end{array}\] for every natural number $m > 0$ and for every integer $n$. Prove that there exists a positive integer $M$ such that $f(M, n) = 1$ for all integers $n$ such that $|n| \leq \frac{(5^{2003}-1)}{2}$ and $f(M, n) = 0$ for all integers n such that $|n| > \frac{5^{2003}-1}{2}.$

In a triangle $ABC$, with $AB\neq AC$ and $A\neq 60^{0},120^{0}$, $D$ is a point on line $AC$ different from $C$. Suppose that the circumcentres and orthocentres of triangles $ABC$ and $ABD$ lie on a circle. Prove that $\angle ABD=\angle ACB$.

Let $ABC$ be an acute-angled triangle with circumcircle $\omega$. Prove that there exists a point $J$ such that for any point $X$ inside $ABC$ if $AX,BX,CX$ intersect $\omega$ in $A_1,B_1,C_1$ and $A_2,B_2,C_2$ be reflections of $A_1,B_1,C_1$ in midpoints of $BC,AC,AB$ respectively then $A_2,B_2,C_2,J$ lie on a circle.

Let $N$ be a positive integer. Suppose given any real $x\in (0,1)$ with decimal representation $0.a_1a_2a_3a_4\cdots$, one can color the digits $a_1,a_2,\cdots$ with $N$ colors so that the following hold: 1. each color is used at least once; 2. for any color, if we delete all the digits in $x$ except those of this color, the resulting decimal number is rational. Find the least possible value of $N$. [i]~Sutanay Bhattacharya[/i]

In the acute-angled triangle $ABC$, $P$ is the midpoint of segment $BC$ and the point $K$ is the foot of the altitude from $A$. $D$ is a point on segment $AP$ such that $\angle BDC=90$. Let $(ADK) \cap BC=E$ and $(ABC) \cap AE=F$. Prove that $\angle AFD=90$.

Kevin is at the point $(19,12)$. He wants to walk to a point on the ellipse $9x^2 + 25y^2 = 8100$, and then walk to $(-24, 0)$. Find the shortest length that he has to walk. [i]Proposed by Kevin Zhao[/i]

In the $ xy$ plane, the line $ l$ touches to 2 parabolas $ y\equal{}x^2\plus{}ax,\ y\equal{}x^2\minus{}2ax$, where $ a$ is positive constant. (1) Find the equation of $ l$. (2) Find the area $ S$ bounded by the parabolas and the tangent line $ l$.

Consider parallelogram $ABCD$ and let $E$ be the midpoint of $BC$. In segment $DE$ a point $F$ is chosen such that $AF$ is perpendicular to $DE$. Prove that $\angle CDE=\angle EFB$.

The sequence $\{y_{n}\}_{n \ge 1}$ is defined by \[y_{1}=y_{2}=1,\;\; y_{n+2}= (4k-5)y_{n+1}-y_{n}+4-2k.\] Determine all integers $k$ such that each term of this sequence is a perfect square.

A farmer divides his herd of $n$ cows among his four sons so that one son gets one-half the herd, a second son, one-fourth, a third son, one-fifth, and the fourth son, 7 cows. Then $n$ is: $ \textbf{(A)}\ 80 \qquad\textbf{(B)}\ 100\qquad\textbf{(C)}\ 140\qquad\textbf{(D)}\ 180\qquad\textbf{(E)}\ 240 $

Let $ABC$ be a triangle with circumcenter $O$ such that $AC = 7$. Suppose that the circumcircle of $AOC$ is tangent to $BC$ at $C$ and intersects the line $AB$ at $A$ and $F$. Let $FO$ intersect $BC$ at $E$. Compute $BE$.

$ABCDE$ is a regular pentagon. Two circles $C_1$ and $C_2$ are drawn through $B$ with centers $A$ and $C$ respectively. Let the other intersection of $C_1$ and $C_2$ be $P$. The circle with center $P$ which passes through $E$ and $D$ intersects $C_2$ at $X$ and $AE$ at $Y$. Prove that $AX = AY$.

Abby and Brian play the following game: They first choose a positive integer $N$. Then they write numbers on a blackboard in turn. Abby starts by writing a $1$. Thereafter, when one of them has written the number $n$, the other writes down either $n + 1$ or $2n$, provided that the number is not greater than $N$. The player who writes $N$ on the blackboard wins. (a) Determine which player has a winning strategy if $N = 2011$. (b) Find the number of positive integers $N\leqslant2011$ for which Brian has a winning strategy. (This is based on ISL 2004, Problem C5.)

Let $H$ be the orthocenter of an acute angled triangle $ABC$. Circumcircle of the triangle $ABC$ and the circle of diameter $[AH]$ intersect at point $E$, different from $A$. Let $M$ be the midpoint of the small arc $BC$ of the circumcircle of the triangle $ABC$ and let $N$ the midpoint of the large arc $BC$ of the circumcircle of the triangle $BHC$ Prove that points $E, H, M, N$ are concyclic.

Which number is larger, $A$ or $B$, where $$A = \dfrac{1}{2015} (1 + \dfrac12 + \dfrac13 + \cdots + \dfrac{1}{2015})$$ and $$B = \dfrac{1}{2016} (1 + \dfrac12 + \dfrac13 + \cdots + \dfrac{1}{2016}) \text{ ?}$$ Prove your answer is correct.

For each positive integer $m$ and $n$ define function $f(m, n)$ by $f(1, 1) = 1$, $f(m+ 1, n) = f(m, n) +m$ and $f(m, n + 1) = f(m, n) - n$. Find the sum of all the values of $p$ such that $f(p, q) = 2004$ for some $q$.

If $ a$, $ b$, $ c$, $ d$, $ e$, $ f$ are nonnegative real numbers satisfying $ a \plus{} b \plus{} c \plus{} d \plus{} e \plus{} f \equal{} 1$ and $ ace \plus{} bdf \geq \frac {1}{108}$, then prove that \[ abc \plus{} bcd \plus{} cde \plus{} de f \plus{} efa \plus{} fab \leq \frac {1}{36} \]

In order to encourage the kids to straighten up their closets and the storage shed, Jerry offers his kids some extra spending money for their upcoming vacation. "I don't care what you do, I just want to see everything look clean and organized." While going through his closet, Joshua finds an old bag of marbles that are either blue or red. The ratio of blue to red marbles in the bag is $17:7$. Alexis also has some marbles of the same colors, but hasn't used them for anything in years. She decides to give Joshua her marbles to put in his marble bag so that all the marbles are in one place. Alexis has twice as many red marbles as blue marbles, and when the twins get all their marbles in one bag, there are exactly as many red marbles and blue marbles, and the total number of marbles is between $200$ and $250$. How many total marbles do the twins have together?

Let the coordinate planes have the reflection property. A ray falls onto one of them. How does the final direction of the ray after reflecting from all three coordinate planes depend on its initial direction?

In a plane are given a straight line $\ell$ and a point $P$ not located on $\ell$. Is there a circle in this plane such that there exist more than three different points $S$ on $\ell$ with the property that the perpendicular bisector of $PS$ is tangent to the circle ? Explain the answer.

Find all pairs of positive integers \((a,b)\) with the following property: there exists an integer \(N\) such that for any integers \(m\ge N\) and \(n\ge N\), every \(m\times n\) grid of unit squares may be partitioned into \(a\times b\) rectangles and fewer than \(ab\) unit squares. [i]Proposed by Holden Mui[/i]