A positive integer $N$ is called [i]balanced[/i], if $N=1$ or if $N$ can be written as a product of an even number of not necessarily distinct primes. Given positive integers $a$ and $b$, consider the polynomial $P$ defined by $P(x)=(x+a)(x+b)$. (a) Prove that there exist distinct positive integers $a$ and $b$ such that all the number $P(1)$, $P(2)$,$\ldots$, $P(50)$ are balanced. (b) Prove that if $P(n)$ is balanced for all positive integers $n$, then $a=b$. [i]Proposed by Jorge Tipe, Peru[/i]

The point $M$ lies inside the rhombus $ABCD$. It is known that $\angle DAB=110^o$, $\angle AMD=80^o$, $\angle BMC= 100^o$. What can the angle $\angle AMB$ be equal?

Let $ABC$ ($AB < AC$) be a triangle with circumcircle $k$. The tangent to $k$ at $A$ intersects the line $BC$ at $D$ and the point $E\neq A$ on $k$ is such that $DE$ is tangent to $k$. The point $X$ on line $BE$ is such that $B$ is between $E$ and $X$ and $DX = DA$ and the point $Y$ on the line $CX$ is such that $Y$ is between $C$ and $X$ and $DY = DA$. Prove that the lines $BC$ and $YE$ are perpendicular.

Let $\Delta ABC$ be an isosceles triangle with $BC = AC$ and $\angle ACB = 40^{\circ}$. Contruct the circle with diameter $\overline{BC}$, and let $D$ and $E$ be the other intersection points of the circle with the sides $\overline{AC}$ and $\overline{AB}$, respectively. Let $F$ be the intersection of the diagonals of the quadrilateral $BCDE$. What is the degree measure of $\angle BFC ?$ $\textbf{(A) } 90 \qquad\textbf{(B) } 100 \qquad\textbf{(C) } 105 \qquad\textbf{(D) } 110 \qquad\textbf{(E) } 120$

Take the square with vertices $(0,0)$, $(1,0)$, $(0,1)$, and $(1,1)$. Choose a random point in this square and draw the line segment from it to $(0,0)$. Choose a second random point in this square and draw the line segment from it to $(1,0)$. What is the probability that the two line segments intersect?

An equilateral triangle and a regular hexagon have equal perimeters. If the area of the triangle is 4, what is the area of the hexagon? $\textbf{(A)}\hspace{.05in}4 \qquad \textbf{(B)}\hspace{.05in}5 \qquad \textbf{(C)}\hspace{.05in}6 \qquad \textbf{(D)}\hspace{.05in}4\sqrt3 \qquad \textbf{(E)}\hspace{.05in}6\sqrt3 $

Let $\mathbb{Z}$ be the set of integers. Determine, with proof, all primes $p$ for which there exists a function $f:\mathbb{Z}\to\mathbb{Z}$ such that for any integer $x,$ $\quad \bullet \ f(x+p)=f(x)\text{ and}$ $\quad \bullet \ p \text{ divides } f(x+f(x))-x.$

We color some cells in $10000 \times 10000$ square, such that every $10 \times 10$ square and every $1 \times 100$ line have at least one coloring cell. What minimum number of cells we should color ?

Are there $10$ different integers such that all the sums made up of $9$ of them are perfect squares?

Alice starts with the number 0. She can apply 100 operations on her number. In each operation, she can either add 1 to her number, or square her number. After applying all operations, her score is the minimum distance from her number to any perfect square. What is the maximum score she can attain?

In a checkered square $101 \times 101$, each cell of the inner square $99 \times 99$ is painted in one of ten colors (cells adjacent to the border of the square, not painted). Could it turn out that in every in a $3\times 3$ square, is exactly one more cell painted the same color as the central cell?

Let $A$ be a set, and let $P (A)$ be the powerset of all non-empty subsets of $A$. (For example, $A = \{1,2,3\}$, then $P (A) = \{\{1\},\{2\} ,\{3\},\{1,2\}, \{1,3\},\{2,3\}, \{1,2,3\}\}$.) A subset $F$ of P $(A)$ is called [i]strong [/i] if the following is true: If $B_1$ and $B_2$ are elements of $F$, then $B_1 \cup B_2$ is also an element of $F$. Suppose that $F$ and $G$ are strong subsets of $P (A)$. a) Is the union $F \cup G$ necessarily strong? b) Is the intersection $F \cap G$ necessarily strong?

A small hard solid sphere of mass m and negligible radius is connected to a thin rod of length L and mass 2m. A second small hard solid sphere, of mass M and negligible radius, is fired perpendicularly at the rod at a distance h above the sphere attached to the rod, and sticks to it. In order for the rod not to rotate after the collision, the second sphere should have a mass M given by which of the following? $\textbf{(A)} M = m\qquad \textbf{(B)} M = 1.5m\qquad \textbf{(C)} M = 2m\qquad \textbf{(D)} M = 3m\qquad \textbf{(E)}\text{Any mass M will work}$

We consider the real sequence ($x_n$) defined by $x_0=0, x_1=1$ and $x_{n+2}=3x_{n+1}-2 x_{n}$ for $n=0,1,2,...$ We define the sequence ($y_n$) by $y_n=x^2_n+2^{n+2}$ for every nonnegative integer $n$. Prove that for every $n>0, y_n$ is the square of an odd integer.

Let $ABC$ be an isosceles triangle with $AC = BC$ and circumcircle $k$. The point $D$ lies on the shorter arc of $k$ over the chord $BC$ and is different from $B$ and $C$. Let $E$ denote the intersection of $CD$ and $AB$. Prove that the line through $B$ and $C$ is a tangent of the circumcircle of the triangle $BDE$. (Karl Czakler)

Which of the following inequalities are satisfied for all real numbers $ a$, $ b$, $ c$, $ x$, $ y$, $ z$ which satisfy the conditions $ x < a$, $ y < b$, and $ z < c$? $ \text{I}. \ xy \plus{} yz \plus{} zx < ab \plus{} bc \plus{} ca$ $ \text{II}. \ x^2 \plus{} y^2 \plus{} z^2 < a^2 \plus{} b^2 \plus{} c^2$ $ \text{III}. \ xyz < abc$ $ \textbf{(A)}\ \text{None are satisfied.} \qquad \textbf{(B)}\ \text{I only} \qquad \textbf{(C)}\ \text{II only} \qquad$ $ \textbf{(D)}\ \text{III only} \qquad \textbf{(E)}\ \text{All are satisfied.}$

Let $c \ge 1$ be an integer. Define a sequence of positive integers by $a_1 = c$ and \[a_{n+1}=a_n^3-4c\cdot a_n^2+5c^2\cdot a_n+c\] for all $n\ge 1$. Prove that for each integer $n \ge 2$ there exists a prime number $p$ dividing $a_n$ but none of the numbers $a_1 , \ldots , a_{n -1}$ . [i]Proposed by Austria[/i]

In a quadrilateral $ABCD$ we have $\angle A = \angle C = 90^o$. Let $E$ be a point in the interior of $ABCD$. Let $M$ be the midpoint of $BE$. Prove that $\angle ADB = \angle EDC$ if and only if $|MA| = |MC|$.

Bricklayer Brenda would take $ 9$ hours to build a chimney alone, and bricklayer Brandon would take $ 10$ hours to build it alone. When they work together they talk a lot, and their combined output is decreased by $ 10$ bricks per hour. Working together, they build the chimney in $ 5$ hours. How many bricks are in the chimney? $ \textbf{(A)}\ 500 \qquad \textbf{(B)}\ 900 \qquad \textbf{(C)}\ 950 \qquad \textbf{(D)}\ 1000 \qquad \textbf{(E)}\ 1900$

For all $n\ge 1$, let $a_n=\sum_{k=1}^{n-1}\frac{\sin(\frac{(2k-1)\pi}{2n})}{\cos^2(\frac{(k-1)\pi}{2n})\cos^2(\frac{k\pi}{2n})}$. Determine $\lim_{n\rightarrow \infty}\frac{a_n}{n^3}$.

[b]p1.[/b] The expression $3 + 2\sqrt2$ can be represented as a perfect square: $3 +\sqrt2 = (1 + \sqrt2)^2$. (a) Represent $29 - 12\sqrt5$ as a prefect square. (b) Represent $10 - 6\sqrt3$ as a prefect cube. [b]p2.[/b] Find all values of the parameter $c$ for which the following system of equations has no solutions. $$x+cy = 1$$ $$cx+9y = 3$$ [b]p3.[/b] The equation $y = x^2 + 2ax + a$ represents a parabola for all real values of $a$. (a) Prove hat each of these parabolas pass through a common point and determine the coordinates of this point. (b) The vertices of the parabolas lie on a curve. Prove that this curve is a parabola and find its equation. [b]p4.[/b] Miranda is a $10$th grade student who is very good in mathematics. In fact she just completed an advanced algebra class and received a grade of A+. Miranda has five sisters, Cathy, Stella, Eva, Lucinda, and Dorothea. Miranda made up a problem involving the ages of the six girls and dared Cathy to solve it. Miranda said: “The sum of our ages is five times my age. (By ’age’ throughout this problem is meant ’age in years’.) When Stella is three times my present age, the sum of my age and Dorothea’s will be equal to the sum of the present ages of the five of us; Eva’s age will be three times her present age; and Lucinda’s age will be twice Stella’s present age, plus one year. How old are Stella and Miranda?” “Well, Miranda, could you tell me something else?” “Sure”, said Miranda, “my age is an odd number”. [b]p5.[/b] Cities $A,B,C$ and $D$ are located in vertices of a square with the area $10, 000$ square miles. There is a straight-line highway passing through the center of a square. Find the sum of squares of the distances from the cities of to the highway. [img]https://cdn.artofproblemsolving.com/attachments/b/4/1f53d81d3bc2a465387ff64de15f7da0949f69.png[/img] [b]p6.[/b] (a) Among three similar coins there is one counterfeit. It is not known whether the counterfeit coin is lighter or heavier than a genuine one (all genuine coins weight the same). Using two weightings on a pan balance, how can the counterfeit be identified and in process determined to be lighter or heavier than a genuine coin? (b) There is one counterfeit coin among $12$ similar coins. It is not known whether the counterfeit coin is lighter or heavier than a genuine one. Using three weightings on a pan balance, how can the counterfeit be identified and in process determined to be lighter or heavier than a genuine coin? PS. You should use hide for answers.

Let $ABCD$ be a square of side paper $10$ and $P$ a point on side $BC$. By folding the paper along the $AP$ line, point $B$ determines the point $Q$, as seen in the figure. The line $PQ$ cuts the side $CD$ at $R$. Calculate the perimeter of the triangle $ PCR$ [img]https://3.bp.blogspot.com/-ZSyCUznwutE/XNY7cz7reQI/AAAAAAAAKLc/XqgQnjm8DQYq6Q7fmCAKJwKt3ihoL8AuQCK4BGAYYCw/s400/may%2B2013%2Bl1.png[/img]

Triangles $ABC$ and $DEF$ share circumcircle $\Omega$ and incircle $\omega$ so that points $A,F,B,D,C,$ and $E$ occur in this order along $\Omega$. Let $\Delta_A$ be the triangle formed by lines $AB,AC,$ and $EF,$ and define triangles $\Delta_B, \Delta_C, \ldots, \Delta_F$ similarly. Furthermore, let $\Omega_A$ and $\omega_A$ be the circumcircle and incircle of triangle $\Delta_A$, respectively, and define circles $\Omega_B, \omega_B, \ldots, \Omega_F, \omega_F$ similarly. (a) Prove that the two common external tangents to circles $\Omega_A$ and $\Omega_D$ and the two common external tangents to $\omega_A$ and $\omega_D$ are either concurrent or pairwise parallel. (b) Suppose that these four lines meet at point $T_A$, and define points $T_B$ and $T_C$ similarly. Prove that points $T_A,T_B$, and $T_C$ are collinear. [i]Nikolai Beluhov[/i]

Consider the set of all pairs of positive integers $(A , B)$ in which $A < B$ . Some of these pairs are to $be$ designated as "black" , while the remainder are to be designated as "white" . Is it possible to designate these pairs in such a way that for any triple of positive integers of form $A, A + D, A + 2D$, in which $D > 0$, the associated pairs $(A, A + D )$ , $(A , A + 2D)$ and $(A + D, A + 2D)$ would include at least one pair of each colour?

Let $P(x)$ be a polynomial with integer coefficients such that $P(0)=1$, and let $c > 1$ be an integer. Define $x_0=0$ and $x_{i+1} = P(x_i)$ for all integers $i \ge 0$. Show that there are infinitely many positive integers $n$ such that $\gcd (x_n, n+c)=1$. [i]Proposed by Milan Haiman and Carl Schildkraut[/i]