There are $m \ge 2$ blue points and $n \ge 2$ red points in three-dimensional space, and no four points are coplanar. Geoff and Nazar take turns, picking one blue point and one red point and connecting the two with a straight-line segment. Assume that Geoff starts first and the one who first makes a cycle wins. Who has the winning strategy?

Let $n$ be a positive integer. On a number line, Azer is at point $0$ in his car which have fuel capacity of $2^n$ units and is initially full. At each positive integer $m$, there is a gas station. Azer only moves to the right with constant speed and doesn't stop anywhere except the gas stations. Each time his car moves to the right by some amount, its fuel decreases by the same amount. Azer may choose to stop at a gas station or pass it. There are thieves at some gas stations. (A station may have multiple thieves) If Azer stops at a station which have $k\ge 0$ thieves while its car have fuel capacity $d$, his cars new fuel capacity becomes $\frac{d}{2^k}$. After that, Azer fulls his cars tank and leaves the station. Find the minimum number of thieves needed to guarantee that Azer will eventually run out of fuel. Proposed by[i] Mehmet Can Baştemir[/i] and [i]Deniz Can Karaçelebi[/i]

Let $a$ and $b$ be positive integers. When $a^{2}+b^{2}$ is divided by $a+b,$ the quotient is $q$ and the remainder is $r.$ Find all pairs $(a,b)$ such that $q^{2}+r=1977$.

How many positive three-digit integers $\underline{a}\underline{b}\underline{c}$ can represent a valid date in $2013$, where either $a$ corresponds to a month and $\underline{b}\underline{c}$ corresponds to the day in that month, or $\underline{a}\underline{b}$ corresponds to a month and $c$ corresponds to the day? For example, 202 is a valid representation for February 2nd, and 121 could represent either January 21st or December 1st. (Note: During the actual test they had to write the number of days in each month so don't feel bad if you have to google that :P)

Given the points $O = (0, 0)$ and $A = (2024, -2024)$ in the plane. For any positive integer $n$, Damian draws all the points with integer coordinates $B_{i,j} = (i, j)$ with $0 \leq i, j \leq n$ and calculates the area of each triangle $OAB_{i,j}$. Let $S(n)$ denote the sum of the $(n+1)^2$ areas calculated above. Find the following limit: \[ \lim_{n \to \infty} \frac{S(n)}{n^3}. \]

Let $ ABCD$ be a trapezoid inscribed in a circle $ \mathcal{C}$ with $ AB\parallel CD$, $ AB\equal{}2CD$. Let $ \{Q\}\equal{}AD\cap BC$ and let $ P$ be the intersection of tangents to $ \mathcal{C}$ at $ B$ and $ D$. Calculate the area of the quadrilateral $ ABPQ$ in terms of the area of the triangle $ PDQ$.

Find, as a function of $\, n, \,$ the sum of the digits of \[ 9 \times 99 \times 9999 \times \cdots \times \left( 10^{2^n} - 1 \right), \] where each factor has twice as many digits as the previous one.

Prove that we can find an infinite set of positive integers of the from $2^n-3$ (where $n$ is a positive integer) every pair of which are relatively prime.

Prove that, for all convex pentagons $P_1 P_2 P_3 P_4 P_5$ with area 1, there are indices $i$ and $j$ (assume $P_7 = P_2$ and $P_6 = P_1$) such that: \[ \text{Area of} \ \triangle P_i P_{i+1} P_{i+2} \le \frac{5 - \sqrt 5}{10} \le \text{Area of} \ \triangle P_j P_{j+1} P_{j+2}\]

We call a set of $6$ points in the plane [i]splittable[/i] if we if can denote its elements by $A,B,C,D,E$ and $F$ in such a way that $\triangle ABC$ and $\triangle DEF$ have the same centroid. [list=a] [*]Construct a splittable set. [*]Show that any set of $7$ points has a subset of $6$ points which is [i]not[/i] splittable. [/list]

Let $n$ be a positive integer. Let $a$ be an integer such that $\gcd (a,n)=1$. Prove that \[\frac{a^{\phi (n)}-1}{n}=\sum_{i\in R}\frac{1}{ai}\left[\frac{ai}{n}\right]\pmod{n}\] where $R$ is the reduced residue system of $n$ with each element a positive integer at most $n$.

We are given a balance with two bowls and a number of weights. (a) Give an example of four integer weights using which one can measure any weight of $1,2,...,40$ grams. (b) Are there four weights using which one can measure any weight of $1,2,...,50$ grams?

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]