Consider a circle $O_1$ with radius $R$ and a point $A$ outside the circle. It is known that $\angle BAC=60^\circ$, where $AB$ and $AC$ are tangent to $O_1$. We construct infinitely many circles $O_i$ $(i=1,2,\dots\>)$ such that for $i>1$, $O_i$ is tangent to $O_{i-1}$ and $O_{i+1}$, that they share the same tangent lines $AB$ and $AC$ with respect to $A$, and that none of the $O_i$ are larger than $O_1$. Find the total area of these circles. I know this problem was easy, but it still appeared in the TST, and so I posted it. It was kind of a disappointment for me.

Does there exist an positive integer $n$, so that for any positive integer $m<1002$, there exists an integer $k$ so that \[\displaystyle \frac{m}{1002} < \frac{k}{n} < \frac {m+1}{1003}\] holds? If $n$ does not exist, prove it; if $n$ exists, determine the minimum value of it. I know this problem was easy, but it still appeared on our TST, and so I posted it here.

More than three quarters of the circumference of a circle is colored black. Prove that there exists a rectangle such that all of its vertices are black. Actually the result holds if "three quarters" is replaced by "one half"...

The absolute value of every number in the sequence $\{a_n\}$ is smaller than 2005, and \[a_{n+6}=a_{n+4}+a_{n+2}-a_n.\] holds for all positive integers n. Prove that $\{a_n\}$ is periodic. Incredibly, this was probably the most difficult problem of our independent study problems in the 1st TST (excluding the final exam).

Prove that there exists infinitely many positive integers $n$ such that $n, n+1$, and $n+2$ can be written as the sum of two perfect squares.

Let $ABCD$ be a convex quadrilateral. Is it possible to find a point $P$ such that the segments drawn between $P$ and the midpoints of the sides of $ABCD$ divide the quadrilateral into four sections of equal area? If $P$ exists, is it unique?

Let $f(x)=Ax^2+Bx+C$, $g(x)=ax^2+bx+c$ be two quadratic polynomial functions with real coefficients that satisfy the relation \[|f(x)| \ge |g(x)|\] for all real $x$. Prove that $|b^2-4ac| \le |B^2-4AC|.$ My solution was nearly complete...

$P$ is a point in the interior of $\triangle ABC$, and $\angle ABP = \angle PCB = 10^\circ$. (a) If $\angle PBC = 10^\circ$ and $\angle ACP = 20^\circ$, what is the value of $\angle BAP$? (b) If $\angle PBC = 20^\circ$ and $\angle ACP = 10^\circ$, what is the value of $\angle BAP$?

$n$ teams take part in a tournament, in which every two teams compete exactly once, and that no draws are possible. It is known that for any two teams, there exists another team which defeated both of the two teams. Find all $n$ for which this is possible.

If $a$, $b$ ,$c$ are three positive real numbers such that $ab+bc+ca = 1$, prove that \[ \sqrt[3]{ \frac{1}{a} + 6b} + \sqrt[3]{\frac{1}{b} + 6c} + \sqrt[3]{\frac{1}{c} + 6a } \leq \frac{1}{abc}. \]

Show that for any tetrahedron, the condition that opposite edges have the same length is equivalent to the condition that the segment drawn between the midpoints of any two opposite edges is perpendicular to the two edges.

Find all positive integer triples $(x,y,z)$ such that $x<y<z$, $\gcd (x,y)=6$, $\gcd (y,z)=10$, $\gcd (x,z)=8$, and lcm$(x,y,z)=2400$. Note that the problems of the TST are not arranged in difficulty (Problem 1 of day 1 was probably the most difficult!)