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]

In the interior of a square $ABCD$ we construct the equilateral triangles $ABK, BCL, CDM, DAN.$ Prove that the midpoints of the four segments $KL, LM, MN, NK$ and the midpoints of the eight segments $AK, BK, BL, CL, CM, DM, DN, AN$ are the 12 vertices of a regular dodecagon.

Let $AH_A, BH_B$ and $CH_C$ be altitudes in $\triangle ABC$. Let $p_A,p_B,p_C$ be the perpendicular lines from vertices $A,B,C$ to $H_BH_C, H_CH_A, H_AH_B$ respectively. Prove that $p_A,p_B,p_C$ are concurrent lines.

In rectangle $ABCD$, $AB=6$ and $BC=16$. Points $P, Q$ are chosen on the interior of side $AB$ such that $AP=PQ=QB$, and points $R, S$ are chosen on the interior of side $CD$ such that $CR=RS=SD$. Find the area of the region formed by the union of parallelograms $APCR$ and $QBSD$. [i]Proposed by Yannick Yao[/i]

In a group of 21 persons, every two person communicate with different radio frequency. It's possible for two person to not communicate (means there's no frequency occupied to connect them). Only one frequency used by each couple, and it's unique for every couple. In every 3 persons, exactly two of them is not communicating to each other. Determine the maximum number of frequency required for this group. Explain your answer.

Positive integers $x,y,z \le 100$ satisfy \begin{align*} 1099x+901y+1110z &= 59800 \\ 109x+991y+101z &= 44556 \end{align*} Compute $10000x+100y+z$. [i]Evan Chen[/i]

Permutation $\pi$ of $\{1,\dots,n\}$ is called [b]stable[/b] if the set $\{\pi (k)-k|k=1,\dots,n\}$ is consisted of exactly two different elements. Prove that the number of stable permutation of $\{1,\dots,n\}$ equals to $\sigma (n)-\tau (n)$ in which $\sigma (n)$ is the sum of positive divisors of $n$ and $\tau (n)$ is the number of positive divisors of $n$. Time allowed for this problem was 75 minutes.

Consider two concentric circles $\Omega$ and $\omega$. Chord $AD$ of the circle $\Omega$ is tangent to $\omega$. Inside the minor disk segment $AD$ of $\Omega$, an arbitrary point $P{}$ is selected. The tangent lines drawn from the point $P{}$ to the circle $\omega$ intersect the major arc $AD$ of the circle $\Omega$ at points $B{}$ and $C{}$. The line segments $BD$ and $AC$ intersect at the point $Q{}$. Prove that the line segment $PQ$ passes through the midpoint of line segment $AD$. [i]Note.[/i] A circle together with its interior is called a disk, and a chord $XY$ of the circle divides the disk into disk segments, a minor disk segment $XY$ (the one of smaller area) and a major disk segment $XY$.

Let $n$ be a positive integer. Determine the minimum number of lines that can be drawn on the plane so that they intersect in exactly $n$ distinct points.