Consider a $3 \times 3$ grid of squares. A circle is inscribed in the lower left corner, the middle square of the top row, and the rightmost square of the middle row, and a circle $O$ with radius $r$ is drawn such that $O$ is externally tangent to each of the three inscribed circles. If the side length of each square is 1, compute $r$.

Suppose that the vertices of a regular polygon of $20$ sides are coloured with three colours - red, blue and green - such that there are exactly three red vertices. Prove that there are three vertices $A,B,C$ of the polygon having the same colour such that triangle $ABC$ is isosceles.

Bob has invented the Very Normal Coin (VNC). When the VNC is flipped, it shows heads $\textstyle\frac{1}{2}$ of the time and tails $\textstyle\frac{1}{2}$ of the time - unless it has yielded the same result five times in a row, in which case it is guaranteed to yield the opposite result. For example, if Bob flips five heads in a row, then the next flip is guaranteed to be tails. Bob flips the VNC an infinite number of times. On the $n$th flip, Bob bets $2^{-n}$ dollars that the VNC will show heads (so if the second flip shows heads, Bob wins $\$0.25$, and if the third flip shows tails, Bob loses $\$0.125$). Assume that dollars are infinitely divisible. Given that the first flip is heads, the expected number of dollars Bob is expected to win can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a, b$. Compute $100a + b$. [i]Proposed by Lewis Chen[/i]

A line passes through the points $\left(-1,3\right)$ and $\left(7,-2\right)$. At what value of $x$ does this line intercept the $x$-axis? $\text{(A) }\frac{7}{5}\qquad\text{(B) }\frac{19}{8}\qquad\text{(C) }\frac{19}{5}\qquad\text{(D) }\frac{27}{5}\qquad\text{(E) }\frac{23}{4}$

Let $ABC$ a acute triangle. (a) Find the locus of all the points $P$ such that, calling $O_{a}, O_{b}, O_{c}$ the circumcenters of $PBC$, $PAC$, $PAB$: \[\frac{ O_{a}O_{b}}{AB}= \frac{ O_{b}O_{c}}{BC}=\frac{ O_{c}O_{a}}{CA}\] (b) For all points $P$ of the locus in (a), show that the lines $AO_{a}$, $BO_{b}$ , $CO_{c}$ are cuncurrent (in $X$); (c) Show that the power of $X$ wrt the circumcircle of $ABC$ is: \[-\frac{ a^{2}+b^{2}+c^{2}-5R^{2}}4\] Where $a=BC$ , $b=AC$ and $c=AB$.

Prove that $$\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c} \ge \dfrac{a}{b}+\dfrac{b}{c}+\dfrac{c}{a}+2(a+b+c)$$ for the all $a,b,c$ positive real numbers satisfying $a^2+b^2+c^2+2abc \le 1$.

Show that there is an infinite number of positive integers $t$ such that none of the equations $$ \begin{cases} x^2 + y^6 = t \\ x^2 + y^6 = t + 1 \\ x^2 - y^6 = t \\ x^2 - y^6 = t + 1 \end{cases}$$ has solutions $(x, y) \in Z \times Z$.

Determine all the integers $a$ and $b$, such that $\sqrt{2010 + 2 \sqrt{2009}}$ be a solution of the equation $x^2 + ax + b = 0$. Prove that for such $a$ and $b$ the number$\sqrt{2010 - 2 \sqrt{2009}}$ is not a solution to the given equation.

(B.Frenkin) Consider the triangles such that all their vertices are vertices of a given regular 2008-gon. What triangles are more numerous among them: acute-angled or obtuse-angled?

Let $\vartriangle BCD$ be an equilateral triangle and $A$ be a point on the circumcircle of $\vartriangle BCD$ such that $A$ is on the minor arc $BD$. Then, let $P$ be the intersection of $\overline{AB}$ with $\overline{CD}$, $Q$ be the intersection of $\overline{AC}$ with $\overline{DB}$, and $R$ be the intersection of $\overline{AD}$ with $\overline{BC}$. Finally, let $X$, $Y$ , and $Z$ be the feet of the altitudes from $P$, $Q$, and $R$, respectively, in triangle $\vartriangle PQR$. Given $BQ = 3 -\sqrt5$ and $BC = 2$, compute the product of the areas $[\vartriangle XCD] \cdot [\vartriangle Y DB] \cdot [\vartriangle ZBC]$.

Let $\Gamma_1$ be a circle with centre $A$ and $\Gamma_2$ be a circle with centre $B$, with $A$ lying on $\Gamma_2$. On $\Gamma_2$ there is a (variable) point $P$ not lying on $AB$. A line through $P$ is a tangent of $\Gamma_1$ at $S$, and it intersects $\Gamma_2$ again in $Q$, with $P$ and $Q$ lying on the same side of $AB$. A different line through $Q$ is tangent to $\Gamma_1$ at $T$. Moreover, let $M$ be the foot of the perpendicular to $AB$ through $P$. Let $N$ be the intersection of $AQ$ and $MT$. Show that $N$ lies on a line independent of the position of $P$ on $\Gamma_2$.

Given a regular hexagon, a circle is drawn circumscribing it and another circle is drawn inscribing it. The ratio of the area of the larger circle to the area of the smaller circle can be written in the form $\frac{m}{n}$ , where m and n are relatively prime positive integers. Compute $m + n$.

Find the sum of all positive integers $x$ such that $$|x^2-x-6|$$ has exactly $4$ positive integer divisors. [i]Proposed by Evan Chang (squareman), USA[/i]

Solve for $z\in \mathbb{C}$ the equation : \[ |z-|z+1||=|z+|z-1|| \]

Prove that, for all $n \in \mathbb{N}$ \begin{align*} \frac{1}{1}+\frac{1}{3}+\frac{1}{5}+\ldots+\frac{1}{2n+1} \not\in \mathbb{Z} \end{align*}

Find all solutions of the equation $$sin\pi \sqrt x+cos\pi \sqrt x=(-1)^{\lfloor \sqrt x \rfloor }$$

Let $ G$ be anon-commutative group. Consider all the one-to-one mappings $ a\rightarrow a'$ of $ G$ onto itself such that $ (ab)'\equal{}b'a'$ (i.e. the anti-automorphisms of $ G$). Prove that this mappings together with the automorphisms of $ G$ constitute a group which contains the group of the automorphisms of $ G$ as direct factor.

Let $XY$ be a segment, which is a diameter of a semi-circle. Let $Z$ be a point on $XY$ and 9 rays from $Z$ are drawn that divide $\angle XZY=180^{\circ}$ into $10$ equal angles. These rays meet the semi-circle at $A_1, A_2, \ldots, A_9$ in this order in the direction from $X$ to $Y$. Prove that the sum of the areas of triangles $ZA_2A_3$ and $ZA_7A_8$ equals the area of the quadrilateral $A_2A_3A_7A_8$.

Find all ordered triples of primes $(p, q, r)$ such that \[ p \mid q^r + 1, \quad q \mid r^p + 1, \quad r \mid p^q + 1. \] [i]Reid Barton[/i]

$k$ is the circumscribed circle of $\Delta ABC$. $M$ and $N$ are arbitrary points on sides $CA$ and $CB$, and $MN$ intersects $k$ in points $U$ and $V$. Prove that the middle points of $BM$,$AN$,$MN$, and $UV$ lie on one circle.

Consider the finite sequence $\left\lfloor \frac{k^2}{1998} \right\rfloor$, for $k=1,2,\ldots, 1997$. How many distinct terms are there in this sequence? [i]Greece[/i]

Suppose that $2002$ numbers, each equal to $1$ or $-1$, are written around a circle. For every two adjacent numbers, their product is taken; it turns out that the sum of all $2002$ such products is negative. Prove that the sum of the original numbers has absolute value less than or equal to $1000$. (The absolute value of $x$ is usually denoted by $|x|$. It is equal to $x$ if $x \ge 0$, and to $-x$ if $x < 0$. For example, $|6| = 6, |0| = 0$, and $|-7| = 7$.)

Daud want to paint some faces of a cube with green paint. At least one face must be painted. How many ways are there for him to paint the cube? Note: Two colorings are considered the same if one can be obtained from the other by rotation.

If $a$, $b$ ,$c$ are three positive real numbers such that $ab+bc+ca = 1$, prove that \[ \sqrt[3]{ \frac{1}{a} + 6b} + \sqrt[3]{\frac{1}{b} + 6c} + \sqrt[3]{\frac{1}{c} + 6a } \leq \frac{1}{abc}. \]

Is it possible that two cross-sections of a tetrahedron by two different cutting planes are two squares, one with a side of length no greater than $1$ and another with a side of length at least $100$? Mikhail Evdokimov