The center $O$ of the circle $\omega$ passing through the vertex $C$ of the isosceles triangle $ABC$ ($AB = AC$) is the interior point of the triangle $ABC$. This circle intersects segments $BC$ and $AC$ at points $D \ne C$ and $E \ne C$, respectively, and the circumscribed circle $\Omega$ of the triangle $AEO$ at the point $F \ne E$. Prove that the center of the circumcircle of the triangle $BDF$ lies on the circle $\Omega$.

Evaluate $ \int_{ \minus{} \pi}^{\pi} \frac {\sin nx}{(1 \plus{} 2009^x)\sin x}\ dx\ (n\equal{}0,\ 1,\ 2,\ \cdots)$.

Simplify the fraction: $\frac{(1^4+4)\cdot (5^4+4)\cdot (9^4+4)\cdot ... (69^4+4)\cdot(73^4+4)}{(3^4+4)\cdot (7^4+4)\cdot (11^4+4)\cdot ... (71^4+4)\cdot(75^4+4)}$.

A Dyck $n$-path is a lattice path of $n$ upsteps $(1, 1)$ and $n$ downsteps $(1, -1)$ that starts at the origin $O$ and never dips below the $x$-axis. A return is a maximal sequence of contiguous downsteps that terminates on the $x$-axis. For example, the Dyck $5$-path illustrated has two returns, of length $3$ and $1$ respectively. Show that there is a one-to-one correspondence between the Dyck $n$-paths with no return of even length and the Dyck $(n - 1)$ paths. \[\begin{picture}(165,70) \put(-5,0){O} \put(0,10){\line(1,0){150}} \put(0,10){\line(1,1){30}} \put(30,40){\line(1,-1){15}} \put(45,25){\line(1,1){30}} \put(75,55){\line(1,-1){45}} \put(120,10){\line(1,1){15}} \put(135,25){\line(1,-1){15}} \put(0,10){\circle{1}}\put(0,10){\circle{2}}\put(0,10){\circle{3}}\put(0,10){\circle{4}} \put(15,25){\circle{1}}\put(15,25){\circle{2}}\put(15,25){\circle{3}}\put(15,25){\circle{4}} \put(30,40){\circle{1}}\put(30,40){\circle{2}}\put(30,40){\circle{3}}\put(30,40){\circle{4}} \put(45,25){\circle{1}}\put(45,25){\circle{2}}\put(45,25){\circle{3}}\put(45,25){\circle{4}} \put(60,40){\circle{1}}\put(60,40){\circle{2}}\put(60,40){\circle{3}}\put(60,40){\circle{4}} \put(75,55){\circle{1}}\put(75,55){\circle{2}}\put(75,55){\circle{3}}\put(75,55){\circle{4}} \put(90,40){\circle{1}}\put(90,40){\circle{2}}\put(90,40){\circle{3}}\put(90,40){\circle{4}} \put(105,25){\circle{1}}\put(105,25){\circle{2}}\put(105,25){\circle{3}}\put(105,25){\circle{4}} \put(120,10){\circle{1}}\put(120,10){\circle{2}}\put(120,10){\circle{3}}\put(120,10){\circle{4}} \put(135,25){\circle{1}}\put(135,25){\circle{2}}\put(135,25){\circle{3}}\put(135,25){\circle{4}} \put(150,10){\circle{1}}\put(150,10){\circle{2}}\put(150,10){\circle{3}}\put(150,10){\circle{4}} \end{picture}\]

Let $\triangle MBT$ be a triangle with $\overline{MB} = 4$ and $\overline{MT} = 7$. Furthermore, let circle $\omega$ be a circle with center $O$ which is tangent to $\overline{MB}$ at $B$ and $\overline{MT}$ at some point on segment $\overline{MT}$. Given $\overline{OM} = 6$ and $\omega$ intersects $ \overline{BT}$ at $I \neq B$, find the length of $\overline{TI}$. [i]Proposed by Chad Yu[/i]

Let $ \alpha $ be a plane and let $ ABC $ be an equilateral triangle situated on a parallel plane whose distance from $ \alpha $ is $ h. $ Find the locus of the points $ M\in\alpha $ for which $$ \left|MA\right| ^2 +h^2 = \left|MB\right|^2 +\left|MC\right|^2. $$

Circles $K_1$ and $K_2$ are externally tangent to each other at $A$ and are internally tangent to a circle $K$ at $A_1$ and $A_2$ respectively. The common tangent to $K_1$ and $K_2$ at $A$ meets $K$ at point $P$. Line $PA_1$ meets $K_1$ again at $B_1$ and $PA_2$ meets $K_2$ again at $B_2$. Show that $B_1B_2$ is a common tangent of $K_1$ and $K_2$.

Let $ABC$ be a triangle with $AB=8$, $BC=7$, and $AC=11$. Let $\Gamma_1$ and $\Gamma_2$ be the two possible circles that are tangent to $AB$, $AC$, and $BC$ when $AC$ and $BC$ are extended, with $\Gamma_1$ having the smaller radius. $\Gamma_1$ and $\Gamma_2$ are tangent to $AB$ to $D$ and $E$, respectively, and $CE$ intersects the perpendicular bisector of $AB$ at a point $F$. What is $\tfrac{CF}{FD}$?

A polynomial $P$ has integer coefficients and $P(3)=4$ and $P(4)=3$. For how many $x$ we might have $P(x)=x$?

Let $\vartriangle ABC$ have $AB = 15$, $AC = 20$, and $BC = 21$. Suppose $\omega$ is a circle passing through $A$ that is tangent to segment $BC$. Let point $D\ne A$ be the second intersection of AB with $\omega$, and let point $E \ne A$ be the second intersection of $AC$ with $\omega$. Suppose $DE$ is parallel to $BC$. If $DE = \frac{a}{b}$ , where $a$, $b$ are relatively prime positive integers, find $a + b$.

Let there be an acute triangle $ABC$ with orthocenter $H$. Let $BH, CH$ hit the circumcircle of $\triangle ABC$ at $D, E$. Let $P$ be a point on $AB$, between $B$ and the foot of the perpendicular from $C$ to $AB$. Let $PH \cap AC = Q$. Now $\triangle AEP$'s circumcircle hits $CH$ at $S$, $\triangle ADQ$'s circumcircle hits $BH$ at $R$, and $\triangle AEP$'s circumcircle hits $\triangle ADQ$'s circumcircle at $J (\not=A)$. Prove that $RS$ is the perpendicular bisector of $HJ$.

Within the real numbers, solve the equation $$x^5 + \lfloor x \rfloor = 20$$ where $\lfloor x \rfloor$ denotes the largest whole number less than or equal to $x$.

Let $ABCDEFGHIJKL$ be a regular dodecagon. Find $\frac{AB}{AF} + \frac{AF}{AB}$.

For $0<k<2$, consider two curves $C_1: y=\sin 2x\ (0\leq x\leq \pi),\ C_2: y=k\cos x\ (0\leqq x\leqq \pi).$ Denote by $S(k)$ the sum of the areas of four parts enclosed by $C_1,\ C_2$ and two lines $x=0,\ x=\pi$. Find the minimum value of $S(k).$ [i]2010 Nagoya Institute of Technology entrance exam[/i]

Let $u,v,w$ be positive real numbers. Prove that there exists a cyclic permutation $(x,y,z)$ of $(u,v,w)$ such that for all positive real numbers $a,b,c$ the following holds: \[ \frac{a}{x\,a + y\,b + z\,c} \;+\; \frac{b}{x\,b + y\,c + z\,a} \;+\; \frac{c}{x\,c + y\,a + z\,b} \;\ge\; \frac{3}{x + y + z}. \]

Point $K$ is marked on the diagonal $AC$ in rectangle $ABCD$ so that $CK = BC$. On the side $BC$, point $M$ is marked so that $KM = CM$. Prove that $AK + BM = CM$.

Let $\{a_n\}_{n=0}^{\infty}$ be a sequence of integers numbers. Let $\Delta^1a_n=a_{n+1}-a_n$ for a non-negative integer $n$. Define $\Delta^Ma_n= \Delta^{M-1}a_{n+1}- \Delta^{M-1}a_n$. A sequence is [i]patriota[/i] if there are positive integers $k,l$ such that $a_{n+k}=\Delta^Ma_{n+l}$ for all non-negative integers $n$. Determine, with proof, whether exists a sequence that the last value of $M$ for which the sequence is [i]patriota[/i] is $2022$.

Prove that there exist infinitely many positive integers $n$ such that the number $\frac{1^2+2^2+\cdots+n^2}{n}$ is a perfect square. Obviously, $1$ is the least integer having this property. Find the next two least integers having this property.

Given an acute triangle $ABC$ with altitudes $AD$ and $BE$ intersecting at $H$, $M$ is the midpoint of $AB$. A nine-point circle of $ABC$ intersects with a circumcircle of $ABH$ on $P$ and $Q$ where $P$ lays on the same side of $A$ (with respect to $CH$). Prove that $ED, PH, MQ$ are concurrent on circumcircle $ABC$

Let $A$ be an acute angle such that $\tan A = 2\cos A$. Find the number of positive integers $n$ less than or equal to $1000$ such that $\sec^n A + \tan^n A$ is a positive integer whose units digit is $9$.

Consider an $m\times n$ board where $m, n \ge 3$ are positive integers, divided into unit squares. Initially all the squares are white. What is the minimum number of squares that need to be painted red such that each $3\times 3$ square contains at least two red squares? Andrei Eckstein and Alexandru Mihalcu

Let ABC with orthocenter $H$ and circumcenter $O$ be an acute scalene triangle satisfying $AB = AM$ where $M$ is the midpoint of $BC$. Suppose $Q$ and $K$ are points on $(ABC)$ distinct from A satisfying $\angle AQH = 90$ and $\angle BAK = \angle CAM$. Let $N$ be the midpoint of $AH$. • Let $I$ be the intersection of $B\text{-midline}$ and $A\text{-altitude}$ Prove that $IN = IO$. • Prove that there is point $P$ on the symmedian lying on circle with center $B$ and radius $BM$ such that $(APN)$ is tangent to $AB$. [i]Proposed by Krutarth Shah[/i]

Let $v$ be a vertex of a cube $C$ with edges of length $4$. Let $S$ be the largest sphere that can be inscribed in $C$. Let $R$ be the region consisting of all points $p$ between $S$ and $C$ such that $p$ is closer to $v$ than to any other vertex of the cube. Find the volume of $R$.

A line $d$ is called [i]faithful[/i] to triangle $ABC$ if $d$ be in plane of triangle $ABC$ and the reflections of $d$ over the sides of $ABC$ be concurrent. Prove that for any two triangles with acute angles lying in the same plane, either there exists exactly one [i]faithful[/i] line to both of them, or there exist infinitely [i]faithful[/i] lines to them.

Let $x_1,x_2,\ldots,x_n $ be real numbers, where $n\ge 2$ is a given integer, and let $\lfloor{x_1}\rfloor,\lfloor{x_2}\rfloor,\ldots,\lfloor{x_n}\rfloor $ be a permutation of $1,2,\ldots,n$. Find the maximum and minimum of $\sum\limits_{i=1}^{n-1}\lfloor{x_{i+1}-x_i}\rfloor$ (here $\lfloor x\rfloor $ is the largest integer not greater than $x$).