The points $X, Y,Z$ are marked on the sides $AB, BC,AC$ of the triangle $ABC$, respectively. Points $A',B', C'$ are on the $XZ, XY, YZ$ sides of the triangle $XYZ$, respectively, so that $\frac{AB}{A'B'} = \frac{AB}{A'B'} =\frac{BC}{B'C'}= 2$ and $ABB'A',BCC'B',ACC'A'$ are trapezoids in which the sides of the triangle $ABC$ are bases. a) Determine the ratio between the area of the trapezium $ABB'A'$ and the area of the triangle $A'B'X$. b) Determine the ratio between the area of the triangle $XYZ$ and the area of the triangle $ABC$.

Let $\omega$ be the circumcircle of a triangle $ABC$. Denote by $M$ and $N$ the midpoints of the sides $AB$ and $AC$, respectively, and denote by $T$ the midpoint of the arc $BC$ of $\omega$ not containing $A$. The circumcircles of the triangles $AMT$ and $ANT$ intersect the perpendicular bisectors of $AC$ and $AB$ at points $X$ and $Y$, respectively; assume that $X$ and $Y$ lie inside the triangle $ABC$. The lines $MN$ and $XY$ intersect at $K$. Prove that $KA=KT$.

Rectangle $ABCD$ is divided into four parts of equal area by five segments as shown in the figure, where $XY = YB + BC + CZ = ZW = WD + DA + AX$, and $PQ$ is parallel to $AB$. Find the length of $AB$ (in cm) if $BC = 19$ cm and $PQ = 87$ cm. [asy] size(250); pair A=origin, B=(96,0), C=(96,22), D=(0,22), W=(16,22), X=(20,0), Y=(80,0), Z=(76,22), P=(24,11), Q=(72,11); draw(P--X--A--D--W--P--Q--Y--B--C--Z--Q^^W--Z^^X--Y); dot(A^^B^^C^^D^^P^^Q^^W^^X^^Y^^Z); pair point=(48,11); label("$A$", A, SW); label("$B$", B, SE); label("$C$", C, NE); label("$D$", D, NW); label("$P$", P, dir(point--P)); label("$Q$", Q, dir(point--Q)); label("$W$", W, N); label("$X$", X, S); label("$Y$", Y, S); label("$Z$", Z, N);[/asy]

Let $ABCD$ be a trapezoid with bases $ AB$ and $CD$ $(AB>CD)$. Diagonals $AC$ and $BD$ intersect in point $ N$ and lines $AD$ and $BC$ intersect in point $ M$. The circumscribed circles of $ADN$ and $BCN$ intersect in point $ P$, different from point $ N$. Prove that the angles $AMP$ and $BMN$ are equal.

Point $M$ is the midpoint of the base $BC$ of trapezoid $ABCD$. On base $AD$, point $P$ is selected. Line $PM$ intersects line $DC$ at point $Q$, and the perpendicular from $P$ on the bases intersects line $BQ$ at point $K$. Prove that $\angle QBC = \angle KDA$.

Construct a trapezoid given the midpoints of the legs, the point of intersection of the diagonals and the foot of the perpendicular, drawn from this point on the larger base.

Let $\omega$ be incircle of $ABC$. $P$ and $Q$ are on $AB$ and $AC$, such that $PQ$ is parallel to $BC$ and is tangent to $\omega$. $AB,AC$ touch $\omega$ at $F,E$. Prove that if $M$ is midpoint of $PQ$, and $T$ is intersection point of $EF$ and $BC$, then $TM$ is tangent to $\omega$. [i]By Ali Khezeli[/i]

Let $ABCD$ be a trapezoid with bases $AB$ and $CD$. Points $P$ and $Q$ lie on diagonals $AC$ and $BD$, respectively and $\angle APD = \angle BQC$. Prove that $\angle AQD = \angle BPC$.

In trapezoid $ ABCD$, $ \overline{AB}$ and $ \overline{CD}$ are perpendicular to $ \overline{AD}$, with $ AB\plus{}CD\equal{}BC$, $ AB<CD$, and $ AD\equal{}7$. What is $ AB\cdot CD$? $ \textbf{(A)}\ 12 \qquad \textbf{(B)}\ 12.25 \qquad \textbf{(C)}\ 12.5 \qquad \textbf{(D)}\ 12.75 \qquad \textbf{(E)}\ 13$

Let $ABCD$ be a trapezoid with $AD \parallel BC$, $AD < BC$. Let $E$ be a point on the side $AB$ and $F$ be point on the side $CD$. The circle $(AEF)$ intersects the segment $AD$ again at $A_1$ and the circle $(CEF)$ intersects these segment $BC$ again at $C_1$. Prove that the lines $A_1 C_1$, $BD$ and $EF$ are concurrent. [i]Proposed by A. Kuznetsov[/i]

Let $O$ be the point of intersection of the diagonals of the trapezoid $ABCD$ with the bases $AB$ and $CD$. It is known that $\angle AOB = \angle DAB = 90^o$. On the sides $AD$ and $BC$ take the points $E$ and $F$ so that $EF\parallel AB$ and $EF = AD$. Find the angle $\angle AOE$.

Given that points $ D,E$ lie on the sidelines $ AB,BC$ of triangle $ ABC$, respectively, point $ P$ is in interior of triangle $ ABC$ such that $ PE \equal{} PC$ and $ \bigtriangleup DEP\sim \bigtriangleup PCA.$ Prove that $ BP$ is tangent of the circumcircle of triangle $ PAD.$

Given is a tetrahedron $ABCD$ in which $AB=CD$ and the sum of measures of the angles $BAD$ and $BCD$ equals $180$ degrees. Prove that the measure of the angle $BAD$ is larger than the measure of the angle $ADC$.

Let $C: y=\ln x$. For each positive integer $n$, denote by $A_n$ the area of the part enclosed by the line passing through two points $(n,\ \ln n),\ (n+1,\ \ln (n+1))$ and denote by $B_n$ that of the part enclosed by the tangent line at the point $(n,\ \ln n)$, $C$ and the line $x=n+1$. Let $g(x)=\ln (x+1)-\ln x$. (1) Express $A_n,\ B_n$ in terms of $n,\ g(n)$ respectively. (2) Find $\lim_{n\to\infty} n\{1-ng(n)\}$.

In trapezoid $ABCD$, $AB < CD$, $AB\perp BC$, $AB\parallel CD$, and the diagonals $AC$, $BD$ are perpendicular at point $P$. There is a point $Q$ on ray $CA$ past $A$ such that $QD\perp DC$. If \[\frac{QP} {AP}+\frac{AP} {QP} = \left( \frac{51}{14}\right)^4 - 2,\]then $\frac{BP} {AP}-\frac{AP}{BP}$ can be expressed in the form $\frac{m}{n}$ for relatively prime positive integers $m,n$. Compute $m+n$. [i]Ray Li.[/i]

Let $ ABCD $ be a quadrilateral, whose diagonals intersect at $ O $. The triangles $ \triangle AOB $, $ \triangle BOC $, $ \triangle COD $ have areas $1, 2, 4$, respectively. Find the area of $ \triangle AOD $ and prove that $ ABCD $ is a trapezoid.

Let $n=3k$ be a positive integer (with $k\geq 2$). An equilateral triangle is divided in $n^2$ unit equilateral triangles with sides parallel to the initial, forming a grid. We will call "trapezoid" the trapezoid which is formed by three equilateral triangles (one base is equal to one and the other is equal to two). We colour the points of the grid with three colours (red, blue and green) such that each two neighboring points have different colour. Finally, the colour of a "trapezoid" will be the colour of the midpoint of its big base. Find the number of all "trapezoids" in the grid (not necessarily disjoint) and determine the number of red, blue and green "trapezoids".

What is the perimeter of trapezoid $ ABCD$? [asy]defaultpen(linewidth(0.8));size(3inch, 1.5inch); pair a=(0,0), b=(18,24), c=(68,24), d=(75,0), f=(68,0), e=(18,0); draw(a--b--c--d--cycle); draw(b--e); draw(shift(0,2)*e--shift(2,2)*e--shift(2,0)*e); label("30", (9,12), W); label("50", (43,24), N); label("25", (71.5, 12), E); label("24", (18, 12), E); label("$A$", a, SW); label("$B$", b, N); label("$C$", c, N); label("$D$", d, SE); label("$E$", e, S);[/asy] $ \textbf{(A)}\ 180\qquad\textbf{(B)}\ 188\qquad\textbf{(C)}\ 196\qquad\textbf{(D)}\ 200\qquad\textbf{(E)}\ 204 $

Find the least positive integer $n$ ($n\geq 3$), such that among any $n$ points (no three are collinear) in the plane, there exist three points which are the vertices of a non-isoscele triangle.

Let $A,B$ and $C$ be three points on a line with $B$ between $A$ and $C$. Let $\Gamma_1,\Gamma_2, \Gamma_3$ be semicircles, all on the same side of $AC$ and with $AC,AB,BC$ as diameters, respectively. Let $l$ be the line perpendicular to $AC$ through $B$. Let $\Gamma$ be the circle which is tangent to the line $l$, tangent to $\Gamma_1$ internally, and tangent to $\Gamma_3$ externally. Let $D$ be the point of contact of $\Gamma$ and $\Gamma_3$. The diameter of $\Gamma$ through $D$ meets $l$ in $E$. Show that $AB=DE$.

Given a trapezoid $ ABCD$ with $ AD\parallel BC, E$ is a moving point on the side $ AB,$ let $ O_{1},O_{2}$ be the circumcenters of triangles $ AED,BEC$, respectively. Prove that the length of $ O_{1}O_{2}$ is a constant value.

Let $ ABCDE$ be convex pentagon such that $ S(ABC) \equal{} S(BCD) \equal{} S(CDE) \equal{} S(DEA) \equal{} S(EAB)$. Prove that there is a point $ M$ inside pentagon such that $ S(MAB) \equal{} S(MBC) \equal{} S(MCD) \equal{} S(MDE) \equal{} S(MEA)$.

In the trapezoid $ABCD, AB // CD$ and the diagonals intersect at $O$. The points $P, Q$ are on $AD, BC$ respectively such that $\angle AP B = \angle CP D$ and $\angle AQB = \angle CQD$. Show that $OP = OQ$.

The orthogonal projection of a tetrahedron onto a plane containing one of its faces is a trapezoid of area $1$, which has only one pair of parallel sides. a) Is it possible that the orthogonal projection of this tetrahedron onto a plane containing another its face is a square of area $1$? b) The same question for a square of area $1/2019$. (Mikhail Evdokimov)

Let $ ABCD$ be a trapezoid with $ AB\parallel{}CD$, $ AB\equal{}11$, $ BC\equal{}5$, $ CD\equal{}19$, and $ DA\equal{}7$. Bisectors of $ \angle A$ and $ \angle D$ meet at $ P$, and bisectors of $ \angle B$ and $ \angle C$ meet at $ Q$. What is the area of hexagon $ ABQCDP$? $ \textbf{(A)}\ 28\sqrt{3}\qquad \textbf{(B)}\ 30\sqrt{3}\qquad \textbf{(C)}\ 32\sqrt{3}\qquad \textbf{(D)}\ 35\sqrt{3}\qquad \textbf{(E)}\ 36\sqrt{3}$