Which of the following is equal to the product \[ \frac {8}{4}\cdot\frac {12}{8}\cdot\frac {16}{12}\cdots\frac {4n \plus{} 4}{4n}\cdots\frac {2008}{2004}? \]$ \textbf{(A)}\ 251 \qquad \textbf{(B)}\ 502 \qquad \textbf{(C)}\ 1004 \qquad \textbf{(D)}\ 2008 \qquad \textbf{(E)}\ 4016$

Solve the following equation in integers: $4n^4+7n^2+3n+6=m^3$.

Let $\mathbb{R}$ be the set of real numbers. Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function such that \[f(x+y)f(x-y)\geqslant f(x)^2-f(y)^2\] for every $x,y\in\mathbb{R}$. Assume that the inequality is strict for some $x_0,y_0\in\mathbb{R}$. Prove that either $f(x)\geqslant 0$ for every $x\in\mathbb{R}$ or $f(x)\leqslant 0$ for every $x\in\mathbb{R}$.

A bus and a cyclist left town $A$ at $10$ o'clock in the same direction, and a motorcyclist left town $B$ to meet them $15$ minutes later. The bus drove past the pedestrian at $10$ o'clock $30$ minutes, met the motorcyclist at $11$ o'clock and arrived in the city of $B$ at $12$ o'clock. The motorcyclist met the cyclist $15$ minutes after meeting the bus and another $15$ minutes later caught up with the pedestrian. At what time did the cyclist and the pedestrian meet? (The speeds and directions of movement of all participants were equal, the pedestrian and the motorcyclist were moving in the direction of city $A$.)

Let $n$ be a positive integer number and let $a_1, a_2, \ldots, a_n$ be $n$ positive real numbers. Prove that $f : [0, \infty) \rightarrow \mathbb{R}$, defined by \[f(x) = \dfrac{a_1 + x}{a_2 + x} + \dfrac{a_2 + x}{a_3 + x} + \cdots + \dfrac{a_{n-1} + x}{a_n + x} + \dfrac{a_n + x}{a_1 + x}, \] is a decreasing function. [i]Dan Marinescu et al.[/i]

A wedding is going to be held in a city with $25$ types of meals, to which some of the $2017$ citizens will be invited. All of the citizens like some meals and each meal is liked by at least one person. A "$suitable$ $list$" is a set of citizens, such that each meal is liked by at least one person in the set. A "$kamber$ $group$" is a set that contains at least one person from each "$suitable$ $list$". Given a "$kamber$ $group$", which has no subset (other than itself) that is also a "$kamber$ $group$", prove that there exists a meal, which is liked by everyone in the group.

Find all pairs $(a,b)$ of non-negative integers such that: $$2017^a=b^6-32b+1$$ [i]proposed by Walther Janous[/i]

Given a triangle $ABC$ with $\angle A = 45^\circ$. Let $A'$ be the antipode of $A$ in the circumcircle of $ABC$. Points $E$ and $F$ on segments $AB$ and $AC$ respectively are such that $A'B = BE$, $A'C = CF$. Let $K$ be the second intersection of circumcircles of triangles $AEF$ and $ABC$. Prove that $EF$ bisects $A'K$.

Find the probability that any given row in Pascal’s Triangle contains a perfect square. [i] (.1 point)[/i]

Given a function $p(x) = ax^5 + bx^4 + cx^3 + dx^2 + ex + f$. Each coefficient $a, b, c, d, e$, and$ f$ is equal to either $ 1$ or $-1$. If $p(2) = 11$, what is the value of $p(3)$?

Two distinct real numbers are written on each vertex of a convex $2012-$gon. Show that we can remove a number from each vertex such that the remaining numbers on any two adjacent vertices are different.

A $2\times 2$ square was cut from a squared sheet of paper. Using only a ruler without divisions and without going beyond the square, divide the diagonal of the square into $6$ equal parts.

Let $ \left(x_{n}\right)$ be a real sequence satisfying $ x_{0}=0$, $ x_{2}=\sqrt[3]{2}x_{1}$, and $ x_{n+1}=\frac{1}{\sqrt[3]{4}}x_{n}+\sqrt[3]{4}x_{n-1}+\frac{1}{2}x_{n-2}$ for every integer $ n\geq 2$, and such that $ x_{3}$ is a positive integer. Find the minimal number of integers belonging to this sequence.

There is a grid of equilateral triangles with a distance 1 between any two neighboring grid points. An equilateral triangle with side length $n$ lies on the grid so that all of its vertices are grid points, and all of its sides match the grid. Now, let us decompose this equilateral triangle into $n^2$ smaller triangles (not necessarily equilateral triangles) so that the vertices of all these smaller triangles are all grid points, and all these small triangles have equal areas. Prove that there are at least $n$ equilateral triangles among these smaller triangles.

Find how many multiples of 360 are of the form $\overline{ab2017cd}$, where a, b, c, d are digits, with a > 0.

When $ \left ( 1 \minus{} \frac{1}{a} \right ) ^6$ is expanded the sum of the last three coefficients is: $ \textbf{(A)}\ 22 \qquad \textbf{(B)}\ 11 \qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ \minus{}10 \qquad \textbf{(E)}\ \minus{}11$

In triangle $ABC$, angle $A$ is twice as large as angle $B$. $AB = 3$ and $AC = 2$. Calculate $BC$.

Let $N$ be the product of the first nine multiples of $19$ (i.e. $N = 19\times38 \times57\times\cdots\times 152\times 171$). What is the last digit of $N$?

Find all functions $f: R \to R$ that satisfy $$f (x^2y) + 2f (y^2) =(x^2 + f (y)) \cdot f (y)$$ for all $x, y \in R$

In triangle $ABC$, $AB=3$, $AC=5$, and $BC=4$. Let $P$ be a point inside triangle $ABC$, and let $D$, $E$, and $F$ be the projections of $P$ onto sides $BC$, $AC$, and $AB$, respectively. If $PD:PE:PF=1:1:2$, then find the area of triangle $DEF$. (Express your answer as a reduced fraction.) (will insert image here later)

Adamand Topher are playing a game in which each of them starts with $2$ pickles. Each turn, they flip a fair coin: if it lands heads, Topher takes $1$ pickle from Adam; if it lands tails, Adam takes $2$ pickles from Topher. (If Topher has only $1$ pickle left, Adam will just take it.) What’s the probability that Topher will have all $4$ pickles before Adam does?

A circle can be circumscribed around the quadrilateral $ABCD$. Point $P$ is the foot of the perpendicular drawn from point $A$ on line $BC$, and respectively $Q$ from $A$ on $DC$, $R$ from $D$ on $AB$ and $T$ from $D$ on $BC$ . Prove that points $P,Q,R$ and $T$ lie on the same circle. (A. Myakishev)

Given a positive integer $n$, an allowed move is to form $2n+1$ or $3n+2$. The set $S_{n}$ is the set of all numbers that can be obtained by a sequence of allowed moves starting with $n$. For example, we can form $5 \rightarrow 11 \rightarrow 35$ so $5, 11$ and $35$ belong to $S_{5}$. We call $m$ and $n$ compatible if $S_{m}$ and $S_{n}$ has a common element. Which members of $\{1, 2, 3, ... , 2002\}$ are compatible with $2003$?

For a given triangle ABC, let X be a variable point on the line BC such that the point C lies between the points B and X. Prove that the radical axis of the incircles of the triangles ABX and ACX passes through a point independent of X. This is a slight extension of the [url=http://www.mathlinks.ro/Forum/viewtopic.php?t=41033]IMO Shortlist 2004 geometry problem 7[/url] and can be found, together with the proposed solution, among the files uploaded at http://www.mathlinks.ro/Forum/viewtopic.php?t=15622 . Note that the problem was proposed by Russia. I could not find the names of the authors, but I have two particular persons under suspicion. Maybe somebody could shade some light on this... Darij

Sohom constructs a square $BERK$ of side length $10$. Darlnim adds points $T$, $O$, $W$, and $N$, which are the midpoints of $\overline{BE}$, $\overline{ER}$, $\overline{RK}$, and $\overline{KB}$, respectively. Lastly, Sylvia constructs square $CALI$ whose edges contain the vertices of $BERK$, such that $\overline{CA}$ is parallel to $\overline{BO}$. Compute the area of $CALI$. [img]https://cdn.artofproblemsolving.com/attachments/0/9/0fda0c273bb73b85f3b1bc73661126630152b3.png[/img]