A segment through the focus $F$ of a parabola with vertex $V$ is perpendicular to $\overline{FV}$ and intersects the parabola in points $A$ and $B$. What is $\cos(\angle AVB)$? $ \textbf{(A)}\ -\frac{3\sqrt{5}}{7} \qquad \textbf{(B)}\ -\frac{2\sqrt{5}}{5} \qquad \textbf{(C)}\ -\frac{4}{5} \qquad \textbf{(D)}\ -\frac{3}{5} \qquad \textbf{(E)}\ -\frac{1}{2} $

Let $ABC$ be a triangle with $AB=5$, $BC=7$, and $CA=8$. Let $D$ be a point on $BC$, and define points $B'$ and $C'$ on line $AD$ (or its extension) such that $BB'\perp AD$ and $CC'\perp AD$. If $B'A=B'C'$, then the ratio $BD:DC$ can be expressed in the form $m:n$, where $m$ and $n$ are relatively prime positive integers. Compute $100m+n$. [i]Proposed by Michael Ren[/i]

For a given triangle ABC, let X be a variable point on the line BC such that the point C lies between the points B and X. Prove that the radical axis of the incircles of the triangles ABX and ACX passes through a point independent of X. This is a slight extension of the [url=http://www.mathlinks.ro/Forum/viewtopic.php?t=41033]IMO Shortlist 2004 geometry problem 7[/url] and can be found, together with the proposed solution, among the files uploaded at http://www.mathlinks.ro/Forum/viewtopic.php?t=15622 . Note that the problem was proposed by Russia. I could not find the names of the authors, but I have two particular persons under suspicion. Maybe somebody could shade some light on this... Darij

A circle which is tangent to the sides $ [AB]$ and $ [BC]$ of $ \triangle ABC$ is also tangent to its circumcircle at the point $ T$. If $ I$ is the incenter of $ \triangle ABC$ , show that $ \widehat{ATI}\equal{}\widehat{CTI}$

Let $A,B,C$ be points on the sides $B_1C_1, C_1A_1,A_1B_1$ of a triangle $A_1B_1C_1$ such that $A_1A,B_1B,C_1C$ are the bisectors of angles of the triangle. We have that $AC = BC$ and $A_1C_1 \neq B_1C_1.$ $(a)$ Prove that $C_1$ lies on the circumcircle of the triangle $ABC$. $(b)$ Suppose that $\angle BAC_1 =\frac{\pi}{6};$ find the form of triangle $ABC$.

Prove that if points $ A_1, B_1, C_1 $ lying on the sides $ BC, CA, AB $ of a triangle $ ABC $ are the orthogonal projections of a point $ P $ of the triangle onto these sides, then $$ AC_1^2 + BA_1^2 + CB_1^2 = AB_1^2 + BC_1^2 + CA_1^2.$$

The diagonals $ AC$ and $ BD$ of a convex quadrilateral $ ABCD$ intersect at point $ M$. The bisector of $ \angle ACD$ meets the ray $ BA$ at $ K$. Given that $ MA \cdot MC \plus{}MA \cdot CD \equal{} MB \cdot MD$, prove that $ \angle BKC \equal{} \angle CDB$.

Vectors coincide with the edges of an arbitrary tetrahedron (possibly non-regular). Is it possible for the sum of these six vectors to equal the zero vector? (Problem from Leningrad)

The angle bisectors at $A$ and $C$ in a non-isosceles triangle $ABC$ with incenter $I$ intersect its circumcircle $k$ at $A_0$ and $C_0$, respectively. The line through $I$, parallel to $AC$, intersects $A_0C_0$ at $P$. Prove that $PB$ is tangent to $k$.

p1. Among the numbers $\frac15$ and $\frac14$ there are infinitely many fractional numbers. Find $999$ decimal numbers between $\frac15$ and $\frac14$ so that the difference between the next fractional number with the previous fraction constant. (i.e. If $x_1, x_2, x_3, x_4,..., x_{999}$ is a fraction that meant, then $x_2 - x_1= x_3 - x_3= ...= x_n - x_{n-1}=...=x_{999}-x_{998}$) p2. The pattern in the image below is: "Next image obtained by adding an image of a black equilateral triangle connecting midpoints of the sides of each white triangle that is left in the previous image." The pattern is continuous to infinity. [img]https://cdn.artofproblemsolving.com/attachments/e/f/81a6b4d20607c7508169c00391541248b8f31e.png[/img] It is known that the area of ​​the triangle in Figure $ 1$ is $ 1$ unit area. Find the total area of ​​the area formed by the black triangles in figure $5$. Also find the total area of the area formed by the black triangles in the $20$th figure. p3. For each pair of natural numbers $a$ and $b$, we define $a*b = ab + a - b$. The natural number $x$ is said to be the [i]constituent [/i] of the natural number $n$ if there is a natural number $y$ that satisfies $x*y = n$. For example, $2$ is a constituent of $6$ because there is a natural number 4 so that $2*4 = 2\cdot 4 + 2 - 4 = 8 + 2 - 4 = 6$. Find all constituent of $2005$. p4. Three people want to eat at a restaurant. To find who pays them to make a game. Each tossing one coin at a time. If the result is all heads or all tails, then they toss again. If not, then "odd person" (i.e. the person whose coin appears different from the two other's coins) who pay. Determine the number of all possible outcomes, if the game ends in tossing: a. First. b. Second. c. Third. d. Tenth. p5. Given the equation $x^2 + 3y^2 = n$, where $x$ and $y$ are integers. If $n < 20$ what number is $n$, and which is the respective pair $(x,y)$ ? Show that it is impossible to solve $x^2 + 3y^2 = 8$ in integers.

Jimmy’s garden has right angled triangle shape that lies on island of circular shape in such a way that the corners of the triangle are on the shore of the island. When he made fences along the garden he realized that the length of the shortest side is $36$ meter shorter than the longest side, and third side required $48$ meter long fence. In the middle of the garden he built a house of circular shape that has the largest possible size. Jimmy measured the distance between the center of his house and the center of the island. What is the square of this distance?

There are 6 lines in space, of which no 3 are parallel, no 3 pass through the same point, and no 3 are contained in the same plane. Prove that among these 6 lines there are 3 mutually oblique lines.

$ n $ lines meet at a point. Each one of the $ 2n $ disjoint angles formed around this point by these lines has either $ 7^{\circ} $ or $ 17^{\circ} . $ [b]a)[/b] Find $ n. $ [b]b)[/b] Prove that among these lines there are at least two perpendicular ones.

Let $a,b,c \in \left[ \frac 12, 1 \right]$. Prove that \[ 2 \leq \frac{ a+b}{1+c} + \frac{ b+c}{1+a} + \frac{ c+a}{1+b} \leq 3 . \] [i]selected by Mircea Lascu[/i]

Let $M$ be the intersection point of the medians $AD$ and $BE$ of a right triangle $ABC$ ($\angle C=90^\circ$). It is known that the circumcircles of triangles $AEM$ and $CDM$ are tangent. Find the angle $\angle BMC.$

$ AB$ is a fixed diameter of a circle whose center is $ O$. From $ C$, any point on the circle, a chord $ CD$ is drawn perpendicular to $ AB$. Then, as $ C$ moves over a semicircle, the bisector of angle $ OCD$ cuts the circle in a point that always: $ \textbf{(A)}\ \text{bisects the arc } AB \qquad\textbf{(B)}\ \text{trisects the arc } AB \qquad\textbf{(C)}\ \text{varies}$ $ \textbf{(D)}\ \text{is as far from }AB \text{ as from } D \qquad\textbf{(E)}\ \text{is equidistant from }B \text{ and } C$

Prove that the product of the sines of two opposite dihedrals in a tetrahedron is proportional to the product of the lengths of the edges of these dihedrals.

Let $ABC$ be a right triangle with$ \angle A = 90^o$. Point $D$ lies on side $BC$ such that $\angle BAD = \angle CAD$. Point $I_a$ is the excenter of the triangle opposite $A$. Prove that $\frac{AD}{DI_a } \le \sqrt{2} -1$

Find the maximum possible value of the inradius of a triangle whose vertices lie in the interior, or on the boundary, of a unit square.

A quadrilateral $ABCD$ without parallel sides is circumscribed around a circle with centre $O$. Prove that $O$ is a point of intersection of middle lines of quadrilateral $ABCD$ (i.e. barycentre of points $A,\,B,\,C,\,D$) iff $OA\cdot OC=OB\cdot OD$.

A circle inscribed within quadrilateral $ABCD$ is tangent to $AB$ at $E$, to $BC$ at $F$, to $CD$ at $G$, and to $DA$ at $H$. Suppose that $AE = 6$, $EB = 30$, $CG = 10$, and $GD = 2$. Compute $EF^2 + F G^2 + GH^2 + HE^2$. .

Given three skew lines. Prove that they are pair-wise perpendicular to their pair-wise perpendiculars.

Let $ABCD$ be a convex quadrilateral with $\angle CBD = 2 \angle ADB$, $\angle ABD = 2 \angle CDB$ and $AB = CB$. Prove that $AD = CD$.

Let $I$ be the center of the incircle of non-isosceles triangle $ABC,A_{1}=AI\cap BC$ and $B_{1}=BI\cap AC.$ Let $l_{a}$ be the line through $A_{1}$ which is parallel to $AC$ and $l_{b}$ be the line through $B_{1}$ parallel to $BC.$ Let $l_{a}\cap CI=A_{2}$ and $l_{b}\cap CI=B_{2}.$ Also $N=AA_{2}\cap BB_{2}$ and $M$ is the midpoint of $AB.$ If $CN\parallel IM$ find $\frac{CN}{IM}$.

Let $ABC$ be an equilateral triangle and $M$ be a point inside the triangle. We denote by $A'$, $B'$, $C'$ the projections of the point $M$ on the sides $BC$, $CA$ and $AB$ respectively. Prove that the lines $AA'$, $BB'$ and $CC'$ are concurrent if and only if $M$ belongs to an altitude of the triangle.