Let $\frac{1}{1-x-x^2-x^3} =\sum^{\infty}_{i=0} a_nx^n$, for what positive integers $n$ does $a_{n-1} = n^2$?

The audience shuffles a deck of $36$ cards, containing $9$ cards in each of the suits spades, hearts, diamonds and clubs. A magician predicts the suit of the cards, one at a time, starting with the uppermost one in the face-down deck. The design on the back of each card is an arrow. An assistant examines the deck without changing the order of the cards, and points the arrow on the back each card either towards or away from the magician, according to some system agreed upon in advance with the magician. Is there such a system which enables the magician to guarantee the correct prediction of the suit of at least (a) $19$ cards; (b) $20$ cards?

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 $R$.