Two equal rectangles of area $10$ are arranged as follows. Find the area of the gray rectangle. [img]https://cdn.artofproblemsolving.com/attachments/7/1/112b07530a2ef42e5b2cf83a2cb9fb11dfc9e6.png[/img]

Let $ABC$ be a scalene triangle. The tangents at the perpendicular foot dropped from $A$ on the line $BC$ and the midpoint of the side $BC$ to the nine-point circle meet at the point $A'$\,; the points $B'$ and $C'$ are defined similarly. Prove that the lines $AA'$, $BB'$ and $CC'$ are concurrent. [i]Gazeta Matematica[/i]

Let $ABC$ be an acute triangle with $AC \neq BC$. Points $H$ and $I$ are the orthocenter and incenter of the triangle, respectively. Line $CH$ and $CI$ meet the circumcircle of triangle $ABC$ again at $D$ and $L$ (other than $C$), respectively. Prove that $\angle CIH=90^{\circ}$ if and only if $\angle IDL=90^{\circ}$.

Let $B$ and $C$ be arbitrary points on sides $AP$ and $PD$ respectively of an acute triangle $APD$. The diagonals of the quadrilateral $ABCD$ meet at $Q$, and $H_1,H_2$ are the orthocenters of triangles $APD$ and $BPC$, respectively. Prove that if the line $H_1H_2$ passes through the intersection point $X \ (X \neq Q)$ of the circumcircles of triangles $ABQ$ and $CDQ$, then it also passes through the intersection point $Y \ (Y \neq Q)$ of the circumcircles of triangles $BCQ$ and $ADQ.$

In all triangles $ABC$ does it hold: $$(b^n+c^p)\tan^{n+p}\frac A2+(c^n+a^p)\tan^{n+p}\frac B2+(a^n+b^p)\tan^{n+p}\frac C2\ge6\sqrt{\left(\frac{4r^2}{R\sqrt3}\right)^{n+p}}$$ where $n,p\in(0,\infty)$. [i]Proposed by Nicolae Papacu[/i]

The triangle $ABC$ is right-angled at $A$. Its incentre is $I$, and $H$ is the foot of the perpendicular from $I$ on $AB$. The perpendicular from $H$ on $BC$ meets $BC$ at $E$, and it meets the bisector of $\angle ABC$ at $D$. The perpendicular from $A$ on $BC$ meets $BC$ at $F$. Prove that $\angle EFD = 45^o$

Let $a$ be a positive number. Assume that the parameterized curve $C: \ x=t+e^{at},\ y=-t+e^{at}\ (-\infty <t< \infty)$ is touched to $x$ axis. (1) Find the value of $a.$ (2) Find the area of the part which is surrounded by two straight lines $y=0, y=x$ and the curve $C.$

Let $C_1$ and $C_2$ be two concentric circles with $C_1$ inside $C_2$. Let $P_1$ and $P_2$ be two points on $C_1$ that are not diametrically opposite. Extend the segment $P_1P_2$ past $P_2$ until it meets the circle $C_2$ in $Q_2$. The tangent to $C_2$ at $Q_2$ and the tangent to $C_1$ at $P_1$ meet in a point $X$. Draw from X the second tangent to $C_2$ which meets $C_2$ at the point $Q_1$. Show that $P_1X$ bisects angle $Q_1P_1Q_2$.

For $\triangle ABC$ and its circumcircle $\omega$, draw the tangents at $B,C$ to $\omega$ meeting at $D$. Let the line $AD$ meet the circle with center $D$ and radius $DB$ at $E$ inside $\triangle ABC$. Let $F$ be the point on the extension of $EB$ and $G$ be the point on the segment $EC$ such that $\angle AFB=\angle AGE=\angle A$. Prove that the tangent at $A$ to the circumcircle of $\triangle AFG$ is parallel to $BC$. [i]Proposed by 61plus[/i]

Let $ABC$ be an acute triangle and let $M$ be the midpoint of $AC$. A circle $\omega$ passing through $B$ and $M$ meets the sides $AB$ and $BC$ at points $P$ and $Q$ respectively. Let $T$ be the point such that $BPTQ$ is a parallelogram. Suppose that $T$ lies on the circumcircle of $ABC$. Determine all possible values of $\frac{BT}{BM}$.

The point $M$ belongs to the side $[AC]$ of the acute-angle triangle $ABC$. Two circles are circumscribed around triangles $ABM$ and $BCM$ . What $M$ position corresponds to the minimal area of those circles intersection?

Let the circles $\omega$ and $\omega^ \prime$ intersect in $A$ and $B$. Tangent to circle$\omega$ at $A$ intersects$\omega^ \prime$ in $C$ and tangent to circle $\omega^ \prime$ at $A$ intersects $\omega$ in $D$. Suppose that $CD$ intersects$\omega$ and $\omega^ \prime$ in $E$ and $F$, respectively (assume that $E$ is between $F$ and $C$). The perpendicular to $AC$ from $E$ intersects $\omega^ \prime$ in point $P$ and perpendicular to $AD$ from $F$ intersects$\omega$ in point $Q$ (The points $A, P$ and $Q$ lie on the same side of the line $CD$). Prove that the points $A, P$ and $Q$ are collinear. Proposed by Mahdi Etesami Fard

$ABC$ is acute-angled triangle, $AA_1,CC_1$ are altitudes. $M$ is centroid. $M$ lies on circumcircle of $A_1BC_1$. Find all values of $\angle B$

Let $ P$ be any point in the interior of a $ \triangle ABC$.Prove That $ \frac{PA}{a}\plus{}\frac{PB}{b}\plus{}\frac{PC}{c}\ge \sqrt{3}$.

[b]p1.[/b] A Slavic dragon has three heads. A knight fights the dragon. If the knight cuts off one dragon’s head three new heads immediately grow. Is it possible that the dragon has $2018$ heads at some moment of the fight? [b]p2.[/b] Peter has two squares $3\times 3$ and $4\times 4$. He must cut one of them or both of them in no more than four parts in total. Is Peter able to assemble a square using all these parts? [b]p3.[/b] Usually, dad picks up Constantine after his music lessons and they drive home. However, today the lessons have ended earlier and Constantine started walking home. He met his dad $14$ minutes later and they drove home together. They arrived home $6$ minutes earlier than usually. Home many minutes earlier than usual have the lessons ended? Please, explain your answer. [b]p4.[/b] All positive integers from $1$ to $2018$ are written on a blackboard. First, Peter erased all numbers divisible by $7$. Then, Natalie erased all remaining numbers divisible by $11$. How many numbers did Natalie remove? Please, explain your answer. [b]p5.[/b] $30$ students took part in a mathematical competition consisting of four problems. $25$ students solved the first problem, $24$ students solved the second problem, $22$ students solved the third, and, finally, $21$ students solved the fourth. Show that there are at least two students who solved all four problems. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In triangle $ABC$, let points $M$ and $N$ be the midpoints of sides $AB$ and $BC$ respectively. It is known that the perimeter of the triangle $MBN$ is $12$ cm, and the perimeter of the quadrilateral $AMNC$ is $20$ cm. Find the length of the segment $MN$.

The points $X, \, \, Y$are selected on the sides $AB$ and $AD$ of the convex quadrilateral $ABCD$, respectively. Find the ratio $AX \, \,: \, \, BX$ if you know that $CX || DA$, $DX || CB$, $BY || CD$ and $CY || BA$.

The set $M$ was formed from the plane by removing three points $A, B, C$, which are vertices of the triangle. What is the smallest number of convex sets whose union is $M$? [hide=original wording] Množina M Vznikla z roviny vyjmutím tří bodů A, B, C, které jsou vrcholy trojúhelníka. Jaký je nejmenší počet konvexních množin, jejichž sjednocením je M?[/hide]

How many rearrangements of $abcd$ are there in which no two adjacent letters are also adjacent letters in the alphabet? For example, no such rearrangements could include either $ab$ or $ba$. $ \textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad\textbf{(E) }4 $

Let the circles $C_1, C_2$, and $C_3$ be orthogonal to the circle $C$ and intersect each other inside $C$ forming acute angles of measures $A, B$, and $C$. Show that $A + B +C < \pi.$

The angle bisector $[AD]$, the median $[BM]$ and the height $[CH]$ of the acute-angled triangle $ABC$ intersect in one point. Prove that the $\angle BAC> 45^o$.

$P$ is a point in the interior of acute triangle $ABC$. $D,E,F$ are the reflections of $P$ across $BC,CA,AB$ respectively. Rays $AP,BP,CP$ meet the circumcircle of $\triangle ABC$ at $L,M,N$ respectively. Prove that the circumcircles of $\triangle PDL,\triangle PEM,\triangle PFN$ meet at a point $T$ different from $P$.

Let $ABCD$ be a parallelogram such that $\angle{ABC} > 90^{\circ}$, and $\mathcal{L}$ the line perpendicular to $BC$ that passes through $B$. Suppose that the segment $CD$ does not intersect $\mathcal{L}$. Of all the circumferences that pass through $C$ and $D$, there is one that is tangent to $\mathcal{L}$ at $P$, and there is another one that is tangent to $\mathcal{L}$ at $Q$ (where $P \neq Q$). If $M$ is the midpoint of $AB$, prove that $\angle{PMD} = \angle{QMD}$.

Given a cube $ABCDA_1B_1C_1D_1$ with edge $5$. On the edge $BB_1$ of the cube , point $K$ such thath $BK=4$. a) Construct a cube section with the plane $a$ passing through the points $K$ and $C_1$ parallel to the diagonal $BD_1$. b) Find the angle between the plane $a$ and the plane $BB_1C_1$.

To the exterior of side $AB$ of square $ABCD$, we have drawn the regular triangle $ABE$. Point $A$ reflected on line $BE$ is $F$, and point $E$ reflected on line $BF$ is $G$. Let the perpendicular bisector of segment $FG$ meet segment $AD$ at $X$. Show that the circle centered at $X$ with radius $XA$ touches line$ FB$.