Let $OFT$ and $NOT$ be two similar triangles (with the same orientation) and let $FANO$ be a parallelogram. Show that \[\vert OF\vert \cdot \vert ON\vert=\vert OA\vert \cdot \vert OT\vert.\]

Prove that you can always pose a circle of radius $S/P$ inside a convex polygon with the perimeter $P$ and area $S$.

In a right $\Delta ABC$ ($\angle C = 90^{\circ} $), $CD$ is the height. Let $r_1$ and $r_2$ be the radii of inscribed circles of $\Delta ACD$ and $\Delta DCB$. Find the radius of inscribed circle of $\Delta ABC$

A triangle $T$ has all integer side lengths and at most one of its side lengths is greater than ten. What is the largest possible area of $T$ ?

A bar of chocolate is made of $10$ distinguishable triangles as shown below: [center][img]https://cdn.artofproblemsolving.com/attachments/3/d/f55b0af0ce320fbfcfdbfab6a5c9c9306bfd16.png[/img][/center] How many ways are there to divide the bar, along the edges of the triangles, into two or more contiguous pieces?

The student drew a right triangle $ABC$ on the board with a right angle at the vertex $B$ and inscribed in it an equilateral triangle $KMP$ such that the points $K, M, P$ lie on the sides $AB, BC, AC$, respectively, and $KM \parallel AC$. Then the picture was erased, leaving only points $A, P$ and $C$. Restore erased points and lines.

Inside a circle with radius $6$ lie four smaller circles with centres $A,B,C$ and $D$. The circles touch each other as shown. The point where the circles with centres $A$ and $C$ touch each other is the centre of the big circle. Calculate the area of quadrilateral $ABCD$. [img]https://1.bp.blogspot.com/-FFsiOOdcjao/XzT_oJYuQAI/AAAAAAAAMVk/PpyUNpDBeEIESMsiElbexKOFMoCXRVaZwCLcBGAsYHQ/s0/2012%2BMohr%2Bp1.png[/img]

In triangle $ABC$, $\angle BAC= 60^o$. Point $P$ is taken inside the triangle so that angles $\angle APB=\angle BPC= \angle CP A=120^o$. It is known that $AP = a$. Find the area of triangle $BPC$.

Let $ ABC$ be a triangle, and let the angle bisectors of its angles $ CAB$ and $ ABC$ meet the sides $ BC$ and $ CA$ at the points $ D$ and $ F$, respectively. The lines $ AD$ and $ BF$ meet the line through the point $ C$ parallel to $ AB$ at the points $ E$ and $ G$ respectively, and we have $ FG \equal{} DE$. Prove that $ CA \equal{} CB$. [i]Original formulation:[/i] Let $ ABC$ be a triangle and $ L$ the line through $ C$ parallel to the side $ AB.$ Let the internal bisector of the angle at $ A$ meet the side $ BC$ at $ D$ and the line $ L$ at $ E$ and let the internal bisector of the angle at $ B$ meet the side $ AC$ at $ F$ and the line $ L$ at $ G.$ If $ GF \equal{} DE,$ prove that $ AC \equal{} BC.$

How many ways are there to color the vertices of a triangle red, green, blue, or yellow such that no two vertices have the same color? Rotations and reflections are considered distinct.

In a triangle $ABC$, let $D$ and $E$ be points of the sides $ BC$ and $AC$ respectively. Segments $AD$ and $BE$ intersect at $O$. Suppose that the line connecting midpoints of the triangle and parallel to $AB$, bisects the segment $DE$. Prove that the triangle $ABO$ and the quadrilateral $ODCE$ have equal areas.

A circle $C$ with radius $3$ has an equilateral triangle inscribed in it. Let $D$ be a circle lying outside the equilateral triangle, tangent to $C$, and tangent to the equilateral triangle at the midpoint of one of its sides. The radius of $D$ can be written in the form $m/n$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.

In the Crelle $[ABCD]$ tetrahedron, we note with $A',B',C',A'',B'',C''$ the tangent points of the hexatangent sphere $\varphi(J,\rho)$, associated with the tetrahedron, with the edges $|BC|,|CA|,|AB|,|DA|,|DB|,|DC|$. Show that these inequalities occur: a) $$2\sqrt3R\ge6\rho\ge A'A''+B'B''+C'C''\ge6\sqrt3r$$ b) $$4R^2\ge12\rho^2\ge(A'A'')^2+(B'B'')^2+(C'C'')^2\ge36r^2$$ c) $$\frac{8R^3}{3\sqrt3}\ge8\rho^3\ge A'A''\cdot B'B''\cdot C'C''\ge24\sqrt3r^3$$ where $r,R$ is the length of the radius of the sphere inscribed and respectively circumscribed to the tetrahedron. [i]Proposed by Marius Olteanu[/i]

In a certain geometry we operate with two types of elements, points and lines, related to each other by the following axioms: [b]I.[/b] Given two points $A$ and $B$, there is a unique line $(AB)$ that passes through both. [b]II. [/b]There are at least two points on a line. There are three points not situated on a straight line. [b]III.[/b] When a point $B$ is located between $A$ and $C$, then $B$ is also between $C$ and $A$. ($A, B, C$ are three different points on a line.) [b]IV.[/b] Given two points $A$ and $C$, there exists at least one point $B$ on the line $(AC)$ of the form that C is between $A$ and $B$. [b]V.[/b] Among three points located on the same straight line, one at most is between the other two. [b]VI.[/b] If $A, B, C$ are three points not lying on the same line and a is a line that does not contain any of the three, when the line passes through a point on segment [AB] , then it goes through one of the $[BC]$ , or it goes through one of the [AC] . (We designate by [AB] the set of points that lie between $A$ and $B$.) From the previous axioms, prove the following propositions: Theorem 1. Between points A and C there is at least one point $B$. Theorem 2. Among three points located on a line, one is always between the two others.

The symmetric difference of two homothetic triangles $T_1$ and $T_2$ consists of six triangles $t_1, \ldots, t_6$ with circumcircles $\omega_1, \omega_2, \ldots, \omega_6$ (counterclockwise, no two intersect). Circle $\Omega_1$ with center $O_1$ is externally tangent to $\omega_1, \omega_3,$ and $\omega_5$; circle $\Omega_2$ with center $O_2$ is externally tangent to $\omega_2, \omega_4,$ and $\omega_6$; circle $\Omega_3$ with center $O_3$ is internally tangent to $\omega_1, \omega_3,$ and $\omega_5$; circle $\Omega_4$ with center $O_4$ is internally tangent to $\omega_2, \omega_4,$ and $\omega_6$. Prove that $O_1O_3 = O_2O_4$. [i]Proposed by Ilya Zamotorin[/i]

There are six lines in the plane. No two of them are parallel and no point lies on more than three lines. What is the minimum possible number of points that lie on at least two lines?

The Duke of Squares left to his three sons a square estate, $100\times 100$ square miles, made up of ten thousand $1\times 1$ square mile square plots. The whole estate was divided among his sons as follows. Each son was assigned a point inside the estate. A $1\times 1$ square plot was bequeathed to the son whose assigned point was closest to the center of this square plot. Is it true that, irrespective of the choice of assigned points, each of the regions bequeathed to the sons is connected (that is, there is a path between every two of its points, never leaving the region)?

Are the following statements true? a) if the four vertices of a rectangle lie on the four sides of a rhombus, then the sides of the rectangle are parallel to the diagonals of the rhombus; b) if the four vertices of a square lie on the four sides of a rhombus that is not a square, then the sides of the square are parallel to the diagonals of the rhombus.

Let $ ABC$ be an acute triangle, $ AD,BE,CZ$ its altitudes and $ H$ its orthocenter. Let $ AI,A \Theta$ be the internal and external bisectors of angle $ A$. Let $ M,N$ be the midpoints of $ BC,AH$, respectively. Prove that: (a) $MN$ is perpendicular to $EZ$ (b) if $ MN$ cuts the segments $ AI,A \Theta$ at the points $ K,L$, then $ KL\equal{}AH$

Prove that if, in parallel projection of one plane onto another plane, the image of a certain square is a square, then the image of every figure is the figure congruent to it.

[b]p1.[/b] Suppose that $20\%$ of a number is $17$. Find $20\%$ of $17\%$ of the number. [b]p2.[/b] Let $A, B, C, D$ represent the numbers $1$ through $4$ in some order, with $A \ne 1$. Find the maximum possible value of $\frac{\log_A B}{C +D}$. Here, $\log_A B$ is the unique real number $X$ such that $A^X = B$. [b]p3. [/b]There are six points in a plane, no four of which are collinear. A line is formed connecting every pair of points. Find the smallest possible number of distinct lines formed. [b]p4.[/b] Let $a,b,c$ be real numbers which satisfy $$\frac{2017}{a}= a(b +c), \frac{2017}{b}= b(a +c), \frac{2017}{c}= c(a +b).$$ Find the sum of all possible values of $abc$. [b]p5.[/b] Let $a$ and $b$ be complex numbers such that $ab + a +b = (a +b +1)(a +b +3)$. Find all possible values of $\frac{a+1}{b+1}$. [b]p6.[/b] Let $\vartriangle ABC$ be a triangle. Let $X,Y,Z$ be points on lines $BC$, $CA$, and $AB$, respectively, such that $X$ lies on segment $BC$, $B$ lies on segment $AY$ , and $C$ lies on segment $AZ$. Suppose that the circumcircle of $\vartriangle XYZ$ is tangent to lines $AB$, $BC$, and $CA$ with center $I_A$. If $AB = 20$ and $I_AC = AC = 17$ then compute the length of segment $BC$. [b]p7. [/b]An ant makes $4034$ moves on a coordinate plane, beginning at the point $(0, 0)$ and ending at $(2017, 2017)$. Each move consists of moving one unit in a direction parallel to one of the axes. Suppose that the ant stays within the region $|x - y| \le 2$. Let N be the number of paths the ant can take. Find the remainder when $N$ is divided by $1000$. [b]p8.[/b] A $10$ digit positive integer $\overline{a_9a_8a_7...a_1a_0}$ with $a_9$ nonzero is called [i]deceptive [/i] if there exist distinct indices $i > j$ such that $\overline{a_i a_j} = 37$. Find the number of deceptive positive integers. [b]p9.[/b] A circle passing through the points $(2, 0)$ and $(1, 7)$ is tangent to the $y$-axis at $(0, r )$. Find all possible values of $ r$. [b]p10.[/b] An ellipse with major and minor axes $20$ and $17$, respectively, is inscribed in a square whose diagonals coincide with the axes of the ellipse. Find the area of the square. PS. You had better use hide for answers.

Inscribe an equilateral triangle of minimum side in a given acute-angled triangle $ABC$ (one vertex on each side).

The side lengths of an equiangular octagon alternate between $20$ and $22$. Find its area.

The sides of a rhombus have length $a$ and the area is $S$. What is the length of the shorter diagonal?

In the triangle $ABC$, let $AM$ and $CN$ internal bisectors, with $M$ in $BC$ and $N$ in $AB$. Prove that if $$\frac{\angle BNM}{\angle MNC}=\frac{\angle BMN}{\angle NMA}$$ then $ABC$ is isosceles.