Let $P$ and $Q$ be points on the sides $BC$ and $CD$ of a convex quadrilateral $ABCD$, respectively, such that $\angle{BAP}=\angle{ DAQ}$. Prove that the triangles $ABP$ and $ADQ$ have equal area if and only if the line joining their orthocenters is perpendicular to $AC.$

$ABCD$ is a trapezium where $AD\parallel BC$. $P$ is the point on the line $AB$ such that $\angle CPD$ is maximal. $Q$ is the point on the line $CD$ such that $\angle BQA$ is maximal. Given that $P$ lies on the segment $AB$, prove that $\angle CPD=\angle BQA$.

Let $ABC$ be an acute triangle. Let $M$ be the midpoint of side $BC$, and let $E$ and $F$ be the feet of the altitudes from $B$ and $C$, respectively. Suppose that the common external tangents to the circumcircles of triangles $BME$ and $CMF$ intersect at a point $K$, and that $K$ lies on the circumcircle of $ABC$. Prove that line $AK$ is perpendicular to line $BC$. [i]Kevin Cong[/i]

Given an acute triangle $ABC$ with altitudes $AD$ and $BE$ intersecting at $H$, $M$ is the midpoint of $AB$. A nine-point circle of $ABC$ intersects with a circumcircle of $ABH$ on $P$ and $Q$ where $P$ lays on the same side of $A$ (with respect to $CH$). Prove that $ED, PH, MQ$ are concurrent on circumcircle $ABC$

Ali Barber, the carpet merchant, has a rectangular piece of carpet whose dimensions are unknown. Unfortunately, his tape measure is broken and he has no other measuring instruments. However, he finds that if he lays it flat on the floor of either of his storerooms, then each corner of the carpet touches a different wall of that room. If the two rooms have dimensions of 38 feet by 55 feet and 50 feet by 55 feet, what are the carpet dimensions?

In acute triangle $ABC$, $D\in BC,E\in CA,F\in AB$. Prove that the necessary and sufficient condition of $AD,BE,CF$ are heights of $\triangle ABC$ is that $S=\frac{R}{2}(EF+FD+DE)$. Note: $S$ is the area of $\triangle ABC$, $R$ is the circumradius of $\triangle ABC$.

Prove that for the sides $a, b, c$, the angles $A, B, C$ and the area $S$ of the triangle holds $$\cot A+ \cot B + \cot C = \frac{a^2+b^2+c^2}{4S}.$$

In rectangle $PQRS$, $PQ=8$ and $QR=6$. Points $A$ and $B$ lie on $\overline{PQ}$, points $C$ and $D$ lie on $\overline{QR}$, points $E$ and $F$ lie on $\overline{RS}$, and points $G$ and $H$ lie on $\overline{SP}$ so that $AP=BQ<4$ and the convex octagon $ABCDEFGH$ is equilateral. The length of a side of this octagon can be expressed in the form $k+m\sqrt{n}$, where $k$, $m$, and $n$ are integers and $n$ is not divisible by the square of any prime. What is $k+m+n$? $\textbf{(A) } 1 \qquad \textbf{(B) } 7 \qquad \textbf{(C) } 21 \qquad \textbf{(D) } 92 \qquad \textbf{(E) } 106$

The triangle $ABC$ has perimeter $36$, and the length of $BC$ is $9$. Point $M$ is the midpoint of $AC$, and $I$ is the incenter. Find the angle $MIC$.

Let $ABC$ be a triangle with $\angle{A}=90^{\circ}$ and $AB<AC$. Let $AD$ be the altitude from $A$ on to $BC$, Let $P,Q$ and $I$ denote respectively the incentres of triangle $ABD,ACD$ and $ABC$. Prove that $AI$ is perpendicular to $PQ$ and $AI=PQ$.

The line $l$ passing through the orthocenter $H$ of a triangle $ABC$ $(BC>AB)$ and parallel to $AC$ meets $AB$ and $BC$ at points $D$ and $E$ respectively. The line passing through the circumcenter of the triangle and parallel to the median $BM$ meets $l$ at point $F$. Prove that the length of segment $HF$ is three times greater than the difference of $FE$ and $DH$ Proposed by: A.Mardanov, K.Mardanova

Let $ABCD$ a convex quadrilateral with $AB=BC=CD$, with $AC$ not equal to $BD$ and $E$ be the intersection point of it's diagonals. Prove that $AE=DE$ if and only if $\angle BAD+\angle ADC = 120$.

On a cellular plane with a cell side equal to $1$, arbitrarily $100 \times 100$ napkin is thrown. It covers some nodes (the node lying on the border of a napkin, is also considered covered). What is the smallest number of lines (going not necessarily along grid lines) you can certainly cover all these nodes?

A square has sides of length $ 10$, and a circle centered at one of its vertices has radius $ 10$. What is the area of the union of the regions enclosed by the square and the circle? $ \textbf{(A)}\ 200 \plus{} 25\pi\qquad \textbf{(B)}\ 100 \plus{} 75\pi\qquad \textbf{(C)}\ 75 \plus{} 100\pi\qquad \textbf{(D)}\ 100 \plus{} 100\pi$ $ \textbf{(E)}\ 100 \plus{} 125\pi$

Let $ABC$ be an acute-angled triangle with $AC > AB$, let $O$ be its circumcentre, and let $D$ be a point on the segment $BC$. The line through $D$ perpendicular to $BC$ intersects the lines $AO, AC,$ and $AB$ at $W, X,$ and $Y,$ respectively. The circumcircles of triangles $AXY$ and $ABC$ intersect again at $Z \ne A$. Prove that if $W \ne D$ and $OW = OD,$ then $DZ$ is tangent to the circle $AXY.$

In the figure, point $P, Q,R,S$ lies on the side of the rectangle $ABCD$. [img]https://1.bp.blogspot.com/-Ff9rMibTuHA/X9PRPbGVy-I/AAAAAAAAMzA/2ytG0aqe-k0fPL3hbSp_zHrMYAfU-1Y_ACLcBGAsYHQ/s426/2020%2BIndonedia%2BMO%2BProvince%2BP2%2Bq1.png[/img] If it is known that the area of the small square is $1$ unit, determine the area of the rectangle $ABCD$.

Let $ABC$ be a triangle. Points $P$ and $Q$ lie on side $BC$ and satisfy $|BP| =|PQ| = |QC| = \frac13 |BC|$. Points $R$ and $S$ lie on side $CA$ and satisfy $|CR| =|RS| = |SA| = 1 3 |CA|$. Finally, points $T$ and $U$ lie on side $AB$ and satisfy $|AT| = |TU| = |UB| =\frac13 |AB|$. Points $P, Q,R, S, T$ and $U$ turn out to lie on a common circle. Prove that $ABC$ is an equilateral triangle.

A finite set of unit circles is given in a plane such that the area of their union $U$ is $S$. Prove that there exists a subset of mutually disjoint circles such that the area of their union is greater that $\frac{2S}{9}.$

Two circles $\omega_1$ and $\omega_2$ intersecting at points $X{}$ and $Y{}$ are inside the circle $\Omega$ and touch it at points $A{}$ and $B{}$, respectively; the segments $AB$ and $XY$ intersect. The line $AB$ intersects the circles $\omega_1$ and $\omega_2$ again at points $C{}$ and $D{}$, respectively. The circle inscribed in the curved triangle $CDX$ touches the side $CD$ at the point $Z{}$. Prove that $XZ$ is a bisector of $\angle AXB{}$.

[b]p1.[/b] A very large lucky number $N$ consists of eighty-eight $8$s in a row. Find the remainder when this number $N$ is divided by $6$. [b]p2.[/b] If $3$ chickens can lay $9$ eggs in $4$ days, how many chickens does it take to lay $180$ eggs in $ 8$ days? [b]p3.[/b] Find the ordered pair $(x, y)$ of real numbers satisfying the conditions $x > y$, $x+y = 10$, and $xy = -119$. [b]p4.[/b] There is pair of similar triangles. One triangle has side lengths $4, 6$, and $9$. The other triangle has side lengths $ 8$, $12$ and $x$. Find the sum of two possible values of $x$. [b]p5.[/b] If $x^2 +\frac{1}{x^2} = 3$, there are two possible values of $x +\frac{1}{x}$. What is the smaller of the two values? [b]p6.[/b] Three flavors (chocolate strawberry, vanilla) of ice cream are sold at Brian’s ice cream shop. Brian’s friend Zerg gets a coupon for $10$ free scoops of ice cream. If the coupon requires Zerg to choose an even number of scoops of each flavor of ice cream, how many ways can he choose his ice cream scoops? (For example, he could have $6$ scoops of vanilla and $4$ scoops of chocolate. The order in which Zerg eats the scoops does not matter.) [b]p7.[/b] David decides he wants to join the West African Drumming Ensemble, and thus he goes to the store and buys three large cylindrical drums. In order to ensure none of the drums drop on the way home, he ties a rope around all of the drums at their mid sections so that each drum is next to the other two. Suppose that each drum has a diameter of $3.5$ feet. David needs $m$ feet of rope. Given that $m = a\pi + b$, where $a$ and $b$ are rational numbers, find sum $a + b$. [b]p8.[/b] Segment $AB$ is the diameter of a semicircle of radius $24$. A beam of light is shot from a point $12\sqrt3$ from the center of the semicircle, and perpendicular to $AB$. How many times does it reflect off the semicircle before hitting $AB$ again? [b]p9.[/b] A cube is inscribed in a sphere of radius $ 8$. A smaller sphere is inscribed in the same sphere such that it is externally tangent to one face of the cube and internally tangent to the larger sphere. The maximum value of the ratio of the volume of the smaller sphere to the volume of the larger sphere can be written in the form $\frac{a-\sqrt{b}}{36}$ , where $a$ and $b$ are positive integers. Find the product $ab$. [b]p10.[/b] How many ordered pairs $(x, y)$ of integers are there such that $2xy + x + y = 52$? [b]p11.[/b] Three musketeers looted a caravan and walked off with a chest full of coins. During the night, the first musketeer divided the coins into three equal piles, with one coin left over. He threw it into the ocean and took one of the piles for himself, then went back to sleep. The second musketeer woke up an hour later. He divided the remaining coins into three equal piles, and threw out the one coin that was left over. He took one of the piles and went back to sleep. The third musketeer woke up and divided the remaining coins into three equal piles, threw out the extra coin, and took one pile for himself. The next morning, the three musketeers gathered around to divide the coins into three equal piles. Strangely enough, they had one coin left over this time as well. What is the minimum number of coins that were originally in the chest? [b]p12.[/b] The diagram shows a rectangle that has been divided into ten squares of different sizes. The smallest square is $2 \times 2$ (marked with *). What is the area of the rectangle (which looks rather like a square itself)? [img]https://cdn.artofproblemsolving.com/attachments/4/a/7b8ebc1a9e3808096539154f0107f3e23d168b.png[/img] [b]p13.[/b] Let $A = (3, 2)$, $B = (0, 1)$, and $P$ be on the line $x + y = 0$. What is the minimum possible value of $AP + BP$? [b]p14.[/b] Mr. Mustafa the number man got a $6 \times x$ rectangular chess board for his birthday. Because he was bored, he wrote the numbers $1$ to $6x$ starting in the upper left corner and moving across row by row (so the number $x + 1$ is in the $2$nd row, $1$st column). Then, he wrote the same numbers starting in the upper left corner and moving down each column (so the number $7$ appears in the $1$st row, $2$nd column). He then added up the two numbers in each of the cells and found that some of the sums were repeated. Given that $x$ is less than or equal to $100$, how many possibilities are there for $x$? [b]p15.[/b] Six congruent equilateral triangles are arranged in the plane so that every triangle shares at least one whole edge with some other triangle. Find the number of distinct arrangements. (Two arrangements are considered the same if one can be rotated and/or reflected onto another.) PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The squares $ABMN$, $BCKL$ and $ACPQ$ are constructed outside triangle $ABC$. The difference between the areas of $AB MN$ and $BCKL$ is $d$. Find the difference between the areas of the squares with sides $NQ$ and $PK$ respectively, if $\angle ABC$ is (a) a right angle; (b) not necessarily a right angle. (A Gerko)

Let $M$ be an arbitrary point on side $BC$ of triangle $ABC$. $W$ is a circle which is tangent to $AB$ and $BM$ at $T$ and $K$ and is tangent to circumcircle of $AMC$ at $P$. Prove that if $TK||AM$, circumcircles of $APT$ and $KPC$ are tangent together.

Let $ABC$ be an acute-angled triangle with circumcircle $\omega$. Prove that there exists a point $J$ such that for any point $X$ inside $ABC$ if $AX,BX,CX$ intersect $\omega$ in $A_1,B_1,C_1$ and $A_2,B_2,C_2$ be reflections of $A_1,B_1,C_1$ in midpoints of $BC,AC,AB$ respectively then $A_2,B_2,C_2,J$ lie on a circle.

The quadrangle $ABCD$ is circumscribed around the circle with the centre $O$. Prove that $$\angle AOB+ \angle COD=180^o. $$

There is an integer in each cell of a $2m\times 2n$ table. We define the following operation: choose three cells forming an L-tromino (namely, a cell $C$ and two other cells sharing a side with $C$, one being horizontal and the other being vertical) and sum $1$ to each integer in the three chosen cells. Find a necessary and sufficient condition, in terms of $m$, $n$ and the initial numbers on the table, for which there exists a sequence of operations that makes all the numbers on the table equal.