Find the necessary and sufficient condition that a trapezoid can be formed out of a given four-bar linkage.

In triangle $ABC$ the angular bisector from $A$ intersects the side $BC$ at the point $D$, and the angular bisector from $B$ intersects the side $AC$ at the point $E$. Furthermore $|AE| + |BD| = |AB|$. Prove that $\angle C = 60^o$ [img]https://1.bp.blogspot.com/-8ARqn8mLn24/XzP3P5319TI/AAAAAAAAMUQ/t71-imNuS18CSxTTLzYXpd806BlG5hXxACLcBGAsYHQ/s0/2018%2BMohr%2Bp5.png[/img]

In $\triangle ABC$ we consider the points $A',B',C'$ in sides $BC,AC,AB$ such that $$3BA'=CA',~3CB'=AB',~3AC'=BA'$$$\triangle DEF$ is defined by the intersections of $AA',BB',CC'$. If the are of $\triangle ABC$ is $2020$ find the area of $\triangle DEF$. [i]Proposed by Alejandro Madrid, Valle[/i]

Is there a six-link broken line in space that passes through all the vertices of a given cube?

We are given $n (n \ge 5)$ circles in a plane. Suppose that every three of them have a common point. Prove that all $n$ circles have a common point.

Given the sequence $(t_n)$ defined as $t_0 = 0$, $t_1 = 6$, $t_{n + 2} = 14t_{n + 1} - t_n$. Prove that for every number $n \ge 1$, $t_n$ is the area of a triangle whose lengths are all numbers integers. Dang Hung Thang, University of Natural Sciences, Hanoi National University.

Find the area of the region surrounded by the two curves $ y=\sqrt{x},\ \sqrt{x}+\sqrt{y}=1$ and the $ x$ axis.

Points $Y,Z$ lie on the smaller arc $BC$ of the circumcircle of an acute triangle $\bigtriangleup ABC$ ($Y$ lies on the smaller arc $BZ$). Let $X$ be a point such that the triangles $\bigtriangleup ABC,\bigtriangleup XYZ$ are similar (in this exact order) with $A,X$ lying on the same side of $YZ$. Lines $XY,XZ$ intersect sides $AB,AC$ at points $E,F$ respectively. Let $K$ be the intersection of lines $BY,CZ$. Prove that one of the intersections of the circumcircles of triangles $\bigtriangleup AEF,\bigtriangleup KBC$ lie on the line $KX$. [i]Proposed by Amirparsa Hosseini Nayeri - Iran[/i]

Given an equilateral triangle $ABC$ and a point $M$ in the plane ($ABC$). Let $A', B', C'$ be respectively the symmetric through $M$ of $A, B, C$. [b]I.[/b] Prove that there exists a unique point $P$ equidistant from $A$ and $B'$, from $B$ and $C'$ and from $C$ and $A'$. [b]II.[/b] Let $D$ be the midpoint of the side $AB$. When $M$ varies ($M$ does not coincide with $D$), prove that the circumcircle of triangle $MNP$ ($N$ is the intersection of the line $DM$ and $AP$) pass through a fixed point.

Each corner cube is removed from this $3\text{ cm}\times 3\text{ cm}\times 3\text{ cm}$ cube. The surface area of the remaining figure is [asy]draw((2.7,3.99)--(0,3)--(0,0)); draw((3.7,3.99)--(1,3)--(1,0)); draw((4.7,3.99)--(2,3)--(2,0)); draw((5.7,3.99)--(3,3)--(3,0)); draw((0,0)--(3,0)--(5.7,0.99)); draw((0,1)--(3,1)--(5.7,1.99)); draw((0,2)--(3,2)--(5.7,2.99)); draw((0,3)--(3,3)--(5.7,3.99)); draw((0,3)--(3,3)--(3,0)); draw((0.9,3.33)--(3.9,3.33)--(3.9,0.33)); draw((1.8,3.66)--(4.8,3.66)--(4.8,0.66)); draw((2.7,3.99)--(5.7,3.99)--(5.7,0.99)); [/asy] $\textbf{(A)}\ 19\text{ sq.cm} \qquad \textbf{(B)}\ 24\text{ sq.cm} \qquad \textbf{(C)}\ 30\text{ sq.cm} \qquad \textbf{(D)}\ 54\text{ sq.cm} \qquad \textbf{(E)}\ 72\text{ sq.cm}$

In a triangle $ABC$, point $P$ is the incenter and $A'$, $B'$, $C'$ its orthogonal projections on $BC$, $CA$, $AB$, respectively. Show that $\angle B'A'C'$ is acute.

Let $ I$ be the incenter of triangle $ ABC.$ Let $ M,N$ be the midpoints of $ AB,AC,$ respectively. Points $ D,E$ lie on $ AB,AC$ respectively such that $ BD\equal{}CE\equal{}BC.$ The line perpendicular to $ IM$ through $ D$ intersects the line perpendicular to $ IN$ through $ E$ at $ P.$ Prove that $ AP\perp BC.$

For what real values of $k>0$ is it possible to dissect a $1 \times k$ rectangle into two similar, but noncongruent, polygons?

Equilateral triangle $\vartriangle ABC$ has side length $6$. Points $D$ and $E$ lie on $\overline{BC}$ such that $BD = CE$ and $B$, $D$, $E$, $C$ are collinear in that order. Points $F$ and $G$ lie on $\overline{AB}$ such that $\overline{FD} \perp \overline{BC}$, and $GF = GA$. If the minimum possible value of the sum of the areas of $\vartriangle BFD$ and $\vartriangle DGE$ can be expressed as $\frac{a\sqrt{b}}{c}$ for positive integers $a, b, c$ with $gcd (a, c) = 1$ and $b$ squarefree, find $a + b + c$.

Given a sequence $(a_n)$, with $a_1 = 4$ and $a_{n+1} = a_n^2-2 (\forall n \in\mathbb{N})$, prove that there is a triangle with side lengths $a_{n-1}, a_n, a_{n+1},$ and that its area is equal to an integer.

Three flies are crawling along the perimeter of the triangle $ABC$ in such a way, that the centre of their masses is a constant point. One of the flies has already passed along all the perimeter. Prove that the centre of the flies' masses coincides with the centre of masses of the triangle $ABC$ . (The centre of masses for the triangle is the point of medians intersection.

The triangle $ABC$ satisfies $0\le AB\le 1\le BC\le 2\le CA\le 3$. What is the maximum area it can have?

For $n>1$ consider an $n\times n$ chessboard and place identical pieces at the centers of different squares. [list=i] [*] Show that no matter how $2n$ identical pieces are placed on the board, that one can always find $4$ pieces among them that are the vertices of a parallelogram. [*] Show that there is a way to place $(2n-1)$ identical chess pieces so that no $4$ of them are the vertices of a parallelogram. [/list]

Let $P$ be a point inside a regular tetrahedron ABCD with edge length $1$. Show that $$d(P,AB)+d(P,AC)+d(P,AD)+d(P,BC)+d(P,BD)+d(P,CD) \ge \frac{3}{2} \sqrt2$$ , with equality only when $P$ is the centroid of $ABCD$. Here $d(P,XY)$ denotes the distance from point $P$ to line $XY$.

Let $(O)$ be a circle, and let $ABP$ be a line segment such that $A,B$ lie on $(O)$ and $P$ is a point outside $(O)$. Let $C$ be a point on $(O)$ such that $PC$ is tangent to $(O)$ and let $D$ be the point on $(O)$ such that $CD$ is a diameter of $(O)$ and intersects $AB$ inside $(O)$. Suppose that the lines $DB$ and $OP$ intersect at $E$. Prove that $AC$ is perpendicular to $CE$.

Let $ABCDEF$ be a hexagon inscribed in a circle $\Omega$ such that triangles $ACE$ and $BDF$ have the same orthocenter. Suppose that segments $BD$ and $DF$ intersect $CE$ at $X$ and $Y$, respectively. Show that there is a point common to $\Omega$, the circumcircle of $DXY$, and the line through $A$ perpendicular to $CE$. [i]Proposed by Michael Ren and Vincent Huang[/i]

Consider a triangle $ABC$, with the points $A_1$, $A_2$ on side $BC$, $B_1,B_2\in\overline{AC}$, $C_1,C_2\in\overline{AB}$ such that $AC_1<AC_2$, $BA_1<BA_2$, $CB_1<CB_2$. Let the circles $AB_1C_1$ and $AB_2C_2$ meet at $A$ and $A^*$. Similarly, let the circles $BC_1A_1$ and $BC_2A_2$ intersect at $B^*\neq B$, let $CA_1B_1$ and $CA_2B_2$ intersect at $C^*\neq C$. Prove that the lines $AA^*$, $BB^*$, $CC^*$ are concurrent.

Let $ABC$ be a triangle and $N$ be a point that differs from $A,B,C$. Let $A_b$ be the reflection of $A$ through $NB$, and $B_a$ be the reflection of $B$ through $NA$. Similarly, we define $B_c, C_b, A_c, C_a$. Let $m_a$ be the line through $N$ and perpendicular to $B_cC_b$. Define similarly $m_b, m_c$. a) Assume that $N$ is the orthocenter of $\triangle ABC$, show that the respective reflection of $m_a, m_b, m_c$ through the bisector of angles $\angle BNC, \angle CNA, \angle ANB$ are the same line. b) Assume that $N$ is the nine-point center of $\triangle ABC$, show that the respective reflection of $m_a, m_b, m_c$ through $BC, CA, AB$ concur.

Given three points $H_1, H_2, H_3$ on a plane. The points are the reflections of the intersection point of the heights of the triangle $\vartriangle ABC$ through its sides. Construct $\vartriangle ABC$.

Let $ABC$ be a non-right-angled triangle, with $AC\ne BC$. Let $F$ be the midpoint of side $BC$. Let $D$ be a point on line $AB$ satisfying$CA=CD$,and let $E$ be a point on line $BC$ satisfying $EB = ED$. The line passing through $A$ and parallel to $ED$ meets line $FD$ at point $I$. Line $AF$ meets line $ED$ at point $J$. Prove that points $C$, $I$ and $J$ are collinear.