Determine all integers $1 \le m, 1 \le n \le 2009$, for which \begin{align*} \prod_{i=1}^n \left( i^3 +1 \right) = m^2 \end{align*}

A straight river that is $264$ meters wide flows from west to east at a rate of $14$ meters per minute. Melanie and Sherry sit on the south bank of the river with Melanie a distance of $D$ meters downstream from Sherry. Relative to the water, Melanie swims at $80$ meters per minute, and Sherry swims at $60$ meters per minute. At the same time, Melanie and Sherry begin swimming in straight lines to a point on the north bank of the river that is equidistant from their starting positions. The two women arrive at this point simultaneously. Find $D$.

$11$ consecutive integers sum to $1331$. What is the largest of the $11$ integers?

If $x$ and $y$ are real numbers with $(x+y)^4=x-y$, what is the maximum possible value of $y$?

Define the sequence $(x_n)$ by $x_0 = 0$ and for all $n \in \mathbb N,$ \[x_n=\begin{cases} x_{n-1} + (3^r - 1)/2,&\mbox{ if } n = 3^{r-1}(3k + 1);\\ x_{n-1} - (3^r + 1)/2, & \mbox{ if } n = 3^{r-1}(3k + 2).\end{cases}\] where $k \in \mathbb N_0, r \in \mathbb N$. Prove that every integer occurs in this sequence exactly once.

Given triangle $ABC$. A circle passes through $B$ and $C$ and intersects sides $AB$ and $AC$ at points $C_1$ and $B_1$ respectively. The line $B_1C_1$ intersects the circle $\omega$, which is the circumcircle of $ABC$, at points $X$ and $Y$. Lines $BB_1$ and $CC_1$ intersect $\omega$ at points $P$ and $Q$ respectively ($P \neq B$ and $Q \neq C$). Prove that $QX=PY$.

$x=\frac{1}{\log_{\frac{1}{2}} \frac{1}{3}}+\frac{1}{\log_{\frac{1}{5}} \frac{1}{3}}$, then $\text{(A)}x\in(-2,-1)\qquad\text{(B)}x\in(1,2)\qquad\text{(C)}x\in(-3,-2)\qquad\text{(D)}x\in(2,3)$

Alice has an integer $N > 1$ on the blackboard. Each minute, she deletes the current number $x$ on the blackboard and writes $2x+1$ if $x$ is not the cube of an integer, or the cube root of $x$ otherwise. Prove that at some point of time, she writes a number larger than $10^{100}$. [i]Proposed by Anant Mudgal and Rohan Goyal[/i]

The squares $OABC$ and $OA_1B_1C_1$ are situated in the same plane and are directly oriented. Prove that the lines $AA_1$ , $BB_1$, and $CC_1$ are concurrent.

Find the maximum value of $\lambda ,$ such that for $\forall x,y\in\mathbb R_+$ satisfying $2x-y=2x^3+y^3,x^2+\lambda y^2\leqslant 1.$

Let $n$ be an integer greater than $1$ and let $X$ be an $n$-element set. A non-empty collection of subsets $A_1, ..., A_k$ of $X$ is tight if the union $A_1 \cup \cdots \cup A_k$ is a proper subset of $X$ and no element of $X$ lies in exactly one of the $A_i$s. Find the largest cardinality of a collection of proper non-empty subsets of $X$, no non-empty subcollection of which is tight. [i]Note[/i]. A subset $A$ of $X$ is proper if $A\neq X$. The sets in a collection are assumed to be distinct. The whole collection is assumed to be a subcollection.

We would like to give a present to one of $100$ children. We do this by throwing a biased coin $k$ times, after predetermining who wins in each possible outcome of this lottery. Prove that we can choose the probability $p$ of throwing heads, and the value of $k$ such that, by distributing the $2^k$ different outcomes between the children in the right way, we can guarantee that each child has the same probability of winning.

Given a parallelogram $ABCD$, in which the angle $\angle ABC$ is obtuse. Line $AD$ intersects the circle a second time $\omega$ circumscribed around triangle $ABC$, at the point $E$. Line $CD$ intersects second time circle $\omega$ at point $F$. Prove that the circumcenter of triangle $DEF$ lies on the circle $\omega$.

Find the least number $ c$ satisfyng the condition $\sum_{i=1}^n {x_i}^2\leq cn$ and all real numbers $x_1,x_2,...,x_n$ are greater than or equal to $-1$ such that $\sum_{i=1}^n {x_i}^3=0$

Consider all the ways of writing exactly ten times each of the numbers $0, 1, 2, \ldots , 9$ in the squares of a $10 \times 10$ board. Find the greatest integer $n$ with the property that there is always a row or a column with $n$ different numbers.

How many positive integers less than 2015 have exactly 9 positive integer divisors?

A sphere is inscribed in the tetrahedron whose vertices are $A=(6,0,0), B=(0,4,0), C=(0,0,2),$ and $D=(0,0,0).$ The radius of the sphere is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

Given is a triangle $\triangle{ABC}$,with $AB<AC<BC$,inscribed in circle $c(O,R)$.Let $D,E,Z$ be the midpoints of $BC,CA,AB$ respectively,and $K$ the foot of the altitude from $A$.At the exterior of $\triangle{ABC}$ and with the sides $AB,AC$ as diameters,we construct the semicircles $c_1,c_2$ respectively.Suppose that $P\equiv DZ\cap c_1 \ , \ S\equiv KZ\cap c_1$ and $R\equiv DE\cap c_2 \ , \ T\equiv KE\cap c_2$.Finally,let $M$ be the intersection of the lines $PS,RT$. [b]i.[/b] Prove that the lines $PR,ST$ intersect at $A$. [b]ii.[/b] Prove that the lines $PR\cap MD$ intersect on $c$. [asy]import graph; size(8cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -4.8592569519241255, xmax = 12.331775417316715, ymin = -3.1864435704043403, ymax = 6.540061585876658; /* image dimensions */ pen aqaqaq = rgb(0.6274509803921569,0.6274509803921569,0.6274509803921569); pen uququq = rgb(0.25098039215686274,0.25098039215686274,0.25098039215686274); draw((0.6699432366054657,3.2576036755978928)--(0.,0.)--(5.,0.)--cycle, aqaqaq); /* draw figures */ draw((0.6699432366054657,3.2576036755978928)--(0.,0.), uququq); draw((0.,0.)--(5.,0.), uququq); draw((5.,0.)--(0.6699432366054657,3.2576036755978928), uququq); draw(shift((0.33497161830273287,1.6288018377989464))*xscale(1.662889476749906)*yscale(1.662889476749906)*arc((0,0),1,78.3788505217281,258.3788505217281)); draw(shift((2.834971618302733,1.6288018377989464))*xscale(2.7093067970187343)*yscale(2.7093067970187343)*arc((0,0),1,-36.95500560847834,143.0449943915217)); draw((0.6699432366054657,3.2576036755978928)--(0.6699432366054657,0.)); draw((-0.9938564482532047,2.628510486065423)--(2.5,0.)); draw((0.6699432366054657,0.)--(0.,3.2576036755978923)); draw((0.6699432366054657,0.)--(5.,3.257603675597893)); draw((2.5,0.)--(3.3807330143335355,4.282570444700163)); draw((-0.9938564482532047,2.628510486065423)--(2.5,4.8400585427926455)); draw((2.5,4.8400585427926455)--(5.,3.257603675597893)); draw((-0.9938564482532047,2.628510486065423)--(3.3807330143335355,4.282570444700163), linewidth(1.2) + linetype("2 2")); draw((0.,3.2576036755978923)--(5.,3.257603675597893), linewidth(1.2) + linetype("2 2")); draw(circle((2.5,1.18355242571055), 2.766007292905304), linewidth(0.4) + linetype("2 2")); draw((2.5,4.8400585427926455)--(2.5,0.), linewidth(1.2) + linetype("2 2")); /* dots and labels */ dot((0.6699432366054657,3.2576036755978928),linewidth(3.pt) + dotstyle); label("$A$", (0.7472169504504719,2.65), NE * labelscalefactor); dot((0.,0.),linewidth(3.pt) + dotstyle); label("$B$", (-0.2,-0.4), NE * labelscalefactor); dot((5.,0.),linewidth(3.pt) + dotstyle); label("$C$", (5.028818057451246,-0.34281415594345044), NE * labelscalefactor); dot((2.5,0.),linewidth(3.pt) + dotstyle); label("$D$", (2.4275434226319077,-0.32665717063401356), NE * labelscalefactor); dot((2.834971618302733,1.6288018377989464),linewidth(3.pt) + dotstyle); label("$E$", (3.073822835009383,1.5637101105701008), NE * labelscalefactor); dot((0.33497161830273287,1.6288018377989464),linewidth(3.pt) + dotstyle); label("$Z$", (0.003995626216375389,1.402140257475732), NE * labelscalefactor); dot((0.6699432366054657,0.),linewidth(3.pt) + dotstyle); label("$K$", (0.6179610679749769,-0.3105001853245767), NE * labelscalefactor); dot((-0.9938564482532047,2.628510486065423),linewidth(3.pt) + dotstyle); label("$P$", (-1.0785223895158957,2.7916409940873033), NE * labelscalefactor); dot((0.,3.2576036755978923),linewidth(3.pt) + dotstyle); label("$S$", (-0.14141724156855653,3.454077391774215), NE * labelscalefactor); dot((5.,3.257603675597893),linewidth(3.pt) + dotstyle); label("$T$", (5.061132028070119,3.3571354799175936), NE * labelscalefactor); dot((3.3807330143335355,4.282570444700163),linewidth(3.pt) + dotstyle); label("$R$", (3.445433497126431,4.375025554412117), NE * labelscalefactor); dot((2.5,4.8400585427926455),linewidth(3.pt) + dotstyle); label("$M$", (2.5567993051074027,4.940520040242407), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */[/asy]

Let $a_1,a_2,\dots, a_n$ be positive integers with product $P,$ where $n$ is an odd positive integer. Prove that $$\gcd(a_1^n+P,a_2^n+P,\dots, a_n^n+P)\le 2\gcd(a_1,\dots, a_n)^n.$$ [i]Proposed by Daniel Liu[/i]

Let $ABC$ be an acute-angled triangle with $AC > BC$, with incircle $\tau$ centered at $I$ which touches $BC$ and $AC$ at points $D$ and $E$, respectively. The point $M$ on $\tau$ is such that $BM \parallel DE$ and $M$ and $B$ lie on the same halfplane with respect to the angle bisector of $\angle ACB$. Let $F$ and $H$ be the intersections of $\tau$ with $BM$ and $CM$ different from $M$, respectively. Let $J$ be a point on the line $AC$ such that $JM \parallel EH$. Let $K$ be the intersection of $JF$ and $\tau$ different from $F$. Prove that $ME \parallel KH$.

$A,B,C,D,E$ are points on a circle $O$ with radius equal to $r$. Chords $AB$ and $DE$ are parallel to each other and have length equal to $x$. Diagonals $AC,AD,BE, CE$ are drawn. If segment $XY$ on $O$ meets $AC$ at $X$ and $EC$ at $Y$ , prove that lines $BX$ and $DY$ meet at $Z$ on the circle.

We have a deck of $n$ playing cards, some of which are turned up and some are turned down. In each step we are allowed to take a set of several cards from the top, turn the set and place it back on the top of the deck. What is the smallest number of steps necessary to make all cards in the deck turned down, independent of the initial configuration?

Prove that, for an arbitrary positive integer $n \in Z_{>0}$, the number $n^2- n + 1$ does not have any prime factors of the form $6k + 5$, for $k \in Z_{>0}$.

Find all functions $f:(0,\infty)\rightarrow (0,\infty)$ such that for any $x,y\in (0,\infty)$, $$xf(x^2)f(f(y)) + f(yf(x)) = f(xy) \left(f(f(x^2)) + f(f(y^2))\right).$$

How many ways are there to fill the $2 \times 2$ grid below with $0$'s and $1$'s such that no row or column has duplicate entries? [asy] size(2cm); draw(unitsquare); draw( (0.5,0)--(0.5,1) ); draw( (0,0.5)--(1,0.5) ); [/asy]