Let $ABC$ be a triangle such that $\angle BAC = 90^\circ$ and $AB < AC$. We divide the interior of the triangle into the following six regions: \begin{align*} S_1=\text{set of all points }\mathit{P}\text{ inside }\triangle ABC\text{ such that }PA<PB<PC \\ S_2=\text{set of all points }\mathit{P}\text{ inside }\triangle ABC\text{ such that }PA<PC<PB \\ S_3=\text{set of all points }\mathit{P}\text{ inside }\triangle ABC\text{ such that }PB<PA<PC \\ S_4=\text{set of all points }\mathit{P}\text{ inside }\triangle ABC\text{ such that }PB<PC<PA \\ S_5=\text{set of all points }\mathit{P}\text{ inside }\triangle ABC\text{ such that }PC<PA<PB \\ S_6=\text{set of all points }\mathit{P}\text{ inside }\triangle ABC\text{ such that }PC<PB<PA\end{align*} Suppose that the ratio of the area of the largest region to the area of the smallest non-empty region is $49 : 1$. Determine the ratio $AC : AB$.

For any point $X$ inside an acute-angled triangle $ABC$ we define $$f(X)=\frac{AX}{A_1X}\cdot \frac{BX}{B_1X}\cdot \frac{CX}{C_1X}$$ where $A_1, B_1$, and $C_1$ are the intersection points of the lines $AX, BX,$ and $CX$ with the sides $BC, AC$, and $AB$, respectively. Let $H, I$, and $G$ be the orthocenter, the incenter, and the centroid of the triangle $ABC$, respectively. Prove that $f(H) \ge f(I) \ge f(G)$ . (D. Bazylev)

Let $H$, $I$, $O$ be the orthocenter, incenter and circumcenter of a triangle. Show that $2 \cdot IO \geq IH$. When does the equality hold ?

Points $E$ and $F$ lie on the sides $AB$ and $AC$ of a triangle $ABC$. Lines $EF$ and $BC$ meet at point $S$. Let $M$ and $N$ be the midpoints of $BC$ and $EF$, respectively. The line passing through $A$ and parallel to $MN$ meets $BC$ at point $K$. Prove that $\frac{BK}{CK}=\frac{FS}{ES}$ . .

Triangle $ABC$ has side-lengths $AB=12$, $BC=24$, and $AC=18$. The line through the incenter of $\triangle ABC$ parallel to $\overline{BC}$ intersects $\overline{AB}$ at $M$ and $\overline{AC}$ at $N$. What is the perimeter of $\triangle AMN$? $ \textbf{(A)}\ 27 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 33 \qquad \textbf{(D)}\ 36 \qquad \textbf{(E)}\ 42 $

Let $ABC$ be a triangle, $D$ be a point on side $BC$, and let $\mathcal{O}$ be the circumcircle of triangle $ABC$. Show that the circles tangent to $\mathcal{O},AD,BD$ and to $\mathcal{O},AD,DC$ are tangent to each other if and only if $\angle BAD=\angle CAD$. [i]Dan Branzei[/i]

Let $\triangle{ABC}$ be a triangle with circumcircle $\tau$. The tangentlines to $\tau$ through $A$ and $B$ intersect at $T$. The circle through $A$, $B$ and $T$ intersects $BC$ and $AC$ again at $D$ and $E$, respectively; $CT$ and $BE$ intersect at $F$. Suppose $D$ is the midpoint of $BC$. Calculate the ratio $BF:BE$. [i](Swiss Mathematical Olympiad 2011, Final round, problem 5)[/i]

It is given that $\log_{6}a+\log_{6}b+\log_{6}c=6,$ where $a,$ $b,$ and $c$ are positive integers that form an increasing geometric sequence and $b-a$ is the square of an integer. Find $a+b+c.$

A wire is cut into two pieces, one of length $a$ and the other of length $b$. The piece of length $a$ is bent to form an equilateral triangle, and the piece of length $b$ is bent to form a regular hexagon. The triangle and the hexagon have equal area. What is $\frac{a}{b}$? ${ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ \frac{\sqrt{6}}{2} \qquad\textbf{(C)}\ \sqrt{3}\qquad\textbf{(D}}\ 2 \qquad\textbf{(E)}\ \frac{3\sqrt{2}}{2} $

Let $M$ be the midpoint of side $BC$ of a triangle $ABC$. Let the median $AM$ intersect the incircle of $ABC$ at $K$ and $L,K$ being nearer to $A$ than $L$. If $AK = KL = LM$, prove that the sides of triangle $ABC$ are in the ratio $5 : 10 : 13$ in some order.

Each side of an arbitrarly triangle is divided into $ 2002$ congruent segments. After that, each vertex is joined with all "division" points on the opposite side. Prove that the number of the regions formed, in which the triangle is divided, is divisible by $ 6$. [i]Proposer[/i]: [b]Dorian Croitoru[/b]

For a positive integer, we define it's [i]set of exponents[/i] the unordered list of all the exponents of the primes, in it`s decomposition. For example, $18=2\cdot 3^{2}$ has it`s set of exponents $1,2$ and $300=2^{2}\cdot 3\cdot 5^{2}$ has it`s set of exponents $1,2,2$. There are given two arithmetical progressions $\big(a_{n}\big)_{n}$ and $\big(b_{n}\big)_{n}$, such that for any positive integer $n$, $a_{n}$ and $b_{n}$ have the same set of exponents. Prove that the progressions are proportional (that is, there is $k$ such that $a_{n}=kb_{n}$ for any $n$). [i]Proposed by A. Golovanov[/i]

Can we partition the positive integers in two sets such that none of the sets contains an infinite arithmetic progression of nonzero ratio ?

An ice cream cone consists of a sphere of vanilla ice cream and a right circular cone that has the same diameter as the sphere. If the ice cream melts, it will exactly fill the cone. Assume that the melted ice cream occupies $ 75\%$ of the volume of the frozen ice cream. What is the ratio of the cone’s height to its radius? $ \textbf{(A)}\ 2: 1 \qquad \textbf{(B)}\ 3: 1 \qquad \textbf{(C)}\ 4: 1 \qquad \textbf{(D)}\ 16: 3 \qquad \textbf{(E)}\ 6: 1$

Let be a natural number $ n, $ two $ \text{n-tuplets} $ of real numbers $ a:=\left( a_1,a_2,\ldots, a_n \right) , b:=\left( b_1,b_2,\ldots, b_n \right) , $ and the function $ f:\mathbb{R}\longrightarrow\mathbb{R}, f(x)=\sum_{i=1}^na_i\cos \left( b_ix \right) $. Prove that if the numbers of $ b $ are all positive and pairwise distinct, [b]a)[/b] then, $ f\ge 0 $ implies that the numbers of $ a $ are all equal. [b]b)[/b] if the numbers of $ a $ are all nonzero and $ f $ is periodic, then the ratio of any two numbers of $ b $ is rational. [i]Marin Tolosi[/i]