The center $O$ of the circle $\omega$ passing through the vertex $C$ of the isosceles triangle $ABC$ ($AB = AC$) is the interior point of the triangle $ABC$. This circle intersects segments $BC$ and $AC$ at points $D \ne C$ and $E \ne C$, respectively, and the circumscribed circle $\Omega$ of the triangle $AEO$ at the point $F \ne E$. Prove that the center of the circumcircle of the triangle $BDF$ lies on the circle $\Omega$.

The area of the triangle bounded by the lines $y = x, y = -x$ and $y = 6$ is $ \mathbf{(A)}\; 12\qquad \mathbf{(B)}\; 12\sqrt2\qquad \mathbf{(C)}\; 24\qquad \mathbf{(D)}\; 24\sqrt2\qquad \mathbf{(E)}\; 36$

A point $M$ is chosen on the arc $AC$ of the circumcircle of the equilateral triangle $ABC$. $P$ is the midpoint of this arc, $N$ is the midpoint of the chord $BM$ and $K$ is the foot of the perpendicular drawn from $P$ to $MC$. Prove that the triangle $ANK$ is equilateral. (I Nagel, Yevpatoria)

In the coordinate plane consider the set $ S$ of all points with integer coordinates. For a positive integer $ k$, two distinct points $A$, $ B\in S$ will be called $ k$-[i]friends[/i] if there is a point $ C\in S$ such that the area of the triangle $ ABC$ is equal to $ k$. A set $ T\subset S$ will be called $ k$-[i]clique[/i] if every two points in $ T$ are $ k$-friends. Find the least positive integer $ k$ for which there exits a $ k$-clique with more than 200 elements. [i]Proposed by Jorge Tipe, Peru[/i]

Given $n$ points in the three dimensional space such, that the arbitrary triangle with the vertices in three of those points contains an angle greater than $120$ degrees. Prove that you can rearrange them to make a polyline (unclosed) with all the angles between the sequent links greater than $120$ degrees.

p1. Two integers $m$ and $n$ are said to be [i]coprime [/i] if there are integers $a$ and $ b$ such that $am + bn = 1$. Show that for each integer $p$, the pair of numbers formed by $21p + 4$ and $14p + 3$ are always coprime. p2. Two farmers, Person $A$ and Person $B$ intend to change the boundaries of their land so that it becomes like a straight line, not curvy as in image below. They do not want the area of ​​their origin to be reduced. Try define the boundary line they should agree on, and explain why the new boundary does not reduce the area of ​​their respective origins. [img]https://cdn.artofproblemsolving.com/attachments/4/d/ec771d15716365991487f3705f62e4566d0e41.png[/img] p3. The system of equations of four variables is given: $\left\{\begin{array}{l} 23x + 47y - 3z = 434 \\ 47x - 23y - 4w = 183 \\ 19z + 17w = 91 \end{array} \right. $ where $x, y, z$, and $w$ are positive integers. Determine the value of $(13x - 14y)^3 - (15z + 16w)^3$ p4. A person drives a motorized vehicle so that the material used fuel is obtained at the following graph. [img]https://cdn.artofproblemsolving.com/attachments/6/f/58e9f210fafe18bfb2d9a3f78d90ff50a847b2.png[/img] Initially the vehicle contains $ 3$ liters of fuel. After two hours, in the journey of fuel remains $ 1$ liter. a. If in $ 1$ liter he can cover a distance of $32$ km, what is the distance taken as a whole? Explain why you answered like that? b. After two hours of travel, is there any acceleration or deceleration? Explain your answer. c. Determine what the average speed of the vehicle is. p5. Amir will make a painting of the circles, each circle to be filled with numbers. The circle's painting is arrangement follows the pattern below. [img]https://cdn.artofproblemsolving.com/attachments/8/2/533bed783440ea8621ef21d88a56cdcb337f30.png[/img] He made a rule that the bottom four circles would be filled with positive numbers less than $10$ that can be taken from the numbers on the date of his birth, i.e. $26 \,\, - \,\, 12 \,\, - \,\,1961$ without recurrence. Meanwhile, the circles above will be filled with numbers which is the product of the two numbers on the circles in underneath. a. In how many ways can he place the numbers from left to right, right on the bottom circles in order to get the largest value on the top circle? Explain. b. On another occasion, he planned to put all the numbers on the date of birth so that the number of the lowest circle now, should be as many as $8$ circles. He no longer cares whether the numbers are repeated or not . i. In order to get the smallest value in the top circle, how should the numbers be arranged? ii. How many arrays are worth considering to produce the smallest value?

$ABC$ is a right triangle with $AC=3$, $BC=4$, $AB=5$. Squares are erected externally on the sides of the triangle. Evaluate the area of hexagon $PQRSTU$.

Two squares $ACXE$ and $CBDY$ are constructed in the exterior of an acute-angled triangle $ABC$. Prove that the intersection of the lines $AD$ and $BE$ lies on the altitude of the triangle from $C$.

In triangle $ ABC$, one has marked the incenter, the foot of altitude from vertex $ C$ and the center of the excircle tangent to side $ AB$. After this, the triangle was erased. Restore it.

$(a)$ Prove that in a triangle the sum of the distances from the centroid to the sides is not less than three times the inradius, and find the cases of equality. $(b)$ Determine the points in a triangle that minimize the sum of the distances to the sides.

Let $ABC$ be a triangle. Let $\Omega$ be the brocard point. Prove that $\left(\frac{A\Omega}{BC}\right)^2+\left(\frac{B\Omega}{AC}\right)^2+\left(\frac{C\Omega}{AB}\right)^2\ge 1$

In quadrilateral $ABCD$, $AC = 5$, $CD = 7$, and $AD = 3$. The angle bisector of $\angle CAD$ intersects $CD$ at $E$. If $\angle CBD = 60^o$ and $\angle AED = \angle BEC$, compute the value of $AE + BE$.

Medians $AA_1, BB_1, CC_1$ and altitudes $AA_2, BB_2, CC_2$ are drawn in triangle $ABC$ . Prove that the length of the broken line $A_1B_2C_1A_2B_1C_2A_1$ is equal to the perimeter of triangle $ABC$.

In the convex hexagon $ABCDEF, AB, BC$ and $CD$ are respectively parallel to $DE, EF$ and $FA$. If $AB = DE$, prove that $BC = EF$ and $CD = FA$.

Circle $o$ contains the circles $m$ , $p$ and $r$, such that they are tangent to $o$ internally and any two of them are tangent between themselves. The radii of the circles $m$ and $p$ are equal to $x$ . The circle $r$ has radius $1$ and passes through the center of the circle $o$. Find the value of $x$ .

Let $ABCD$ be a trapezium such that $AB\parallel CD$. Let $E$ be the intersection of diagonals $AC$ and $BD$. Suppose that $AB = BE$ and $AC = DE$. Prove that the internal angle bisector of $\angle BAC$ is perpendicular to $AD$.

Triangle $ABC$ has $AB = 4$, $BC = 5$, and $CA = 6$. Points $A'$, $B'$, $C'$ are such that $B'C'$ is tangent to the circumcircle of $ABC$ at $A$, $C'A'$ is tangent to the circumcircle at $B$, and $A'B'$ is tangent to the circumcircle at $C$. Find the length $B'C'$.

Let $ABC$ be an acute triangle and let $M$ be the midpoint of side $BC$. Let $D,E$ be the excircles of triangles $AMB,AMC$ respectively, towards $M$. Circumcirscribed circle of triangle $ABD$ intersects line $BC$ at points $B$ and $F$. Circumcirscribed circles of triangle $ACE$ intersects line $BC$ at points $C$ and $G$. Prove that $BF=CG$. by Petru Braica, Romania

Let $ABCD$ be a trapezoid, where $AB$ and $CD$ are parallel. Let $P$ be a point on the side $BC$. Show that the parallels to $AP$ and $PD$ intersect through $C$ and $B$ to $DA$, respectively.

Let \( AL \) be the bisector of triangle \( ABC \), \( O \) the center of its circumcircle, and \( D \) and \( E \) the midpoints of \( BL \) and \( CL \), respectively. Points \( P \) and \( Q \) are chosen on segments \( AD \) and \( AE \) such that \( APLQ \) is a parallelogram. Prove that \( PQ \perp AO \). [i]Proposed by Mykhailo Plotnikov[/i]

Tracey baked a square cake whose surface is dissected in a $ 10 \times 10$ grid. In some of the fields she wants to put a strawberry such that for each four fields that compose a rectangle whose edges run in parallel to the edges of the cake boundary there is at least one strawberry. What is the minimum number of required strawberries?

Let $ABC$ be an acute triangle with orthocentre $H$, and let $M$ be the midpoint of $AC$. The point $C_1$ on $AB$ is such that $CC_1$ is an altitude of the triangle $ABC$. Let $H_1$ be the reflection of $H$ in $AB$. The orthogonal projections of $C_1$ onto the lines $AH_1$, $AC$ and $BC$ are $P$, $Q$ and $R$, respectively. Let $M_1$ be the point such that the circumcentre of triangle $PQR$ is the midpoint of the segment $MM_1$. Prove that $M_1$ lies on the segment $BH_1$.

Given triangle $ ABC$. The incircle of triangle $ ABC$ is tangent to $ BC,CA,AB$ at $ D,E,F$ respectively. Construct point $ G$ on $ EF$ such that $ DG$ is perpendicular to $ EF$. Prove that $ \frac{FG}{EG} \equal{} \frac{BF}{CE}$.

A sphere sits inside a cubic box, tangent on all $6$ sides of the box. If a side of the box is $5$, and the volume of the sphere is $x\pi$ , find $x$.

On the hypotenuse $ BC $ of an isosceles right triangle $ ABC $ let $ M,N $ such that $ BM^2-MN^2+NC^2=0. $ Show that $ \angle MAN= 45^{\circ } . $ [i]Cristinel Mortici[/i]