Let $ABC$ be a triangle whose side lengths are, as usual, denoted by $a=|BC|,$ $b=|CA|,$ $c=|AB|.$ Denote by $m_a,m_b,m_c$, respectively, the lengths of the medians which connect $A,B,C$, respectively, with the centers of the corresponding opposite sides. (a) Prove that $2m_a<b+c$. Deduce that $m_a+m_b+m_c<a+b+c$. (b) Give an example of (i) a triangle in which $m_a>\sqrt{bc}$; (ii) a triangle in which $m_a\le \sqrt{bc}$.

Nine circles are drawn around an arbitrary triangle as in the figure. All circles tangent to the same side of the triangle have equal radii. Three lines are drawn, each one connecting one of the triangle’s vertices to the center of one of the circles touching the opposite side, as in the figure. Show that the three lines are concurrent. (N. Beluhov)

Let $ ABCD$ be a unit square (that is, the labels $ A, B, C, D$ appear in that order around the square). Let $ X$ be a point outside of the square such that the distance from $ X$ to $ AC$ is equal to the distance from $ X$ to $ BD$, and also that $ AX \equal{} \frac {\sqrt {2}}{2}$. Determine the value of $ CX^2$.

Consider a prism with triangular base. The total area of the three faces containing a particular vertex $A$ is $K$. Show that the maximum possible volume of the prism is $\sqrt{\frac{K^3}{54}}$ and find the height of this largest prism.

A triangle $ABC$ is given, in which the segment $BC$ touches the incircle and the corresponding excircle in points $M$ and $N$. If $\angle BAC = 2 \angle MAN$, show that $BC = 2MN$. (N.Beluhov)

Let $ABC$ be a triangle and $D$ be a point on $BC$ such that $AB+BD=AC+CD$. The line $AD$ intersects the incircle of triangle $ABC$ at $X$ and $Y$ where $X$ is closer to $A$ than $Y$ i. Suppose $BC$ is tangent to the incircle at $E$, prove that: 1) $EY$ is perpendicular to $AD$; 2) $XD=2IM$ where $I$ is the incentre and $M$ is the midpoint of $BC$.

Let $\overline{BE}$ and $\overline{CF}$ be altitudes of triangle $ABC$, and let $D$ be the antipodal point of $A$ on the circumcircle of $ABC$. The lines $\overleftrightarrow{DE}$ and $\overleftrightarrow{DF}$ intersect $\odot(ABC)$ again at $Y$ and $Z$, respectively. Show that $\overleftrightarrow{YZ}$, $\overleftrightarrow{EF}$ and $\overleftrightarrow{BC}$ intersect at a point.

Let $C_{1}$ and $C_{2}$ be concentric circles, with $C_{2}$ in the interior of $C_{1}$. From a point $A$ on $C_{1}$, draw the tangent $AB$ to $C_{2}$ $(B \in C_{2})$. Let $C$ be the second point of intersection of $AB$ and $C_{1}$,and let $D$ be the midpoint of $AB$. A line passing through $A$ intersects $C_{2}$ at $E$ and $F$ in such a way that the perpendicular bisectors of $DE$ and $CF$ intersect at a point $M$ on $AB$. Find, with proof, the ratio $AM/MC$. This question is taken from Mathematical Olympiad Challenges , the 9-th exercise in 1.3 Power of a Point.

A broken line consists of $31$ segments. It has no self intersections, and its start and end points are distinct. All segments are extended to become straight lines. Find the least possible number of straight lines.

Let $ABC$ be an acute-angled triangle with height intersection $H$. Let $G$ be the intersection of parallel of $AB$ through $H$ with the parallel of $AH$ through $B$. Let $I$ be the point on the line $GH$, so that $AC$ bisects segment $HI$. Let $J$ be the second intersection of $AC$ and the circumcircle of the triangle $CGI$. Show that $IJ = AH$

Suppose $I$ and $I_A$ are the incenter and the $A$-excenter of triangle $ABC$, respectively. Let $M$ be the midpoint of arc $BAC$ on the circumcircle, and $D$ be the foot of the perpendicular from $I_A$ to $BC$. The line $MI$ intersects the circumcircle again at $T$ . For any point $X$ on the circumcircle of triangle $ABC$, let $XT$ intersect $BC$ at $Y$ . Prove that $A, D, X, Y$ are concyclic.

A point $A$ is outside a circle $\mathcal K$ with center $O$. Line $AO$ intersects the circle at $B$ and $C$, and a tangent through $A$ touches the circle in $D$. Let $E$ be an arbitrary point on the line $BD$ such that $D$ lies between $B$ and $E$. The circumcircle of the triangle $DCE$ meets line $AO$ at $C$ and $F$ and line $AD$ at $D$ and $G$. Prove that the lines $BD$ and $FG$ are parallel.

Let $W$ be a regular octahedron and $O$ be its center. In a plane $P$ containing $O$ circles $k_1(O,r_1)$ and $k_2(O,r_2)$ are chosen so that $k_1 \subset P\cap W \subset k_2$. Prove that $\frac{r_1}{r_2}\le \frac{\sqrt3}{2}$

Let $a$, $b$, $c$ be side length of a triangle. Prove the inequality \begin{align*} \sqrt{a^2 + ab + b^2} + \sqrt{b^2 + bc + c^2} + \sqrt{c^2 + ca + a^2} \leq \sqrt{5a^2 + 5b^2 + 5c^2 + 4ab + 4 bc + 4ca}.\end{align*}

Let $ABCDA'B'C'D'$ be a cube (where $ABCD$ is a square and $AA'\parallel BB'\parallel CC'\parallel DD'$). Furthermore, let $\mathcal R$ be a rotation (with respect some line) that maps vertex $A$ to $B.$ Find the set of all images $X=\mathcal R(C)$ such that $X$ lies on the surface of the cube for some rotation $\mathcal R(A)=B.$

[b]p1.[/b] The expression $3 + 2\sqrt2$ can be represented as a perfect square: $3 +\sqrt2 = (1 + \sqrt2)^2$. (a) Represent $29 - 12\sqrt5$ as a prefect square. (b) Represent $10 - 6\sqrt3$ as a prefect cube. [b]p2.[/b] Find all values of the parameter $c$ for which the following system of equations has no solutions. $$x+cy = 1$$ $$cx+9y = 3$$ [b]p3.[/b] The equation $y = x^2 + 2ax + a$ represents a parabola for all real values of $a$. (a) Prove hat each of these parabolas pass through a common point and determine the coordinates of this point. (b) The vertices of the parabolas lie on a curve. Prove that this curve is a parabola and find its equation. [b]p4.[/b] Miranda is a $10$th grade student who is very good in mathematics. In fact she just completed an advanced algebra class and received a grade of A+. Miranda has five sisters, Cathy, Stella, Eva, Lucinda, and Dorothea. Miranda made up a problem involving the ages of the six girls and dared Cathy to solve it. Miranda said: “The sum of our ages is five times my age. (By ’age’ throughout this problem is meant ’age in years’.) When Stella is three times my present age, the sum of my age and Dorothea’s will be equal to the sum of the present ages of the five of us; Eva’s age will be three times her present age; and Lucinda’s age will be twice Stella’s present age, plus one year. How old are Stella and Miranda?” “Well, Miranda, could you tell me something else?” “Sure”, said Miranda, “my age is an odd number”. [b]p5.[/b] Cities $A,B,C$ and $D$ are located in vertices of a square with the area $10, 000$ square miles. There is a straight-line highway passing through the center of a square. Find the sum of squares of the distances from the cities of to the highway. [img]https://cdn.artofproblemsolving.com/attachments/b/4/1f53d81d3bc2a465387ff64de15f7da0949f69.png[/img] [b]p6.[/b] (a) Among three similar coins there is one counterfeit. It is not known whether the counterfeit coin is lighter or heavier than a genuine one (all genuine coins weight the same). Using two weightings on a pan balance, how can the counterfeit be identified and in process determined to be lighter or heavier than a genuine coin? (b) There is one counterfeit coin among $12$ similar coins. It is not known whether the counterfeit coin is lighter or heavier than a genuine one. Using three weightings on a pan balance, how can the counterfeit be identified and in process determined to be lighter or heavier than a genuine coin? PS. You should use hide for answers.

In the figure, $[ABCD]$ is a square of side $1$. The points $E, F, G$ and $H$ are such that $[AFB], [BGC], [CHD]$ and $[DEA]$ are right-angled triangles. Knowing that the circles inscribed in each of these triangles and the circle inscribed in the square $[EFGH]$ has all the same radius, what is the measure of the radius of the circles? [img]https://1.bp.blogspot.com/-l37AEXa7_-c/X4KaJwe6HQI/AAAAAAAAMk4/14wvIipf26cRge_GqKSRwH32bp291vX4QCLcBGAsYHQ/s0/2018%2Bportugal%2Bp2.png[/img]

Problems 7, 8 and 9 are about these kites. To promote her school's annual Kite Olympics, Genevieve makes a small kite and a large kite for a bulletin board display. The kites look like the one in the diagram. For her small kite Genevieve draws the kite on a one-inch grid. For the large kite she triples both the height and width of the entire grid. [asy] for (int a = 0; a < 7; ++a) { for (int b = 0; b < 8; ++b) { dot((a,b)); } } draw((3,0)--(0,5)--(3,7)--(6,5)--cycle);[/asy] What is the number of square inches in the area of the small kite? $ \text{(A)}\ 21\qquad\text{(B)}\ 22\qquad\text{(C)}\ 23\qquad\text{(D)}\ 24\qquad\text{(E)}\ 25 $

Let $D_1$, $D_2$, ..., $D_n$ be closed discs in the plane. (A closed disc is the region limited by a circle, taken jointly with this circle.) Suppose that every point in the plane is contained in at most $2003$ discs $D_i$. Prove that there exists a disc $D_k$ which intersects at most $7\cdot 2003 - 1 = 14020$ other discs $D_i$.

Consider a tetrahedron $OABC$ with $ABC$ equilateral. Let $S$ be the area of the triangle of sides $OA$, $OB$ and $OC$. Show that $V\leqslant \dfrac12 RS$ where $R$ is the circumradius and $V$ is the volume of the tetrahedron. [i]Stere Ianuș[/i]

Let $ABC$ be an actue triangle with $AB<AC$. Let $\Gamma$ be its circumcircle, $I$ its incenter and $P$ is a point on $\Gamma$ such that $\angle API=90^{\circ}$. Let $Q$ be a point on $\Gamma$ such that $$QB\cdot\tan \angle B=QC\cdot \tan \angle C$$ Consider a point $R$ such that $PR$ is tangent to $\Gamma$ and $BR=CR$. Prove that the points $A, Q, R$ are colinear.

Let $ABCD$ be a convex quadrilateral with $AB = AD$ and $CB = CD$. The bisector of $\angle BDC$ intersects $BC$ at $L$, and $AL$ intersects $BD$ at $M$, and it is known that $BL = BM$. Determine the value of $2\angle BAD + 3\angle BCD$.

Points $A_1$, $B_1$ and $C_1$ are taken on the respective edges $SA$, $SB$, $SC$ of a regular triangular pyramid $SABC$ so that the planes $A_1B_1C_1$ and $ABC$ are parallel. Let $O$ be the center of the sphere passing through $A$, $B$, $C_1$ and $S$. Prove that the line $SO$ is perpendicular to the plane $A_1B_1C$.

Let $A=(a_{jk})$ be a $10\times 10$ array of positive real numbers such that the sum of numbers in row as well as in each column is $1$. Show that there exists $j<k$ and $l<m$ such that \[a_{jl}a_{km}+a_{jm}a_{kl}\ge \frac{1}{50}\]

Dürer's $n \times m$ garden is surgically divided into $n \times m$ unit squares, and in the middle of one of these squares, he planted his favourite petunia. Dürer's gardener struggles with a mole, trying to drive him out of the magnificent garden, so he builds an underground wall on the edge of the garden. The only problem is that the mole managed to stay inside the walls.. When the mole meets a wall, it changes it's direction as if it was "reflected", that is, proceeding his route in the direction that includes the same angle with the wall as his direction before. The mole starts beneath the petunia, in a direction that includes a $45^o$ angle with the walls. Is it possible for the mole to cross the petunia in a direction perpendicular to it's original direction? (Think in terms of $n,m$.)