The base of the pyramid is a triangle $ABC$, in which $\angle ACB= 30^o$, and the length of the median from the vertex $B$ is twice less than the side $AC$ and is equal to $\alpha$ . All side edges of the pyramid are inclined to the plane of the base at an angle $a$. Determine the cross-sectional area of ​​the pyramid with a plane passing through the vertex $B$ parallel to the edge $AD$ and inclined to the plane of the base at an angle of $\beta$,

There is a point $D$ on side $AC$ of acute triangle $\triangle ABC$. Let $AM$ be the median drawn from $A$ (so $M$ is on $BC$) and $CH$ be the altitude drawn from $C$ (so $H$ is on $AB$). Let $I$ be the intersection of $AM$ and $CH$, and let $K$ be the intersection of $AM$ and line segment $BD$. We know that $AK=8$, $BK=8$, and $MK=6$. Find the length of $AI$.

In triangle $ABC$ we inscribe a square such that one of the sides of the square lies on the side $AC$, and the other two vertices lie on sides $AB$ and $BC$. Furthermore we know that $AC = 5$, $BC = 4$ and $AB = 3$. This square cuts out three smaller triangles from $\vartriangle ABC$. Express the sum of reciprocals of the inradii of these three small triangles as a fraction $p/q$ in lowest terms (i.e. with $p$ and $q$ coprime). What is $p + q$?

Let $ ABCD$ be a trapezoid inscribed in a circle $ \mathcal{C}$ with $ AB\parallel CD$, $ AB\equal{}2CD$. Let $ \{Q\}\equal{}AD\cap BC$ and let $ P$ be the intersection of tangents to $ \mathcal{C}$ at $ B$ and $ D$. Calculate the area of the quadrilateral $ ABPQ$ in terms of the area of the triangle $ PDQ$.

Let $ABC$ be a triangle such that midpoints of three altitudes are collinear. If the largest side of triangle is $10$, what is the largest possible area of the triangle? $ \textbf{(A)}\ 20 \qquad\textbf{(B)}\ 25 \qquad\textbf{(C)}\ 30 \qquad\textbf{(D)}\ 40 \qquad\textbf{(E)}\ 50 $

Let $\triangle{ABC}$ be an equilateral triangle. Let $E_{AB}$ be the ellipse with foci $A, B$ passing through $C,$ and in the parallel manner define $E_{BC}, E_{AC}.$ Let $\triangle{GHI}$ be a (nondegenerate) triangle with vertices where two ellipses intersect such that the edges of $\triangle{GHI}$ do not intersect those of $\triangle{ABC}.$ Compute the ratio of the largest sides of $\triangle{GHI}$ and $\triangle{ABC}.$

Given two circles ${\omega _1},{\omega _2}$, $S$ denotes all $\Delta ABC$ satisfies that ${\omega _1}$ is the circumcircle of $\Delta ABC$, ${\omega _2}$ is the $A$- excircle of $\Delta ABC$ , ${\omega _2}$ touches $BC,CA,AB$ at $D,E,F$. $S$ is not empty, prove that the centroid of $\Delta DEF$ is a fixed point.

Given a parabola $C : y=1-x^2$ in $xy$-palne with the origin $O$. Take two points $P(p,\ 1-p^2),\ Q(q,\ 1-q^2)\ (p<q)$ on $C$. (1) Express the area $S$ of the part enclosed by two segments $OP,\ OQ$ and the parabalola $C$ in terms of $p,\ q$. (2) If $q=p+1$, then find the minimum value of $S$. (3) If $pq=-1$, then find the minimum value of $S$.

Let $ABCD$ be a parallelogram with $\angle ABC = 105^\circ$. Inside the parallelogram, there is a point $E$ such that triangle $BEC$ is equilateral and $\angle CED = 135^\circ$. Let $K$ be the midpoint of side $AB$. Determine the measure of angle $\angle BKC$.

Let $ M,N,P $ on the sides $ AB,BC,CA, $ respectively, of a triangle $ ABC $ such that $ AM=BN=CP $ and such that $$ AB\cdot \overrightarrow{AT} +BC\cdot \overrightarrow{BT} +CA\cdot \overrightarrow{CT} =0, $$ where $ T $ is the centroid of $ MNP. $ Prove that $ ABC $ is equilateral.

In trapezoid $ABCD$, $AD$ is parallel to $BC$. Knowing that $AB=AD+BC$, prove that the bisector of $\angle A$ also bisects $CD$.

[u]Set 1[/u] [b]B1.[/b] Evaluate $2 + 0 - 2 \times 0$. [b]B2.[/b] It takes four painters four hours to paint four houses. How many hours does it take forty painters to paint forty houses? [b]B3.[/b] Let $a$ be the answer to this question. What is $\frac{1}{2-a}$? [u]Set 2[/u] [b]B4.[/b] Every day at Andover is either sunny or rainy. If today is sunny, there is a $60\%$ chance that tomorrow is sunny and a $40\%$ chance that tomorrow is rainy. On the other hand, if today is rainy, there is a $60\%$ chance that tomorrow is rainy and a $40\%$ chance that tomorrow is sunny. Given that today is sunny, the probability that the day after tomorrow is sunny can be expressed as n%, where n is a positive integer. What is $n$? [b]B5.[/b] In the diagram below, what is the value of $\angle DD'Y$ in degrees? [img]https://cdn.artofproblemsolving.com/attachments/0/8/6c966b13c840fa1885948d0e4ad598f36bee9d.png[/img] [b]B6.[/b] Christina, Jeremy, Will, and Nathan are standing in a line. In how many ways can they be arranged such that Christina is to the left of Will and Jeremy is to the left of Nathan? Note: Christina does not have to be next to Will and Jeremy does not have to be next to Nathan. For example, arranging them as Christina, Jeremy, Will, Nathan would be valid. [u]Set 3[/u] [b]B7.[/b] Let $P$ be a point on side $AB$ of square $ABCD$ with side length $8$ such that $PA = 3$. Let $Q$ be a point on side $AD$ such that $P Q \perp P C$. The area of quadrilateral $PQDB$ can be expressed in the form $m/n$ for relatively prime positive integers $m$ and $n$. Compute $m + n$. [b]B8.[/b] Jessica and Jeffrey each pick a number uniformly at random from the set $\{1, 2, 3, 4, 5\}$ (they could pick the same number). If Jessica’s number is $x$ and Jeffrey’s number is $y$, the probability that $x^y$ has a units digit of $1$ can be expressed as $m/n$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [b]B9.[/b] For two points $(x_1, y_1)$ and $(x_2, y_2)$ in the plane, we define the taxicab distance between them as $|x_1 - x_2| + |y_1 - y_2|$. For example, the taxicab distance between $(-1, 2)$ and $(3,\sqrt2)$ is $6-\sqrt2$. What is the largest number of points Nathan can find in the plane such that the taxicab distance between any two of the points is the same? [u]Set 4[/u] [b]B10.[/b] Will wants to insert some × symbols between the following numbers: $$1\,\,\,2\,\,\,3\,\,\,4\,\,\,6$$ to see what kinds of answers he can get. For example, here is one way he can insert $\times$ symbols: $$1 \times 23 \times 4 \times 6 = 552.$$ Will discovers that he can obtain the number $276$. What is the sum of the numbers that he multiplied together to get $276$? [b]B11.[/b] Let $ABCD$ be a parallelogram with $AB = 5$, $BC = 3$, and $\angle BAD = 60^o$ . Let the angle bisector of $\angle ADC$ meet $AC$ at $E$ and $AB$ at $F$. The length $EF$ can be expressed as $m/n$, where $m$ and $n$ are relatively prime positive integers. What is $m + n$? [b]B12.[/b] Find the sum of all positive integers $n$ such that $\lfloor \sqrt{n^2 - 2n + 19} \rfloor = n$. Note: $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$. [u]Set 5[/u] [b]B13.[/b] This year, February $29$ fell on a Saturday. What is the next year in which February $29$ will be a Saturday? [b]B14.[/b] Let $f(x) = \frac{1}{x} - 1$. Evaluate $$f\left( \frac{1}{2020}\right) \times f\left( \frac{2}{2020}\right) \times f\left( \frac{3}{2020}\right) \times \times ... \times f\left( \frac{2019}{2020}\right) .$$ [b]B15.[/b] Square $WXYZ$ is inscribed in square $ABCD$ with side length $1$ such that $W$ is on $AB$, $X$ is on $BC$, $Y$ is on $CD$, and $Z$ is on $DA$. Line $W Y$ hits $AD$ and $BC$ at points $P$ and $R$ respectively, and line $XZ$ hits $AB$ and $CD$ at points $Q$ and $S$ respectively. If the area of $WXYZ$ is $\frac{13}{18}$ , then the area of $PQRS$ can be expressed as $m/n$ for relatively prime positive integers $m$ and $n$. What is $m + n$? PS. You had better use hide for answers. Last sets have been posted [url=https://artofproblemsolving.com/community/c4h2777424p24371574]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $\triangle ABC$ be an acute-angled triangle with orthocentre $H$ and circumcentre $O$. Let $R$ be the radius of the circumcircle. \begin{align*} \text{Let }\mathit{A'}\text{ be the point on }\mathit{AO}\text{ (extended if necessary) for which }\mathit{HA'}\perp\mathit{AO}. \\ \text{Let }\mathit{B'}\text{ be the point on }\mathit{BO}\text{ (extended if necessary) for which }\mathit{HB'}\perp\mathit{BO}. \\ \text{Let }\mathit{C'}\text{ be the point on }\mathit{CO}\text{ (extended if necessary) for which }\mathit{HC'}\perp\mathit{CO}.\end{align*} Prove that $HA'+HB'+HC'<2R$ (Note: The orthocentre of a triangle is the intersection of the three altitudes of the triangle. The circumcircle of a triangle is the circle passing through the triangle’s three vertices. The circummcentre is the centre of the circumcircle.)

In acute-angled triangle $ABC$, altitude of $A$ meets $BC$ at $D$, and $M$ is midpoint of $AC$. Suppose that $X$ is a point such that $\measuredangle AXB = \measuredangle DXM =90^\circ$ (assume that $X$ and $C$ lie on opposite sides of the line $BM$). Show that $\measuredangle XMB = 2\measuredangle MBC$.Proposed by Davood Vakili

In a convex quadrilateral $ABCD$, $\angle ABC = \angle CDA$. Let $X \neq C$ be the intersection of the circumcircle of $\triangle BCD$ and circle with diameter $AC$. Prove that the tangent to the circumcircle of $\triangle BCD$ at $X$, the tangent to the circumcircle of $\triangle ABD$ at $A$ concur on $BD$.

Let $ABC$ be a triangle each of whose angles is greater than $30^{\circ}$. Suppose a circle centered with $P$ cuts segments $BC$ in $T,Q; CA$ in $K,L$ and $AB$ in $M,N$ such that they are on a circle in counterclockwise direction in that order.Suppose further $PQK,PLM,PNT$ are equilateral. Prove that: $a)$ The radius of the circle is $\frac{2abc}{a^2+b^2+c^2+4\sqrt{3}S}$ where $S$ is area. $b) a\cdot AP=b\cdot BP=c\cdot PC.$

We are given $3n$ points $A_1,A_2, \ldots , A_{3n}$ in the plane, no three of them collinear. Prove that one can construct $n$ disjoint triangles with vertices at the points $A_i.$

[b]Problem Section #4 b) Let $A$ be a unit square. What is the largest area of a triangle whose vertices lie on the perimeter of $A$? Justify your answer.

Let $ABC$ and $DBC$ be triangles with incircles touching at a point $P$ on $BC.$ Points $A,D$ lie on the same side of $BC$ and $DB < AB < DC < AC.$ The bisector of $\angle BDC$ meets line $AP$ at $X,$ and the altitude from $A$ meets $DP$ at $Y.$ Point $Z$ lies on line $XY$ so $ZP \perp BC.$ Show the reflection of $A$ over $BC$ is on line $ZD.$ [i]Proposed by squareman (Evan Chang), USA[/i]

[b]p1.[/b] Compute $2020 \cdot \left( 2^{(0\cdot1)} + 9 - \frac{(20^1)}{8}\right)$. [b]p2.[/b] Nathan has five distinct shirts, three distinct pairs of pants, and four distinct pairs of shoes. If an “outfit” has a shirt, pair of pants, and a pair of shoes, how many distinct outfits can Nathan make? [b]p3.[/b] Let $ABCD$ be a rhombus such that $\vartriangle ABD$ and $\vartriangle BCD$ are equilateral triangles. Find the angle measure of $\angle ACD$ in degrees. [b]p4.[/b] Find the units digit of $2019^{2019}$. [b]p5.[/b] Determine the number of ways to color the four vertices of a square red, white, or blue if two colorings that can be turned into each other by rotations and reflections are considered the same. [b]p6.[/b] Kathy rolls two fair dice numbered from $1$ to $6$. At least one of them comes up as a $4$ or $5$. Compute the probability that the sumof the numbers of the two dice is at least $10$. [b]p7.[/b] Find the number of ordered pairs of positive integers $(x, y)$ such that $20x +19y = 2019$. [b]p8.[/b] Let $p$ be a prime number such that both $2p -1$ and $10p -1$ are prime numbers. Find the sum of all possible values of $p$. [b]p9.[/b] In a square $ABCD$ with side length $10$, let $E$ be the intersection of $AC$ and $BD$. There is a circle inscribed in triangle $ABE$ with radius $r$ and a circle circumscribed around triangle $ABE$ with radius $R$. Compute $R -r$ . [b]p10.[/b] The fraction $\frac{13}{37 \cdot 77}$ can be written as a repeating decimal $0.a_1a_2...a_{n-1}a_n$ with $n$ digits in its shortest repeating decimal representation. Find $a_1 +a_2 +...+a_{n-1}+a_n$. [b]p11.[/b] Let point $E$ be the midpoint of segment $AB$ of length $12$. Linda the ant is sitting at $A$. If there is a circle $O$ of radius $3$ centered at $E$, compute the length of the shortest path Linda can take from $A$ to $B$ if she can’t cross the circumference of $O$. [b]p12.[/b] Euhan and Minjune are playing tennis. The first one to reach $25$ points wins. Every point ends with Euhan calling the ball in or out. If the ball is called in, Minjune receives a point. If the ball is called out, Euhan receives a point. Euhan always makes the right call when the ball is out. However, he has a $\frac34$ chance of making the right call when the ball is in, meaning that he has a $\frac14$ chance of calling a ball out when it is in. The probability that the ball is in is equal to the probability that the ball is out. If Euhan won, determine the expected number of wrong callsmade by Euhan. [b]p13.[/b] Find the number of subsets of $\{1, 2, 3, 4, 5, 6,7\}$ which contain four consecutive numbers. [b]p14.[/b] Ezra and Richard are playing a game which consists of a series of rounds. In each round, one of either Ezra or Richard receives a point. When one of either Ezra or Richard has three more points than the other, he is declared the winner. Find the number of games which last eleven rounds. Two games are considered distinct if there exists a round in which the two games had different outcomes. [b]p15.[/b] There are $10$ distinct subway lines in Boston, each of which consists of a path of stations. Using any $9$ lines, any pair of stations are connected. However, among any $8$ lines there exists a pair of stations that cannot be reached from one another. It happens that the number of stations is minimized so this property is satisfied. What is the average number of stations that each line passes through? [b]p16.[/b] There exist positive integers $k$ and $3\nmid m$ for which $$1 -\frac12 + \frac13 - \frac14 +...+ \frac{1}{53}-\frac{1}{54}+\frac{1}{55}=\frac{3^k \times m}{28\times 29\times ... \times 54\times 55}.$$ Find the value $k$. [b]p17.[/b] Geronimo the giraffe is removing pellets from a box without replacement. There are $5$ red pellets, $10$ blue pellets, and $15$ white pellets. Determine the probability that all of the red pellets are removed before all the blue pellets and before all of the white pellets are removed. [b]p18.[/b] Find the remainder when $$70! \left( \frac{1}{4 \times 67}+ \frac{1}{5 \times 66}+...+ \frac{1}{66\times 5}+ \frac{1}{67\times 4} \right)$$ is divided by $71$. [b]p19.[/b] Let $A_1A_2...A_{12}$ be the regular dodecagon. Let $X$ be the intersection of $A_1A_2$ and $A_5A_{11}$. Given that $X A_2 \cdot A_1A_2 = 10$, find the area of dodecagon. [b]p20.[/b] Evaluate the following infinite series: $$\sum^{\infty}_{n=1}\sum^{\infty}_{m=1} \frac{n \sec^2m -m \tan^2 n}{3^{m+n}(m+n)}$$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In convex pentagon $ABCDE, AC$ is parallel to $DE, AB$ is perpendicular to $AE$, and $BC$ is perpendicular to $CD$. If $H$ is the orthocentre of triangle $ABC$ and $M$ is the midpoint of segment $DE$, prove that $AD, CE$ and $HM$ are concurrent.

Points $A$, $B$ and $C$ lie on one side of the angle with the vertex at point $O$, and points $A'$, $B'$ and $C'$ lie on the other. It is known that$ B$ is the midpoint of the segment $AC$, $B'$ is the midpoint of the segment $A'C'$, and lines $AA'$, $BB'$ and $CC'$ are parallel (fig.). Prove that the centers of the circles circumscribed around the triangles $OAC$, $OA'C$ and $OBB'$ lie on the same straight line. [img]https://cdn.artofproblemsolving.com/attachments/d/6/92831077781bc45f25e9f71077034f84753a59.png[/img]

Let $ H$ be the piont of midpoint of the cord $ PQ$ that is on the circle centered the origin $ O$ with radius $ 1.$ Suppose the length of the cord $ PQ$ is $ 2\sin \frac {t}{2}$ for the angle $ t\ (0\leq t\leq \pi)$ that is formed by half-ray $ OH$ and the positive direction of the $ x$ axis. Answer the following questions. (1) Express the coordiante of $ H$ in terms of $ t$. (2) When $ t$ moves in the range of $ 0\leq t\leq \pi$, find the minimum value of $ x$ coordinate of $ H$. (3) When $ t$ moves in the range of $ 0\leq t\leq \frac {\pi}{2}$, find the area $ S$ of the region bounded by the curve drawn by the point $ H$ and the $ x$ axis and the $ y$ axis.

Archimedes drew a square with side length $36$ cm into the sand and he also drew a circle of radius $36$ cm around each vertex of the square. If the total area of the grey parts is $n \cdot \pi$ cm$^2$, what is the value of $n$? [i]Do not disturb my circles![/i] [img]https://cdn.artofproblemsolving.com/attachments/e/7/a755007990625c74fc2e59b999f0a3eddb2371.png[/img]

Solve in none-negative integers ${x^3} + 7{x^2} + 35x + 27 = {y^3}$.