If possible, construct an equilateral triangle whose three vertices are on three given circles.

Let $M$ be the point of intersection of diagonals $AC$ and $BD$ of the convex quadrilateral $ABCD$. Let $K$ be the point of intersection of the extension of side $AB$ (beyond$A$) with the bisector of the angle $ACD$. Let $L$ be the intersection of $KC$ and $BD$. If $MA \cdot CD = MB \cdot LD$, prove that the angle $BKC$ is equal to the angle $CDB$.

Triangle $ABC$ lies in a regular decagon as shown in the figure. What is the ratio of the area of the triangle to the area of the entire decagon? Write the answer as a fraction of integers. [img]https://1.bp.blogspot.com/-Ld_-4u-VQ5o/Xzb-KxPX0wI/AAAAAAAAMWg/-qPtaI_04CQ3vvVc1wDTj3SoonocpAzBQCLcBGAsYHQ/s0/2007%2BMohr%2Bp1.png[/img]

[b]p1.[/b] At a certain point in time, $20\%$ of seniors, $30\%$ of juniors, and $50\%$ of sophomores at a school had a cold. If the number of sick students was the same for each grade, the fraction of sick students across all three grades can be written as $\frac{a}{b}$ , where a and b are relatively prime positive integers. Find $a + b$. [b]p2.[/b] The average score on Mr. Feng’s recent test is a $63$ out of $100$. After two students drop out of the class, the average score of the remaining students on that test is now a $72$. What is the maximum number of students that could initially have been in Mr. Feng’s class? (All of the scores on the test are integers between $0$ and $100$, inclusive.) [b]p3.[/b] Madeline is climbing Celeste Mountain. She starts at $(0, 0)$ on the coordinate plane and wants to reach the summit at $(7, 4)$. Every hour, she moves either $1$ unit up or $1$ unit to the right. A strawberry is located at each of $(1, 1)$ and $(4, 3)$. How many paths can Madeline take so that she encounters exactly one strawberry? [b]p4.[/b] Let $E$ be a point on side $AD$ of rectangle $ABCD$. Given that $AB = 3$, $AE = 4$, and $\angle BEC = \angle CED$, the length of segment $CE$ can be written as $\sqrt{a}$ for some positive integer $a$. Find $a$. [b]p5.[/b] Lucy has some spare change. If she were to convert it into quarters and pennies, the minimum number of coins she would need is $66$. If she were to convert it into dimes and pennies, the minimum number of coins she would need is $147$. How much money, in cents, does Lucy have? [b]p6.[/b] For how many positive integers $x$ does there exist a triangle with altitudes of length $20$, $22$, and $x$? [b]p7.[/b] Compute the number of positive integers $x$ for which $\frac{x^{20}}{x+22}$ is an integer. [b]p8.[/b] Vincent the Bug is crawling along an octagonal prism. He starts on a fixed vertex $A$, visits all other vertices exactly once by traveling along the edges, and returns to $A$. Find the number of paths Vincent could have taken. [b]p9.[/b] Point $U$ is chosen inside square $ALEX$ so that $\angle AUL = 90^o$. Given that $UL = 56$ and $UE = 65$, what is the sum of all possible values for the area of square $ALEX$? [b]p10.[/b] Miranda has prepared $8$ outfits, no two of which are the same quality. She asks her intern Andrea to order these outfits for the new runway show. Andrea first randomly orders the outfits in a list. She then starts removing outfits according to the following method: she chooses a random outfit which is both immediately preceded and immediately succeeded by a better outfit and then removes it. Andrea repeats this process until there are no outfits that can be removed. Given that the expected number of outfits in the final routine can be written as $\frac{a}{b}$ for some relatively prime positive integers $a$ and $b$, find $a + b$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Consider a convex $n$-gon $A_1A_2 \dots A_n$. (Note: In a convex polygon, all interior angles are less than $180 \circ$.) Let $h$ be a positive number. Using the sides of the polygon as bases, we draw $n$ rectangles, each of height $h$, so that each rectangle is either entirely inside the $n$-gon or partially overlaps the inside of the $n$-gon. As an example, the left figure below shows a pentagon with a correct configuration of rectangles, while the right figure shows an incorrect configuration of rectangles (since some of the rectangles do not overlap with the pentagon):

Yamin and Tamim are playing a game with subsets of $\{1, 2, \ldots, n\}$ where $n \geq 3$. [list] [*] Tamim starts the game with the empty set. [*] On Yamin's turn, he adds a proper non-empty subset of $\{1, 2, \ldots, n\}$ to his collection $F$ of blocked sets. [*] On Tamim's turn, he adds or removes a positive integer less than or equal to $n$ to or from their set but Tamim can never add or remove an element so that his set becomes one of the blocked sets in $F$. [/list] Tamim wins if he can make his set to be $\{1, 2, \ldots, n\}$. Yamin wins if he can stop Tamim from doing so. Yamin goes first and they alternate making their moves. Does Tamim have a winning strategy? [i]Proposed by Ahmed Ittihad Hasib[/i]

Calculate the following indefinite integrals. [1] $\int \tan ^ 3 x dx$ [2] $\int a^{mx+n}dx\ (a>0,a\neq 1, mn\neq 0)$ [3] $\int \cos ^ 5 x dx$ [4] $\int \sin ^ 2 x\cos ^ 3 x dx$ [5]$ \int \frac{dx}{\sin x}$

An equilateral triangle has has sides $1$ inch long. An ant walks around the triangle maintaining a distance of $1$ inch from the triangle at all times. How far does the ant walk?

find all the integer $n\geq1$ such that $\lfloor\sqrt{n}\rfloor \mid n$

Prove that there are not intgers $a$ and $b$ with conditions, i) $16a-9b$ is a prime number. ii) $ab$ is a perfect square. iii) $a+b$ is also perfect square.

The angle $POQ$ is given ($OP$ and $OQ$ are rays). Let $M$ and $N$ be points inside the angle $POQ$ such that $\angle POM = \angle QON$ and $\angle POM < \angle PON$. Consider two circles: one touches the rays $OP$ and $ON$, the other touches the rays $OM$ and $OQ$. Denote by $B$ and $C$ the points of their intersection. Prove that $\angle POC = \angle QOB$.

Let $ABCD$ be a quadrilateral circumscribed around a circle $\omega$ with center $I$. Assume $P$ and $Q$ are distinct points and are isogonal conjugates such that $P, Q$, and $I$ are collinear. Show that $ABCD$ is a kite, that is, it has two disjoint pairs of consecutive equal sides.

Evaluate $ \int_0^1 \frac{t}{(1\plus{}t^2)(1\plus{}2t\minus{}t^2)}\ dt$.

The set of real values $a$ such that the equation $x^4-3ax^3+(2a^2+4a)x^2-5a^2x+3a^2$ has exactly two nonreal solutions is the set of real numbers between $x$ and $y,$ where $x<y.$ If $x+y$ can be written as $\tfrac{m}{n}$ for relatively prime positive integers $m,n,$ find $m+n.$

In the coordinate plane the points with both coordinates being rational numbers are called rational points. For any positive integer $n$, is there a way to use $n$ colours to colour all rational points, every point is coloured one colour, such that any line segment with both endpoints being rational points contains the rational points of every colour?

Given triangle $ABC$ with $A$-excenter $I_A$, the foot of the perpendicular from $I_A$ to $BC$ is $D$. Let the midpoint of segment $I_AD$ be $M$, $T$ lies on arc $BC$(not containing $A$) satisfying $\angle BAT=\angle DAC$, $I_AT$ intersects the circumcircle of $ABC$ at $S\neq T$. If $SM$ and $BC$ intersect at $X$, the perpendicular bisector of $AD$ intersects $AC,AB$ at $Y,Z$ respectively, prove that $AX,BY,CZ$ are concurrent.

Let $ \Gamma(I,r)$ and $ \Gamma(O,R)$ denote the incircle and circumcircle, respectively, of a triangle $ ABC$. Consider all the triangels $ A_iB_iC_i$ which are simultaneously inscribed in $ \Gamma(O,R)$ and circumscribed to $ \Gamma(I,r)$. Prove that the centroids of these triangles are concyclic.

Positive numbers $a,\ b,\ c$ satisfy $\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}=3$. Prove the inequality \[\dfrac{1}{\sqrt{a^3+1}}+\dfrac{1}{\sqrt{b^3+1}}+\dfrac{1}{\sqrt{c^3+1}}\le \dfrac{3}{\sqrt{2}}. \] [i](N. Alexandrov)[/i]

$\boxed{A4}$Let $m_1,m_2,m_3,n_1,n_2$ and $n_3$ be positive real numbers such that \[(m_1-n_1)(m_2-n_2)(m_3-n_3)=m_1m_2m_3-n_1n_2n_3\] Prove that \[(m_1+n_1)(m_2+n_2)(m_3+n_3)\geq8m_1m_2m_3\]

Let $a$ be a real number. Let $(f_n(x))_{n\ge 0}$ be a sequence of polynomials such that $f_0(x)=1$ and $f_{n+1}(x)=xf_n(x)+f_n(ax)$ for all non-negative integers $n$. a) Prove that $f_n(x)=x^nf_n\left(x^{-1}\right)$ for all non-negative integers $n$. b) Find an explicit expression for $f_n(x)$.

Let $ ABC$ be a triangle and let $ P$ be a point on the angle bisector $ AD$, with $ D$ on $ BC$. Let $ E$, $ F$ and $ G$ be the intersections of $ AP$, $ BP$ and $ CP$ with the circumcircle of the triangle, respectively. Let $ H$ be the intersection of $ EF$ and $ AC$, and let $ I$ be the intersection of $ EG$ and $ AB$. Determine the geometric place of the intersection of $ BH$ and $ CI$ when $ P$ varies.

Find the largest $k$ such that for every positive integer $n$ we can find at least $k$ numbers in the set $\{n+1, n+2, ... , n+16\}$ which are coprime with $n(n+17)$.

Given the equation $$ a^b\cdot b^c=c^a $$ in positive integers $a$, $b$, and $c$. [i](i)[/i] Prove that any prime divisor of $a$ divides $b$ as well. [i](ii)[/i] Solve the equation under the assumption $b\ge a$. [i](iii)[/i] Prove that the equation has infinitely many solutions. [i](I. Voronovich)[/i]

In square $ABCE$, $AF=2FE$ and $CD=2DE$. What is the ratio of the area of $\triangle BFD$ to the area of square $ABCE$? [asy] size((100)); draw((0,0)--(9,0)--(9,9)--(0,9)--cycle); draw((3,0)--(9,9)--(0,3)--cycle); dot((3,0)); dot((0,3)); dot((9,9)); dot((0,0)); dot((9,0)); dot((0,9)); label("$A$", (0,9), NW); label("$B$", (9,9), NE); label("$C$", (9,0), SE); label("$D$", (3,0), S); label("$E$", (0,0), SW); label("$F$", (0,3), W); [/asy] $ \textbf{(A)}\ \frac{1}{6}\qquad\textbf{(B)}\ \frac{2}{9}\qquad\textbf{(C)}\ \frac{5}{18}\qquad\textbf{(D)}\ \frac{1}{3}\qquad\textbf{(E)}\ \frac{7}{20} $

Let $a, b$ be reals with $a \neq 0$ and let $$P(x)=ax^4-4ax^3+(5a+b)x^2-4bx+b.$$ Show that all roots of $P(x)$ are real and positive if and only if $a=b$.