$ABC$ is a triangle: $A=(0,0)$, $B=(36,15)$ and both the coordinates of $C$ are integers. What is the minimum area $\triangle ABC$ can have? $ \textbf{(A)}\ \frac{1}{2} \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ \frac{3}{2} \qquad\textbf{(D)}\ \frac{13}{2} \qquad\textbf{(E)}\ \text{there is no minimum} $

[b]9.[/b] Show that if $f(z) = 1+a_1 z+a_2z^2+\dots$ for $\mid z \mid\leq 1$ and $\frac{1}{2\pi}\int_{0}^{2\pi}\mid f(e^{i\phi}) \mid^{2} d\phi < \left (1+\frac{\mid a_1\mid ^2} {4} \right )^2$, then $f(z)$ has a root in the disc $\mid z \mid \leq 1$.[b](F. 4)[/b]

How many ways can a set of $ n $ items be partitioned into two sets?

Let $AXYZB$ be a convex pentagon inscribed in a semicircle of diameter $AB$. Denote by $P$, $Q$, $R$, $S$ the feet of the perpendiculars from $Y$ onto lines $AX$, $BX$, $AZ$, $BZ$, respectively. Prove that the acute angle formed by lines $PQ$ and $RS$ is half the size of $\angle XOZ$, where $O$ is the midpoint of segment $AB$.

Is there a colouring of all positive integers in three colours so that for each positive integer the numbers of its divisors of any two colours differ at most by $2?$

If $ab\ne0$ and $|a|\ne|b|$ the number of distinct values of $x$ satisfying the equation \[\dfrac{x-a}{b}+\dfrac{x-b}{a}=\dfrac{b}{x-a}+\dfrac{a}{x-b}\] is: $\text{(A)}\ \text{zero}\qquad \text{(B)}\ \text{one}\qquad \text{(C)}\ \text{two}\qquad \text{(D)}\ \text{three}\qquad \text{(E)}\ \text{four}$

A cubic inch of the newly discovered material madelbromium weighs $5$ ounces. How many pounds will a cubic yard of madelbromium weigh?

Suppose $S = \{1, 2, 3, x\}$ is a set with four distinct real numbers for which the difference between the largest and smallest values of $S$ is equal to the sum of elements of $S.$ What is the value of $x?$ $$ \mathrm a. ~ {-1}\qquad \mathrm b.~{-3/2}\qquad \mathrm c. ~{-2} \qquad \mathrm d. ~{-2/3} \qquad \mathrm e. ~{-3} $$

Find all possible values of the number $$\frac{a + b}{c}+\frac{a + c}{b}+\frac{b + c}{a},$$ where $a, b, c$ are positive integers, and $\frac{a + b}{c},\frac{a + c}{b},\frac{b + c}{a}$ are also positive integers.

Simplify $\sqrt{7+\sqrt{33}}-\sqrt{7-\sqrt{33}}$.

Let $n$ be a positive integer. Prove that if the number overbrace $\underbrace{\hbox{99...9}}_{\hbox{n}}$ is divisible by $n$, then the number $\underbrace{\hbox{11...1}}_{\hbox{n}}$ is also divisible by $n$. (H. Nestra)

This problem consists of four parts. 1. Show that for any nonzero integers $m,n,$ and prime $p$, we have $v_p(mn)=v_p(m)+v_p(n).$ 2. Show that if an off prime $p$, a positive integer $k$ and integers $a,b$ satisfy $p \nmid ~^\text{'}~p|a-b$ and $p\nmid k$, then $v_p(a^k-b^k)=v_p(a-b).$ 3. Show that if $p$ is an off prime with $p|a-b$ and $p\nmid a,b$, then $v_p(a^p-b^p)=v_p(a-b)+1)$. 4. Show that if an odd prime $p$, a positive integer $k$ and integers $a,b$ satisfy $p\nmid a,b ~^\text{'}~ p|a-b$, then $v_p(a^k-b^k)=v_p(a-b)$. Proposed by LTE.

Let $n$ be a positive integer. There is a board of $(n + 1) \times (n + 1)$ whose squares are numbered in a diagonal pattern, as as the picture shows. Chepito starts from the lower left square, and moving only up or to the right until he reaches the upper right box. During his tour, Chepito writes down the number of each box on the which made a change of direction, and in the end calculates the sum of all the numbers entered. Determine the maximum value of this sum. [img]https://cdn.artofproblemsolving.com/attachments/e/d/f9dc43092a1407d6fe6f1b2c741af015079946.png[/img]

Given $$a=\dfrac{1}{2+\dfrac{1}{3+\dfrac{1}{...+\dfrac{...}{99}}}}, \,\,and\,\,\, b=\dfrac{1}{2+\dfrac{1}{3+\dfrac{1}{...+\dfrac{...}{99+\dfrac{1}{100}}}}}$$ Prove that $$|a-b| <\frac{1}{99! 100!}$$ (G Galperin, Moscow)

Sides of a triangle form an arithmetic sequence with common difference $2$, and its area is $6 \text{ cm }^2$. Find its sides.

Perry the painter wants to paint his floor, but he decides to leave a 1 foot border along the edges. After painting his floor, Perry notices that the area of the painted region is the same as the area of the unpainted region. Perry's floor measures $a$ x $b$ feet, where $a>b$ and both $a$ and $b$ are positive integers. Find all possible ordered pairs $(a, b)$. [i]2016 CCA Math Bonanza Team #2[/i]

Find all real functions of a real numbers, such that for any $x$, $y$ and $z$ holds the equality $$ f(x)f(y)f(z)-f(xyz)=xy+yz+xz+x+y+z.$$

According to the estimated number of participants who gave a correct solution, this was the hardest (!) problem from today's paper. So here is this great German killer - be warned! Given a circle k and three pairwisely distinct points A, B, C on this circle. Let h and g be the perpendiculars to the line BC at the points B and C. The perpendicular bisector of the segment AB meets the line h at a point F; the perpendicular bisector of the segment AC meets the line g at a point G. Prove that the product $BF\cdot CG$ is independent from the position of the point A, as long as the points B and C stay fixed. The actual problem behind the problem: Why on hell should the points B and C stay fixed? Darij

In triangle $ABC$ let $F$ denote the midpoint of side $BC$. Let the circle passing through point $A$ and tangent to side $BC$ at point $F$ intersect sides $AB$ and $AC$ at points $M$ and $N$, respectively. Let the line segments $CM$ and $BN$ intersect in point $X$. Let $P$ be the second point of intersection of the circumcircles of triangles $BMX$ and $CNX$. Prove that points $A, F$ and $P$ are collinear. Proposed by Imolay András, Budapest

The graphs of $y=3(x-h)^2+j$ and $y=2(x-h)^2+k$ have $y$-intercepts of $2013$ and $2014$, respectively, and each graph has two positive integer $x$-intercepts. Find $h$.

There is a set with three elements: (2,3,5). It has got an interesting property: (2*3) mod 5=(2*5) mod 3=(3*5) mod 2. Prove that it is the only one set with such property.

Triangle $A_0B_0C_0$ is given in the plane. Consider all triangles $ABC$ such that: (i) The lines $AB,BC,CA$ pass through $C_0,A_0,B_0$, respectvely, (ii) The triangles $ABC$ and $A_0B_0C_0$ are similar. Find the possible positions of the circumcenter of triangle $ABC$.

Let $n$ be a positive integer. A \emph{pseudo-Gangnam Style} is a dance competition between players $A$ and $B$. At time $0$, both players face to the north. For every $k\ge 1$, at time $2k-1$, player $A$ can either choose to stay stationary, or turn $90^{\circ}$ clockwise, and player $B$ is forced to follow him; at time $2k$, player $B$ can either choose to stay stationary, or turn $90^{\circ}$ clockwise, and player $A$ is forced to follow him. After time $n$, the music stops and the competition is over. If the final position of both players is north or east, $A$ wins. If the final position of both players is south or west, $B$ wins. Determine who has a winning strategy when: (a) $n=2013^{2012}$ (b) $n=2013^{2013}$

For $x,y,z > 0$ and $xyz=1$, prove that \[\frac{x^{9}+y^{9}}{x^{6}+x^{3}y^{3}+y^{6}}+\frac{x^{9}+z^{9}}{x^{6}+x^{3}z^{3}+z^{6}}+\frac{y^{9}+z^{9}}{y^{6}+y^{3}z^{3}+z^{6}}\geq 2\]

Let $ABC$ be an acute-angled triangle with $AC < AB$ and circumradius $R$. Furthermore, let $D$ be the foot ofthe altitude from $A$ on $BC$ and let $T$ denote the point on the line $AD$ such that $AT = 2R$ holds with $D$ lying between $A$ and $T$. Finally, let $S$ denote the mid-point of the arc $BC$ on the circumcircle that does not include $A$. Prove: $\angle AST = 90^\circ$. (Karl Czakler)