Suppose that $a,b,c,d$ are positive real numbers satisfying $(a+c)(b+d)=ac+bd$. Find the smallest possible value of $$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}.$$ [i]Israel[/i]

The midpoint of the hypotenuse $AB$ of the right triangle $ABC$ is $K$. The point $M$ on the side $BC$ is taken such that $BM = 2 \cdot MC$. Prove that $\angle BAM = \angle CKM$.

Let $ P $ be a point in the interior of a triangle $ ABC $, and let $ D, E, F $ be the point of intersection of the line $ AP $ and the side $ BC $ of the triangle, of the line $ BP $ and the side $ CA $, and of the line $ CP $ and the side $ AB $, respectively. Prove that the area of the triangle $ ABC $ must be $ 6 $ if the area of each of the triangles $ PFA, PDB $ and $ PEC $ is $ 1 $.

A Haiku is a Japanese poem of seventeen syllables, in three lines of five, seven, and five. Ada has been told To write down five haikus plus Two more every hour. Such that she needs to Write down five in the first hour Seven, nine, so on. Ada has so far Forty haikus and writes down Seven every hour. At which hour after She begins will she not have Enough haikus done? [i]Proposed by Ada Tsui[/i]

Define the sequences $ a_n, b_n, c_n$ as follows. $ a_0 \equal{} k, b_0 \equal{} 4, c_0 \equal{} 1$. If $ a_n$ is even then $ a_{n \plus{} 1} \equal{} \frac {a_n}{2}$, $ b_{n \plus{} 1} \equal{} 2b_n$, $ c_{n \plus{} 1} \equal{} c_n$. If $ a_n$ is odd, then $ a_{n \plus{} 1} \equal{} a_n \minus{} \frac {b_n}{2} \minus{} c_n$, $ b_{n \plus{} 1} \equal{} b_n$, $ c_{n \plus{} 1} \equal{} b_n \plus{} c_n$. Find the number of positive integers $ k < 1995$ such that some $ a_n \equal{} 0$.

The figure below is a $ 4 \times 4$ grid of points. [asy]unitsize(15); for ( int x = 1; x <= 4; ++x ) for ( int y = 1; y <= 4; ++y ) dot((x, y));[/asy]Each pair of horizontally adjacent or vertically adjacent points are distance 1 apart. In the plane of this grid, how many circles of radius 1 pass through exactly two of these grid points?

If $ a_2 \neq 0$ and $ r$ and $ s$ are the roots of $ a_0 \plus{} a_1x \plus{} a_2x^2 \equal{} 0$, then the equality $ a_0 \plus{} a_1x \plus{} a_2x^2 \equal{} a_0\left (1 \minus{} \frac {x}{r} \right ) \left (1 \minus{} \frac {x}{s} \right )$ holds: $ \textbf{(A)}\ \text{for all values of }x, a_0\neq 0$ $ \textbf{(B)}\ \text{for all values of }x$ $ \textbf{(C)}\ \text{only when }x \equal{} 0$ $ \textbf{(D)}\ \text{only when }x \equal{} r \text{ or }x \equal{} s$ $ \textbf{(E)}\ \text{only when }x \equal{} r \text{ or }x \equal{} s, a_0 \neq 0$

Given an integer $n\geq 2$. There are $N$ distinct circle on the plane such that any two circles have two distinct intersections and no three circles have a common intersection. Initially there is a coin on each of the intersection points of the circles. Starting from $X$, players $X$ and $Y$ alternatively take away a coin, with the restriction that one cannot take away a coin lying on the same circle as the last coin just taken away by the opponent in the previous step. The one who cannot do so will lost. In particular, one loses where there is no coin left. For what values of $n$ does $Y$ have a winning strategy?

Find the smallest positive integer $a$ such that for some integers $b$, $c$ the polynomial $ax^2 - bx + c$ has two distinct zeros in the interval $(0,1)$.

Prove that there are no integers $m$ and $n$ such that \[19m^2+95mn+2000n^2=1995.\]

How many three-digit positive integers have an odd number of even digits? $\textbf{(A) }150\qquad\textbf{(B) }250\qquad\textbf{(C) }350\qquad\textbf{(D) }450\qquad\textbf{(E) }550$

On a circumference of a circle, seven points are selected, at which different positive integers are assigned to each of them. Then fit simultaneously, each number is replaced by the least common multiple of the two neighboring numbers to it. If the same number $n$ is obtained in each of the seven points, determine the smallest possible value for $n$. [hide=original wording]Sobre una circunferencia de un círculo, se seleccionan siete puntos, a los cuales se le asignan enteros positivos distintos a cada uno de ellos. Luego, en forma simultánea, cada número se reemplaza por el mínimo común múltiplo de los dos números vecinos a él. Si se obtiene el mismo número n en cada uno de los siete puntos, determine el menor valor posible para n.[/url]

Given are $50$ distinct sets of positive integers, each of size $30$, such that every $30$ of them have a common element. Prove that all of them have a common element.

In a $999 \times 999$ square table some cells are white and the remaining ones are red. Let $T$ be the number of triples $(C_1,C_2,C_3)$ of cells, the first two in the same row and the last two in the same column, with $C_1,C_3$ white and $C_2$ red. Find the maximum value $T$ can attain. [i]Proposed by Merlijn Staps, The Netherlands[/i]

Let $n \geq 3$ be an integer. John and Mary play the following game: First John labels the sides of a regular $n$-gon with the numbers $1, 2,\ldots, n$ in whatever order he wants, using each number exactly once. Then Mary divides this $n$-gon into triangles by drawing $n-3$ diagonals which do not intersect each other inside the $n$-gon. All these diagonals are labeled with number $1$. Into each of the triangles the product of the numbers on its sides is written. Let S be the sum of those $n - 2$ products. Determine the value of $S$ if Mary wants the number $S$ to be as small as possible and John wants $S$ to be as large as possible and if they both make the best possible choices.

Define $ x\otimes y \equal{} x^3 \minus{} y$. What is $ h\otimes (h\otimes h)$? $ \textbf{(A) } \minus{} h\qquad \textbf{(B) } 0\qquad \textbf{(C) } h\qquad \textbf{(D) } 2h\qquad \textbf{(E) } h^3$