Two circles $(O_1,R_1)$,$(O_2,R_2)$ lie each external to the other. Find : a) the minimum length of the segment connecting points of the circles b) the max length of the segment connecting points of the circles

Hiven an acute triangle $ABC$, consider the midpoints $M$ and $N$ of the sides $AB$ and $AC$, respectively. If point $S$ is variable on side $BC$, prove that $$(MB - MS)(NC - NS) \le 0$$

Let $\alpha,\beta,\gamma$ be positive real numbers such that $\alpha+\beta+\gamma < \pi$, $\alpha+\beta > \gamma$,$ \beta+\gamma > \alpha$, $\gamma + \alpha > \beta.$ Prove that with the segments of lengths $\sin \alpha, \sin \beta, \sin \gamma $ we can construct a triangle and that its area is not greater than \[A=\dfrac 18\left( \sin 2\alpha+\sin 2\beta+ \sin 2\gamma \right).\] [i]Proposed by Soviet Union[/i]

Bubble Boy and Bubble Girl live in bubbles of unit radii centered at $(20, 13)$ and $(0, 10)$ respectively. Because Bubble Boy loves Bubble Girl, he wants to reach her as quickly as possible, but he needs to bring a gift; luckily, there are plenty of gifts along the $x$-axis. Assuming that Bubble Girl remains stationary, find the length of the shortest path Bubble Boy can take to visit the $x$-axis and then reach Bubble Girl (the bubble is a solid boundary, and anything the bubble can touch, Bubble Boy can touch too)

In an acute triangle $ABC$, the angle bisector$AD$ and altitude $BE$ are drawn. Prove that angle $CED$ is greater than $45^o$.

Let $ABC$ be a triangle with side-lengths $a, b, c$, inscribed in a circle with radius $R$ and let $I$ be ir's incenter. Let $P_1, P_2$ and $P_3$ be the areas of the triangles $ABI, BCI$ and $CAI$, respectively. Prove that $$\frac{R^4}{P_1^2}+\frac{R^4}{P_2^2}+\frac{R^4}{P_3^2}\ge 16$$

Let $l$ denote the length of the smallest diagonal of all rectangles inscribed in a triangle $T$ . (By inscribed, we mean that all four vertices of the rectangle lie on the boundary of $T$ .) Determine the maximum value of $\frac{l^2}{S(T)}$ taken over all triangles ($S(T )$ denotes the area of triangle $T$ ).

a) Let $ABC$ be a triangle and a point $I$ lies inside the triangle. Assume $\angle IBA > \angle ICA$ and $\angle IBC >\angle ICB$. Prove that, if extensions of $BI$, $CI$ intersect $AC$, $AB$ at $B'$, $C'$ respectively, then $BB' < CC'$. b) Let $ABC$ be a triangle with $AB < AC$ and angle bisector $AD$. Prove that for every point $I, J$ on the segment $[AD]$ and $I \ne J$, we always have $\angle JBI > \angle JCI$. c) Let $ABC$ be a triangle with $AB < AC$ and angle bisector $AD$. Choose $M, N$ on segments $CD$ and $BD$, respectively, such that $AD$ is the bisector of angle $\angle MAN$. On the segment $[AD]$ take an arbitrary point $I$ (other than $D$). The lines $BI$, $CI$ intersect $AM$, $AN$ at $B', C'$. Prove that $BB' < CC'$.

The incircle of the triangle $ABC$ touches its sides $AB,BC,CA$ at points $C_1,A_1,B_1$ respectively. If $r$ is the inradius of $\vartriangle ABC, P,P_1$ are the perimeters of $\vartriangle ABC, \vartriangle A_1B_1C_1$ respectively, prove that $P+P_1 \ge 9 \sqrt3 r$. I. Voronovich

The quadrilateral $ABCD$ is such that $AB=AD=1$ and $\angle A=90^\circ$. If $CB=c$, $CA=b$, and $CD=a$, then prove that $(2-a^2-c^2 )^2+(2b^2-a^2-c^2 )^2=4a^2 c^2$ and $(a-c)^2\leq 2b^2\leq (a+c)^2$.

Consider in a plane $ P$ the points $ O,A_1,A_2,A_3,A_4$ such that \[ \sigma(OA_iA_j) \geq 1 \quad \forall i, j \equal{} 1, 2, 3, 4, i \neq j.\] where $ \sigma(OA_iA_j)$ is the area of triangle $ OA_iA_j.$ Prove that there exists at least one pair $ i_0, j_0 \in \{1, 2, 3, 4\}$ such that \[ \sigma(OA_iA_j) \geq \sqrt{2}.\]

Let $P$ be a regular hexagon. For a point $A$, let $d_1\le d_2\le ...\le d_6$ the distances from $A$ to the six vertices of $P$, ordered by magnitude. Find the locus of all points $A$ in the interior or on the boundary of $P$ such that: (a) $d_3$ takes the smallest possible value. (b) $d_4$ takes the smallest possible value.

If $a, b$ and $c$ are the length of the sides of a triangle, show that $$\frac32 \le \frac{b + c}{b + c + 2a}+ \frac{a + c}{a + c + 2b}+ \frac{a + b}{a + b + 2c}\le \frac53.$$

Numbers $a, b, c$ are the length of the medians of some triangle. If $ab + bc + ac = 1$ prove that a) $a^2b + b^2c + c^2a > \frac13$ b) $a^2b + b^2c + c^2a > \frac12$ (I. Bliznets)

In acute triangles $ABC$, $AD, BE ,CF$ are altitudes, with $D, E, F$ on the sides $BC, CA, AB$, respectively. Prove that $$DE + DF \le BC$$

Let $I$ be an incenter of a triangle $ABC, A_1,B_1,C_1$ be intersection points of the circumcircle of the triangle $ABC$ and the lines $AI, BI, Cl$ respectively. Prove that a) $\frac{AI}{IA_1}+ \frac{BI}{IB_1}+ \frac{CI}{IC_1}\ge 3$ b) $AI \cdot BI \cdot CI \le I_1A_1\cdot I_2B_1 \cdot I_1C_1$ (D. Pirshtuk)

Let $ABC$ be a triangle with incircle $(I)$, tangent to $BC$, $CA$, $AB$ at $D, E, F$ respectively. On the line $DF$, take points $M, P$ such that $CM \parallel AB$, $AP \parallel BC$. On the line $DE$, take points $N$, $Q$ such that $BN \parallel AC$, $AQ \parallel BC$. Denote $X$ as intersection of $PE$, $QF$ and $K$ as the midpoint of $BC$. Prove that if $AX = IK$ then $\angle BAC \le 60^o$.

Let $ a,\, b,\, c$ be side lengths of a triangle and $ a+b+c = 3$. Find the minimum of \[ a^{2}+b^{2}+c^{2}+\frac{4abc}{3}\]

In the plane, there are $n \geqslant 6$ pairwise disjoint disks $D_{1}, D_{2}, \ldots, D_{n}$ with radii $R_{1} \geqslant R_{2} \geqslant \ldots \geqslant R_{n}$. For every $i=1,2, \ldots, n$, a point $P_{i}$ is chosen in disk $D_{i}$. Let $O$ be an arbitrary point in the plane. Prove that \[O P_{1}+O P_{2}+\ldots+O P_{n} \geqslant R_{6}+R_{7}+\ldots+R_{n}.\] (A disk is assumed to contain its boundary.)

Let $ABC$ be a triangle and $\Gamma$ the $A$- exscribed circle whose center is $J$ . Let $D$ and $E$ be the touchpoints of $\Gamma$ with the lines $AB$ and $AC$, respectively. Let $S$ be the area of the quadrilateral $ADJE$, Find the maximum value that $\frac{S}{AJ^2}$ has and when equality holds.

Let $X$ be a point inside an equilateral triangle $ABC$ such that $BX+CX <3 AX$. Prove that $$3\sqrt3 \left( \cot \frac{\angle AXC}{2}+ \cot \frac{\angle AXB}{2}\right) +\cot \frac{\angle AXC}{2} \cot \frac{\angle AXB}{2} >5$$

Consider $ 16 $ pairwise distinct natural numbers smaller than $ 1597. $ [b]a)[/b] Prove that among these, there are three numbers having the property that the sum of any two of them is bigger than the third. [b]b)[/b] If one of these numbers is $ 1597, $ is still true the fact from subpoint [b]a)[/b]? [i]Teodor Radu[/i]

Let $ \,ABC\,$ be a triangle and $ \,P\,$ an interior point of $ \,ABC\,$. Show that at least one of the angles $ \,\angle PAB,\;\angle PBC,\;\angle PCA\,$ is less than or equal to $ 30^{\circ }$.

You are given a convex polygon with perimeter $24\sqrt{3} + 4\pi$. If there exists a pair of points dividing the perimeter in half such that the distance between them is equal to $24$, Prove that there exists a pair of points dividing the perimeter in half such that the distance between them does not exceed $12$.

Natural numbers from $1$ to $200$ were divided into $50$ sets. Prove that one of them contains three numbers that are the lengths of the sides some triangle.