Sketch the graph of $x^3 + xy + y^3 = 3$.

Three brothers own a painting company called Tiga Abdul Enterprise. They are hired to paint a building. Wahab says, “I can paint this building in $3$ months if I work alone”. Wahib says, “I can paint this building in $2$ months if I work alone”. Wahub says, “I can paint this building in k months if I work alone”. If they work together, they can finish painting the building in $1$ month only. What is $k$?

For each integer $n\ge 1,$ compute the smallest possible value of \[\sum_{k=1}^{n}\left\lfloor\frac{a_k}{k}\right\rfloor\] over all permutations $(a_1,\dots,a_n)$ of $\{1,\dots,n\}.$ [i]Proposed by Shahjalal Shohag, Bangladesh[/i]

Find all real solutions to the following system of equations: \[\begin{cases} x+y+z+t=5\\xy+yz+zt+tx=4\\xyz+yzt+ztx+txy=3\\xyzt=-1\end{cases}\]

Two treasure-hunters found a treasure containing coins of value $a_1< a_2 < ... < a_{2003}$ (the quantity of coins of each value is unlimited). The first treasure-hunter forms all the possible sets of different coins containing odd number of elements, and takes the most valuable coin of each such set. The second treasure-hunter forms all the possible sets of different coins containing even number of elements, and takes the most valuable coin of each such set. Which one of them is going to have more money and how much more? (H. Nestra)

The number $N$ is the product of two primes. The sum of the positive divisors of $N$ that are less than $N$ is $2014$. Find $N$.

An hourglass is formed from two identical cones. Initially, the upper cone is filled with sand and the lower one is empty. The sand flows at a constant rate from the upper to the lower cone. It takes exactly one hour to empty the upper cone. How long does it take for the depth of sand in the lower cone to be half the depth of sand in the upper cone? (Assume that the sand stays level in both cones at all times.)

What is the value of \[\displaystyle\frac{1}{3}\cdot\displaystyle\frac{2}{4}\cdot\displaystyle\frac{3}{5}\cdots\displaystyle\frac{18}{20}\cdot\displaystyle\frac{19}{21}\cdot\displaystyle\frac{20}{22}?\] $\textbf{(A)} ~\displaystyle\frac{1}{462}\qquad\textbf{(B)} ~\displaystyle\frac{1}{231}\qquad\textbf{(C)} ~\displaystyle\frac{1}{132}\qquad\textbf{(D)} ~\displaystyle\frac{2}{213}\qquad\textbf{(E)} ~\displaystyle\frac{1}{22}\qquad$

Let $ABC$ be an acute triangle. Let $ H $ be the foot of perpendicular from $ A $ to $ BC $. $ D, E $ are the points on $ AB, AC $ and let $ F, G $ be the foot of perpendicular from $ D, E $ to $ BC $. Assume that $ DG \cap EF $ is on $ AH $. Let $ P $ be the foot of perpendicular from $ E $ to $ DH $. Prove that $ \angle APE = \angle CPE $.

A wire figure is held in different ways in a bundle of parallel light rays, so that different shadow figures are created in a plane perpendicular to the light rays. In this way one can form: (a) an isosceles triangle; (b) an isosceles triangle with altitude from the apex; (c) a rectangle containing an isosceles triangle; (d) a rhombus with one diagonal. The wire figure consists of eight straight pieces of iron wire, with each piece connected to both ends are attached to at least one other piece. Determine a figure corresponding to the above description is satisfactory, and indicate the direction of the light rays at which the shadow figures (a) to (d) arise. [hide=original wording]Men houdt een draadfiguur op verschillende manieren in een bundel evenwijdige lichtstralen, waardoor er in een vlak loodrecht op de lichtstralen verschillende schaduwfiguren ontstaan. Op deze wijze kan men vormen: (a) een gelijkbenige driehoek; (b) een gelijkbenige driehoek met hoogtelijn uit de top; (c) een rechthoek met daarin een gelijkbenige driehoek; (d) een ruit met één diagonaal. De draadfiguur bestaat uit acht rechte stukjes ijzerdraad, waarbij ieder stukje aan beide uiteinden aan tenminste één ander stukje vastzit. Bepaal een figuur die aan bovenstaande beschrijving voldoet, en geef de richting van de lichtstralen aan waarbij de schaduwfiguren (a) tot en met (d) ontstaan.[/hide]

Let $A_n$ be the number of arranged n-tuples of natural numbers $(a_1,a_2…a_n)$, such that $\frac{1}{a_1} +\frac{1}{a_2} +...+\frac{1}{a_n} =1$. Find the parity of $A_{68}$.

Jamal wants to store $ 30$ computer files on floppy disks, each of which has a capacity of $ 1.44$ megabytes (MB). Three of his files require $ 0.8$ MB of memory each, $ 12$ more require $ 0.7$ MB each, and the remaining $ 15$ require $ 0.4$ MB each. No file can be split between floppy disks. What is the minimal number of floppy disks that will hold all the files? $ \textbf{(A)}\ 12 \qquad \textbf{(B)}\ 13 \qquad \textbf{(C)}\ 14 \qquad \textbf{(D)}\ 15 \qquad \textbf{(E)}\ 16$

Given a tetrahedron $ T$. Valentin wants to find two its edges $ a,b$ with no common vertices so that $ T$ is covered by balls with diameters $ a,b$. Can he always find such a pair? [i]A. Zaslavsky[/i]

Bill divides a $28 \times 30$ rectangular board into two smaller rectangular boards with a single straightcut, so that the side lengths of both boards are positive whole numbers. How many different pairs of rectangular boards, up to congruence and arrangement, can Bill possibly obtain? (For instance, a cut that is $1$ unit away from either of the edges with length $28$ will result in the same pair of boards: either way, one would end up with a $1 \times 28$ board and a $29 \times 28$ board.)

Find $ [\sqrt{19992000}]$ where $ [x]$ is the greatest integer less than or equal to $ x$.

Let $a\geq 2$ be an integer. Prove that there exists a positive integer $b$ with the following property: For each positive integer $n$, there is a prime number $p$ (possibly depending on $a,b,n$) such that $a^n + b$ is divisible by $p$, but not divisible by $p^2$.

A flame test was performed to confirm the identity of a metal ion in solution. The result was a green flame. Which of the following metal ions is indicated? ${ \textbf{(A)}\ \text{copper} \qquad\textbf{(B)}\ \text{sodium} \qquad\textbf{(C)}\ \text{strontium} \qquad\textbf{(D)}\ \text{zinc} } $

The sequence of digits \[123456789101112131415161718192021\ldots\] is obtained by writing the positive integers in order. If the $10^n$th digit in this sequence occurs in the part of the sequence in which the $m$-digit numbers are placed, define $f(n)$ to be $m$. For example, $f(2) = 2$ because the $100^{\text{th}}$ digit enters the sequence in the placement of the two-digit integer $55$. Find the value of $f(2007)$.

In the accompanying figure , $y=f(x)$ is the graph of a one-to-one continuous function $f$ . At each point $P$ on the graph of $y=2x^2$ , assume that the areas $OAP$ and $OBP$ are equal . Here $PA,PB$ are the horizontal and vertical segments . Determine the function $f$. [asy] Label f; xaxis(0,60,blue); yaxis(0,60,blue); real f(real x) { return (x^2)/60; } draw(graph(f,0,53),red); label("$y=x^2$",(30,15),E); real f(real x) { return (x^2)/25; } draw(graph(f,0,38),red); label("$y=2x^2$",(37,37^2/25),E); real f(real x) { return (x^2)/10; } draw(graph(f,0,25),red); label("$y=f(x)$",(24,576/10),W); label("$O(0,0)$",(0,0),S); dot((20,400/25)); dot((20,400/60)); label("$P$",(20,400/25),E); label("$B$",(20,400/60),SE); dot(((4000/25)^(0.5),400/25)); label("$A$",((4000/25)^(0.5),400/25),W); draw((20,400/25)..((4000/25)^(0.5),400/25)); draw((20,400/25)..(20,400/60)); [/asy]

Call a convex polyhedron a [i]footballoid [/i] if it has the following properties. (1) Any face is either a regular pentagon or a regular hexagon. (2) All neighbours of a pentagonal face are hexagonal (a [i]neighbour [/i] of a face is a face that has a common edge with it). Find all possibilities for the number of pentagonal and hexagonal faces of a footballoid.

The vertices of a convex $n$-gon are colored so that adjacent vertices have different colors. Prove that if $n$ is odd, then the polygon can be divided into triangles with non-intersecting diagonals such that no diagonal has its endpoints the same color.

Which of the following operations has the same effect on a number as multiplying by $\dfrac{3}{4}$ and then dividing by $\dfrac{3}{5}$? $\text{(A)}\ \text{dividing by }\dfrac{4}{3} \qquad \text{(B)}\ \text{dividing by }\dfrac{9}{20} \qquad \text{(C)}\ \text{multiplying by }\dfrac{9}{20} \qquad \text{(D)}\ \text{dividing by }\dfrac{5}{4} \qquad \text{(E)}\ \text{multiplying by }\dfrac{5}{4}$

Find all non-negative integer solutions of the equation \[ 2^x\plus{}2009\equal{}3^y5^z.\]

Find all primes $p,q$ such that $p$ divides $30q-1$ and $q$ divides $30p-1$.

In triangle $ABC$, $AB=12$, $AC=7$, and $BC=10$. If sides $AB$ and $AC$ are doubled while $BC$ remains the same, then: $\textbf{(A)}\ \text{The area is doubled} \qquad\\ \textbf{(B)}\ \text{The altitude is doubled} \qquad\\ \textbf{(C)}\ \text{The area is four times the original area} \qquad\\ \textbf{(D)}\ \text{The median is unchanged} \qquad\\ \textbf{(E)}\ \text{The area of the triangle is 0}$