Suppose $a_1, a_2, ... , a_{100}$ are positive real numbers such that $$a_k =\frac{ka_{k-1}}{a_{k-1} - (k - 1)}$$ for $k = 2, 3, ... , 100$. Given that $a_{20} = a_{23}$, compute $a_{100}$.

If $AB$ and $CD$ are perpendicular diameters of circle $Q$, $P$ in $\overline{AQ}$, and $\measuredangle QPC = 60^\circ$, then the length of $PQ$ divided by the length of $AQ$ is [asy] size(200); defaultpen(linewidth(0.7)+fontsize(10)); pair A=(-1,0), B=(1,0), C=(0,1), D=(0,-1), Q=origin, P=(-0.5,0); draw(P--C--D^^A--B^^Circle(Q,1)); label("$A$", A, W); label("$B$", B, E); label("$C$", C, N); label("$D$", D, S); label("$P$", P, S); label("$Q$", Q, SE); label("$60^\circ$", P+0.0.5*dir(30), dir(30));[/asy] $ \textbf{(A)} \ \frac{\sqrt{3}}{2} \qquad \textbf{(B)} \ \frac{\sqrt{3}}{3} \qquad \textbf{(C)} \ \frac{\sqrt{2}}{2} \qquad \textbf{(D)} \ \frac12 \qquad \textbf{(E)} \ \frac23 $

For $a_1,..., a_5 \in R$, $$\frac{a_1}{k^2 + 1}+ ... +\frac{a_5}{k^2 + 5}=\frac{1}{k^2}$$ for all $k \in \{2, 3, 4, 5, 6\}$. Calculate $$\frac{a_1}{2}+... +\frac{a_5}{6}.$$

Find the locus of points on the surface of a cube that serve as the vertex of the smallest angle that subtends the diagonal.

Several policemen try to catch a thief who has $2m$ accomplices. To that end they place the accomplices under surveillance. In the beginning, the policemen shadow nobody. Every morning each policeman places under his surveillance one of the accomplices. Every evening the thief stops trusting one of his accomplices The thief is caught if by the $m$-th evening some policeman shadows exactly those $m$ accomplices who are still trusted by the thief. Prove that to guarantee the capture of the thief at least $2^m$ policemen are needed.

Each third-grade classroom at Pearl Creek Elementary has $18$ students and $2$ pet rabbits. How many more students than rabbits are there in all $4$ of the third-grade classrooms? ${{ \textbf{(A)}\ 48\qquad\textbf{(B)}\ 56\qquad\textbf{(C)}\ 64\qquad\textbf{(D)}\ 72}\qquad\textbf{(E)}\ 80} $

Let us call a continuous function $ f : [a,b] \rightarrow \mathbb{R}^2 \;\textit{reducible}$ if it has a double arc (that is, if there are $ a \leq \alpha < \beta \leq \gamma < \delta \leq b$ such that there exists a strictly monotone and continuous $ h : [\alpha,\beta] \rightarrow [\gamma,\delta]$ for which $ f(t)\equal{}f(h(t))$ is satisfied for every $ \alpha \leq t \leq \beta$); otherwise $ f$ is irreducible. Construct irreducible $ f : [a,b] \rightarrow \mathbb{R}^2$ and $ g : [c,d] \rightarrow \mathbb{R}^2$ such that $ f([a,b])\equal{}g([c,d])$ and (a) both $ f$ and $ g$ are rectifiable but their lengths are different; (b) $ f$ is rectifiable but $ g$ is not. [i]A. Csaszar[/i]

given $p_1,p_2,...$ be a sequence of integer and $p_1=2$, for positive integer $n$, $p_{n+1}$ is the least prime factor of $np_1^{1!}p_2^{2!}...p_n^{n!}+1 $ prove that all primes appear in the sequence (Proposed by Beatmania)

Quadrilateral $ABCD$ is inscribed in a circle of radius $R$, and also circumscribed around a circle of radius $r$. It is known that $\angle ADB = 45^o$. Find the area of triangle $AIB$, where point $I$ is the center of the circle inscribed in $ABCD$. (Hryhoriy Filippovskyi)

Let $f (x)$ be a function mapping real numbers to real numbers. Given that $f (f (x)) =\frac{1}{3x}$, and $f (2) =\frac19$, find $ f\left(\frac{1}{6}\right)$. [i]Proposed by Zachary Perry[/i]

Is there a positive integer $ k$ such that none of the digits $ 3,4,5,6$ appear in the decimal representation of the number $ 2002!\cdot k$?

How many primes $p$ are there such that $39p + 1$ is a perfect square? $ \textbf{a)}\ 0 \qquad\textbf{b)}\ 1 \qquad\textbf{c)}\ 2 \qquad\textbf{d)}\ 3 \qquad\textbf{e)}\ \text{None of above} $

Suppose $x$ and $y$ are inversely proportional and positive. If $x$ increases by $p\%$, then $y$ decreases by $ \textbf{(A)}\ p\%\qquad\textbf{(B)}\ \frac{p}{1+p}\%\qquad\textbf{(C)}\ \frac{100}{p}\%\qquad\textbf{(D)}\ \frac{p}{100+p}\%\qquad\textbf{(E)}\ \frac{100p}{100+p}\%$

Let $(a_n)$ be a sequence of positive real numbers. Show that $$ \limsup_{n \to \infty} n \left(\frac{1 +a_{n+1}}{a_n } -1 \right) \geq 1$$ and prove that $1$ cannot be replaced by any larger number.

Let $n\ge5$ be an integer. A trapezoid with base lengths of $1$ and $r$ is tiled by $n$ (not necessarily congruent) equilateral triangles. In terms of $n$, find the maximum possible value of $r$. [i]Linus Tang[/i]

Consider a function $ f:\mathbb{R}\longrightarrow\mathbb{R} . $ Show that: [b]a)[/b] $ f $ is nondecreasing, if $ f+g $ is nondecreasing for any increasing function $ g:\mathbb{R}\longrightarrow\mathbb{R} . $ [b]b)[/b] $ f $ is nondecreasing, if $ f\cdot g $ is nondecreasing for any increasing function $ g:\mathbb{R}\longrightarrow\mathbb{R} . $ [i]Cristian Mangra[/i]

A point $H$ lies on the side $AB$ of regular polygon $ABCDE$. A circle with center $H$ and radius $HE$ meets the segments $DE$ and $CD$ at points $G$ and $F$ respectively. It is known that $DG=AH$. Prove that $CF=AH$.

Mr. Earl E. Bird leaves his house for work at exactly 8:00 A.M. every morning. When he averages $ 40$ miles per hour, he arrives at his workplace three minutes late. When he averages $ 60$ miles per hour, he arrives three minutes early. At what average speed, in miles per hour, should Mr. Bird drive to arrive at his workplace precisely on time? $ \textbf{(A)}\ 45 \qquad \textbf{(B)}\ 48 \qquad \textbf{(C)}\ 50 \qquad \textbf{(D)}\ 55 \qquad \textbf{(E)}\ 58$

Do there exist positive integers $a$ and $b$ such that $a^n+n^b$ and $b^n+n^a$ are relatively prime for all natural $n$?

Let $A=(a_{ij})$ be the $n\times n$ matrix, where $a_{ij}$ is the remainder of the division of $i^j+j^i$ by $3$ for $i,j=1,2,\ldots,n$. Find the greatest $n$ for which $\det A\ne0$.

Determine all real values of the parameter $a$ for which the equation \[16x^4 -ax^3 + (2a + 17)x^2 -ax + 16 = 0\] has exactly four distinct real roots that form a geometric progression.

Prove that the equation $4x^3-3x+1=2y^2$ has at least $31$ solutions in positive integers $x$ and $y$ with $x\leq 2005$.

Proce that number $$\underbrace{11...11}_{2n \,\, digits}-\underbrace{22 ... 22}_{n \,\, digits}$$ is, for every natural $n$, a perfect square.

Find all $\{a_n\}_{n\ge 0}$ that satisfies the following conditions. (1) $a_n\in \mathbb{Z}$ (2) $a_0=0, a_1=1$ (3) For infinitly many $m$, $a_m=m$ (4) For every $n\ge2$, $\{2a_i-a_{i-1} | i=1, 2, 3, \cdots , n\}\equiv \{0, 1, 2, \cdots , n-1\}$ $\mod n$

Suppose $n > 1$ is an odd integer satisfying $n \mid 2^{\frac{n-1}{2}} + 1$. Prove [color=red]or disprove[/color] that $n$ is prime. [i] Note: unfortunately, the original form of this problem did not include the red text, rendering it unsolvable. We sincerely apologize for this error and are taking concrete steps to prevent similar issues from reoccurring, including computer-verifying problems where possible. All teams will receive full credit for the question.[/i]