Let $O, I$ be the circumcenter and incenter of triangle $ABC$. The incircle of $\triangle ABC$ touches $BC, CA, AB$ at points $D, E, F$ repsectively. $FD$ meets $CA$ at $P$, $ED$ meets $AB$ at $Q$. $M$ and $N$ are midpoints of $PE$ and $QF$ respectively. Show that $OI \perp MN$.

What is the last digit of $2021^{2021}$? [i]Proposed by Yifan Kang[/i]

There is a white table with a pile of $2008$ coins and there are two empty black tables. At each move, the uppermost coin on a table is transferred to an empty table or to the top of the pile on a non-empty table. What is the least number of moves required to reverse the pile at the beginning on the white table? $ \textbf{(A)}\ 6016 \qquad\textbf{(B)}\ 6017 \qquad\textbf{(C)}\ 6022 \qquad\textbf{(D)}\ 6023 \qquad\textbf{(E)}\ 6024 $

Let $U$ be the center of the circle inscribed in the triangle $ABC$, $O_1$, $O_2$ and $O_3$ the centers of the circles circumscribed by the triangles $BCU$, $CAU$ and $ABU$ respectively. Prove that the circles circumscribed around the triangles $ABC$ and $O_1O_2O_3$ have the same center.

Determine a necessary and sufficient condition for the affixes of three complex numbers $z_1$ , $z_2$ and $z_3$ are the vertices of an equilateral triangle.

If $a$ , $b$ are integers and $s=a^3+b^3-60ab(a+b)\geq 2012$ , find the least possible value of $s$.

Let $n\geq 2$ is positive integer. Find the best constant $C(n)$ such that \[\sum_{i=1}^{n}x_{i}\geq C(n)\sum_{1\leq j<i\leq n}(2x_{i}x_{j}+\sqrt{x_{i}x_{j}})\] is true for all real numbers $x_{i}\in(0,1),i=1,...,n$ for which $(1-x_{i})(1-x_{j})\geq\frac{1}{4},1\leq j<i \leq n.$

The equation $x-\frac{7}{x-3}=3-\frac{7}{x-3}$ has: $ \textbf{(A)}\ \text{infinitely many integral roots} \qquad\textbf{(B)}\ \text{no root} \qquad\textbf{(C)}\ \text{one integral root}\qquad$ $\textbf{(D)}\ \text{two equal integral roots} \qquad\textbf{(E)}\ \text{two equal non-integral roots} $

Let $ABC$ be an acute triangle and let $M$ be the midpoint of $AC$. A circle $\omega$ passing through $B$ and $M$ meets the sides $AB$ and $BC$ at points $P$ and $Q$ respectively. Let $T$ be the point such that $BPTQ$ is a parallelogram. Suppose that $T$ lies on the circumcircle of $ABC$. Determine all possible values of $\frac{BT}{BM}$.

In $\vartriangle ABC$ points $D, E$, and $F$ lie on side $\overline{BC}$ such that $\overline{AD}$ is an angle bisector of $\angle BAC$, $\overline{AE}$ is a median, and $\overline{AF}$ is an altitude. Given that $AB = 154$ and $AC = 128$, and $9 \times DE = EF,$ find the side length $BC$.

There are pairs of square numbers with the following two properties: (1) Their decimal representations have the same number of digits, with the first digit starting is different from $0$ . (2) If one appends the second to the decimal representation of the first, the decimal representation results another square number. Example: $16$ and $81$; $1681 = 41^2$. Prove that there are infinitely many pairs of squares with these properties.

Can an asymptotically stable equilibrium position become unstable in the Lyapunov sense under linearization?

Aumann, Bill, and Charlie each roll a fair $6$-sided die with sides labeled $1$ through $6$ and look at their individual rolls. Each flips a fair coin and, depending on the outcome, looks at the roll of either the player to his right or the player to his left, without anyone else knowing which die he observed. Then, at the same time, each of the three players states the expected value of the sum of the rolls based on the information he has. After hearing what everyone said, the three players again state the expected value of the sum of the rolls based on the information they have. Then, for the third time, after hearing what everyone said, the three players again state the expected value of the sum of the rolls based on the information they have. Prove that Aumann, Bill, and Charlie say the same number the third time.

Let $P_1$, $P_2$ and $P_3$ be polynomials of degree two with positive coefficient leader and real roots . Prove that if each pair of polynomials has a common root , then the polynomial $P_1 + P_2 + P_3$ has also real roots.

A uniform circular ring of radius $R$ is fixed in place. A particle is placed on the axis of the ring at a distance much greater than $R$ and allowed to fall towards the ring under the influence of the ring’s gravity. The particle achieves a maximum speed $v$. The ring is replaced with one of the same (linear) mass density but radius $2R$, and the experiment is repeated. What is the new maximum speed of the particle? (a) $\frac{1}{2}v$ (b) $\frac{1}{\sqrt{2}}v$ (c) $v$ (d) $\sqrt{2}v$ (e) $2v$

Suppose $A=\{1,2,\dots,2002\}$ and $M=\{1001,2003,3005\}$. $B$ is an non-empty subset of $A$. $B$ is called a $M$-free set if the sum of any two numbers in $B$ does not belong to $M$. If $A=A_1\cup A_2$, $A_1\cap A_2=\emptyset$ and $A_1,A_2$ are $M$-free sets, we call the ordered pair $(A_1,A_2)$ a $M$-partition of $A$. Find the number of $M$-partitions of $A$.