Two sequences of nonzero reals $a_1, a_2, a_3, \dots$ and $b_2, b_3, \dots$ are such that $b_n=\prod_{i=1}^{n} a_i$ and $a_n=\frac{b_n^2}{3b_n-3}$ for all integers $n > 1$. Given that $a_1=\frac{1}{2}$, find $\lvert b_{60}\rvert$. [i]Proposed by Andrew Zhao[/i]

Vera has several identical matches, from which she makes a triangle. Vera wants any two sides of this triangle to differ in length by at least $10$ matches, but it turned out that it is impossible to add such a triangle from the available matches (it is impossible to leave extra matches). What is the maximum number of matches Vera can have?

Let $A$ be a point outside the circle $\omega$ . The tangents from $A$ touch the circle at $B$ and $C$. Let $P$ be an arbitrary point on extension of $AC$ towards $C$, $Q$ the projection of $C$ onto $PB$ and $E$ the second intersection point of the circumcircle of $ABP$ with the circle $\omega$ . Prove that $\angle PEQ = 2\angle APB$

[b]BJ2. [/b] Positive real numbers $ a,b,c$ satisfy inequality $ \frac {3}{2}\geq a \plus{} b \plus{} c$. Find the smallest possible value for $ S \equal{} abc \plus{} \frac {1}{abc}$

Omkar, Krit1, Krit2, and Krit3 are sharing $x > 0$ pints of soup for dinner. Omkar always takes $1$ pint of soup (unless the amount left is less than one pint, in which case he simply takes all the remaining soup). Krit1 always takes $\frac16$ of what is left, Krit2 always takes $\frac15$ of what is left, and Krit3 always takes $\frac14$ of what is left. They take soup in the order of Omkar, Krit1, Krit2, Krit3, and then cycle through this order until no soup remains. Find all $x$ for which everyone gets the same amount of soup.

A circle passing through the midpoint $M$ of the side $BC$ and the vertex $A$ of the triangle $ABC$ intersects the segments $AB$ and $AC$ for the second time in the points $P$ and $Q$, respectively. Prove that if $\angle BAC=60^{\circ}$, then $AP+AQ+PQ<AB+AC+\frac{1}{2} BC$.

Given $ a,b, c $ positive real numbers satisfying $ a+b+c=1 $. Prove that \[ \dfrac{1}{\sqrt{ab+bc+ca}}\ge \sqrt{\dfrac{2a}{3(b+c)}} +\sqrt{\dfrac{2b}{3(c+a)}} + \sqrt{\dfrac{2c}{3(a+b)}} \ge \sqrt{a} +\sqrt{b}+\sqrt{c} \]

Prove that if $f(x)$ is a polynomial with integer coefficients and there exists an integer $k$ such that none of $f(1),\ldots,f(k)$ is divisible by $k$, then $f(x)$ has no integral root.

A function $ f$ is defined by $ f(z) \equal{} (4 \plus{} i) z^2 \plus{} \alpha z \plus{} \gamma$ for all complex numbers $ z$, where $ \alpha$ and $ \gamma$ are complex numbers and $ i^2 \equal{} \minus{} 1$. Suppose that $ f(1)$ and $ f(i)$ are both real. What is the smallest possible value of $ | \alpha | \plus{} |\gamma |$? $ \textbf{(A)} \; 1 \qquad \textbf{(B)} \; \sqrt {2} \qquad \textbf{(C)} \; 2 \qquad \textbf{(D)} \; 2 \sqrt {2} \qquad \textbf{(E)} \; 4 \qquad$

The graph of ${(x^2 + y^2 - 1)}^3 = x^2 y^3$ is a heart-shaped curve, shown in the figure below. [asy] import graph; unitsize(10); real f(real x) { return sqrt(cbrt(x^4) - 4 x^2 + 4); } real g(real x) { return (cbrt(x^2) + f(x))/2; } real h(real x) { return (cbrt(x^2) - f(x)) / 2; } real xmax = 1.139028; draw(graph(g, -xmax, xmax) -- reverse(graph(h, -xmax, xmax)) -- cycle); xaxis("$x$", -1.5, 1.5, above = true); yaxis("$y$", -1.5, 1.5, above = true); [/asy] For how many ordered pairs of integers $(x, y)$ is the point $(x, y)$ inside or on this curve?

The eqquality $(x+m)^2-(x+n)^2=(m-n)^2$, where $m$ and $n$ are [i]unequal[/i] non-zero constants, is satisfied by $x=am+bn$, where: $ \textbf{(A)}\ a = 0, b \text{ } \text{has a unique non-zero value}\qquad$ $\textbf{(B)}\ a = 0, b \text{ } \text{has two non-zero values}\qquad$ $\textbf{(C)}\ b = 0, a \text{ } \text{has a unique non-zero value}\qquad$ $\textbf{(D)}\ b = 0, a \text{ } \text{has two non-zero values}\qquad$ $\textbf{(E)}\ a \text{ } \text{and} \text{ } b \text{ } \text{each have a unique non-zero value} $

In a simple graph, there exist two vertices $A,B$ such that there are exactly $100$ shortest paths from $A$ to $B$. Find the minimum number of edges in the graph. [i]CSJL[/i]

Let $K$ and $L$ be positive integers. On a board consisting of $2K \times 2L$ unit squares an ant starts in the lower left corner square and walks to the upper right corner square. In each step it goes horizontally or vertically to a neighbouring square. It never visits a square twice. At the end some squares may remain unvisited. In some cases the collection of all unvisited squares forms a single rectangle. In such cases, we call this rectangle [i]MEMOrable[/i]. Determine the number of different MEMOrable rectangles. [i]Remark: Rectangles are different unless they consist of exactly the same squares.[/i]

Find all functions $f(x)$ with the domain of all positive real numbers, such that for any positive numbers $x$ and $y$, we have $f(x^y)=f(x)^{f(y)}$.

Let $A_1C_2B_1B_2C_1A_2$ be a cyclic convex hexagon inscribed in circle $\Omega$, centered at $O$. Let $\{ P \} = A_2B_2 \cap A_1B_1$ and $\{ Q \} = A_2C_2 \cap A_1C_1$. Let $\Gamma_1$ be a circle tangent to $OB_1$ and $OC_1$ at $B_1,C_1$ respectively. Similarly, define $\Gamma_2$ to be the circle tangent to $OB_2,OC_2$ at $B_2, C_2$ respectively. Prove that there is a homothety that sends $\Gamma_1$ to $\Gamma_2$, whose center lies on $PQ$

Prove that $S_1 = (n + 1)^2 + (n + 2)^2 +...+ (n + 5)^2$ is divisible by $5$ for every $n$. Prove that for no $n$: $\sum_{\ell=1}^5 (n+\ell)^2$ is a perfect square. Let $S_2=(n + 6)^2 + (n + 7)^2 + ... + (n + 10)^2$. Prove that $S_1 \cdot S_2$ is divisible by $150$.

The table below displays some of the results of last summer's Frostbite Falls Fishing Festival, showing how many contestants caught $n$ fish for various values of $n$. \[ \begin{array}{|c|c|c|c|c|c|c|c|c|} \hline n & 0 & 1 & 2 & 3 & \dots & 13 & 14 & 15 \\ \hline \text{number of contestants who caught} \ n \ \text{fish} & 9 & 5 & 7 & 23 & \dots & 5 & 2 & 1 \\ \hline \end{array} \] In the newspaper story covering the event, it was reported that (a) the winner caught 15 fish; (b) those who caught 3 or more fish averaged 6 fish each; (c) those who caught 12 or fewer fish averaged 5 fish each. What was the total number of fish caught during the festival?

A number $ B$ is obtained from a positive integer number $ A$ by permuting its decimal digits. The number $ A\minus{}B\equal{}11...1$ ($ n$ of $ 1's$). Find the smallest possible positive value of $ n$.

Consider the number $\left(101^2-100^2\right)\cdot\left(102^2-101^2\right)\cdot\left(103^2-102^2\right)\cdot...\cdot\left(200^2-199^2\right)$. [list=a] [*] Determine its units digit. [*] Determine its tens digit. [/list]

A dance with 2018 couples takes place in Havana. For the dance, 2018 distinct points labeled $0, 1,\ldots, 2017$ are marked in a circumference and each couple is placed on a different point. For $i\geq1$, let $s_i=i\ (\textrm{mod}\ 2018)$ and $r_i=2i\ (\textrm{mod}\ 2018)$. The dance begins at minute $0$. On the $i$-th minute, the couple at point $s_i$ (if there's any) moves to point $r_i$, the couple on point $r_i$ (if there's any) drops out, and the dance continues with the remaining couples. The dance ends after $2018^2$ minutes. Determine how many couples remain at the end. Note: If $r_i=s_i$, the couple on $s_i$ stays there and does not drop out.

The set $ S$ is a finite subset of $ [0,1]$ with the following property: for all $ s\in S$, there exist $ a,b\in S\cup\{0,1\}$ with $ a, b\neq s$ such that $ s \equal{}\frac{a\plus{}b}{2}$. Prove that all the numbers in $ S$ are rational.

$(CZS 5)$ A convex quadrilateral $ABCD$ with sides $AB = a, BC = b, CD = c, DA = d$ and angles $\alpha = \angle DAB, \beta = \angle ABC, \gamma = \angle BCD,$ and $\delta = \angle CDA$ is given. Let $s = \frac{a + b + c +d}{2}$ and $P$ be the area of the quadrilateral. Prove that $P^2 = (s - a)(s - b)(s - c)(s - d) - abcd \cos^2\frac{\alpha +\gamma}{2}$

A triangle $ABC$ with $AB < AC$ is inscribed in a circle $\omega$. Circles $\gamma_1$ and $\gamma_2$ touch the lines $AB$ and $AC$, and their centres lie on the circumference of $\omega$. Prove that $C$ lies on a common external tangent to $\gamma_1$ and $\gamma_2$.

Let $[ABC]$ be a triangle and $X, Y$ and $Z$ points on the sides $[AB], [BC]$ and $[AC]$, respectively. Prove that circumcircles of triangles $AXZ, BXY$ and $CYZ$ intersect at a point.

There is a board of $2008\times 2008$ and $2008$ pieces, one in each row and each column of the board. It is allowed to do one of the following movements: a) Take two steps to the right and $10$ up. b) Take two steps to the right and $6$ steps down. c) Take two steps to the left and $6$ steps up. d) Take two steps to the left and $10$ steps down. If the path down cannot be completed, it is skipped to the upper part along the same column and the route continues normally, similarly in the other directions. In each play you will move a checker using any of the allowed operations. Would it be possible that at some point, after a finite number of played, the pieces are located forming a square of side $44$ in the upper left corner of the board and the remaining $72$ are in the last row in the first $72$ boxes?