Triangle $ ABC$ has $ AB \equal{} 2 \cdot AC$. Let $ D$ and $ E$ be on $ \overline{AB}$ and $ \overline{BC}$, respectively, such that $ \angle{BAE} \equal{} \angle{ACD}.$ Let $ F$ be the intersection of segments $ AE$ and $ CD$, and suppose that $ \triangle{CFE}$ is equilateral. What is $ \angle{ACB}$? $ \textbf{(A)}\ 60^{\circ}\qquad \textbf{(B)}\ 75^{\circ}\qquad \textbf{(C)}\ 90^{\circ}\qquad \textbf{(D)}\ 105^{\circ}\qquad \textbf{(E)}\ 120^{\circ}$

$ABC$ is a triangle, and $E$ and $F$ are points on the segments $BC$ and $CA$ respectively, such that $\frac{CE}{CB}+\frac{CF}{CA}=1$ and $\angle CEF=\angle CAB$. Suppose that $M$ is the midpoint of $EF$ and $G$ is the point of intersection between $CM$ and $AB$. Prove that triangle $FEG$ is similar to triangle $ABC$.

Given $ k \in [0,1,2,3]$ and a positive integer $ n$, let $ f_k(n)$ be the number of sequences $ x_1,...,x_n,$ where $ x_i \in [\minus{}1,0,1]$ for $ i\equal{}1,...,n,$ and $ x_1\plus{}...\plus{}x_n \equiv k$ mod 4 a) Prove that $ f_1(n) \equal{} f_3(n)$ for all positive integers $ n$. (b) Prove that $ f_0(n) \equal{} [{3^n \plus{} 2 \plus{} [\minus{}1]^n}] / 4$ for all positive integers $ n$.

Find the smallest real number $M$ such that there exist four complex numbers $a,b,c,d$ with $|a|=|b|=|c|=|d|=1$, and for any complex number $z$, if $|z| = 1$, then\[|az^3+bz^2+cz+d|\le M.\]

The incircle $ (I)$ of a given scalene triangle $ ABC$ touches its sides $ BC$, $ CA$, $ AB$ at $ A_1$, $ B_1$, $ C_1$, respectively. Denote $ \omega_B$, $ \omega_C$ the incircles of quadrilaterals $ BA_1IC_1$ and $ CA_1IB_1$, respectively. Prove that the internal common tangent of $ \omega_B$ and $ \omega_C$ different from $ IA_1$ passes through $ A$.

$ABCDE$ is a regular pentagon. Determine the smallest value of the expression \[\frac{|PA|+|PB|}{|PC|+|PD|+|PE|},\] where $P$ is an arbitrary point lying in the plane of the pentagon $ABCDE$.

$2021$ copies of each of the number from $1$ to $5$ are initially written on the board.Every second Alice picks any two f these numbers, say $a$ and $b$ and writes $\frac{ab}{c}$.Where $c$ is the length of the hypoteneus with sides $a$ and $b$.Alice stops when only one number is left.If the minnimum number she could write was $x$ and the maximum number she could write was $y$ then find the greatest integer lesser than $2021^2xy$. [hide=PS]Does any body know how to use floors and ceiling function?cuz actuall formation used ceiling,but since Idk how to use ceiling I had to do it like this :(]

Let $a, b,$ and $c$ be positive real numbers such that $ab + bc + ca = 3$. Show that \[ \dfrac{bc}{1 + a^4} + \dfrac{ca}{1 + b^4} + \dfrac{ab}{1 + c^4} \geq \dfrac{3}{2}. \]

Let $ABC$ be an acute, scalene triangle with altitudes $AD, BE, CF$ with $D \in BC, E \in CA$ and $F \in AB$. Let $H, O, I$ be the orthocenter, circumcenter, incenter of triangle $ABC$ respectively and let $M, N, P$ be the midpoint of segments $BC, CA, AB$ respectively. Let $X, Y, Z$ be the intersection of pairs of lines $(AI, NP), (BI, PM)$ and $(CI, MN)$ respectively. a) Prove that the circumcircle of triangles $AXD, BYE, CZF$ have two common points that lie on line $OH$. b) Lines $XP, YM, ZN$ meet the circumcircle of triangles $AXD, BYE, CZF$ again at $X', Y', Z'$ ($X' \neq X, Y' \neq Y, Z' \neq Z$). Let $J$ be the reflection of $I$ across $O$. Prove that $X', Y', Z'$ lie on a line perpendicular to $HJ$.

Let $S$ denote the sum of all rational numbers of the form $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive divisors of $1300$. If $S$ can be expressed in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, then find $m+n$. [i]Proposed by Ephram Chun[/i]

We say that a subset $A$ of $\mathbb N$ is good if for some positive integer $n$, the equation $x-y=n$ admits infinitely many solutions with $x,y\in A$. If $A_1,A_2,\ldots,A_{100}$ are sets whose union is $\mathbb N$, prove that at least one of the $A_i$s is good.

For a real number $a$, let $F_a(x)=\sum_{n\geq 1}n^ae^{2n}x^{n^2}$ for $0\leq x<1$. Find a real number $c$ such that \begin{align*} \lim_{x\to 1^-}F_a(x)e^{-1/(1-x)}&=0 \ \ \ \text{for all $a<c$, and}\\ \lim_{x\to 1^-}F_a(x)e^{-1/(1-x)}&=\infty \ \ \ \text{for all $a>c$.} \end{align*}

Suppose $x$ is a random real number between $1$ and $4$, and $y$ is a random real number between $1$ and $9$. If the expected value of \[ \left\lceil \log_2 x \right\rceil - \left\lfloor \log_3 y \right\rfloor \] can be expressed as $\frac mn$ where $m$ and $n$ are relatively prime positive integers, compute $100m + n$. [i]Proposed by Lewis Chen[/i]

[hide=R stands for Ramanujan, P stands for Pascal]they had two problem sets under those two names[/hide] [b]R1.[/b] What is the distance between the points $(6, 0)$ and $(-2, 0)$? [b]R2 / P1.[/b] Angle $X$ has a degree measure of $35$ degrees. What is the supplement of the complement of angle $X$? [i]The complement of an angle is $90$ degrees minus the angle measure. The supplement of an angle is $180$ degrees minus the angle measure. [/i] [b]R3.[/b] A cube has a volume of $729$. What is the side length of the cube? [b]R4 / P2.[/b] A car that always travels in a straight line starts at the origin and goes towards the point $(8, 12)$. The car stops halfway on its path, turns around, and returns back towards the origin. The car again stops halfway on its return. What are the car’s final coordinates? [b]R5.[/b] A full, cylindrical soup can has a height of $16$ and a circular base of radius $3$. All the soup in the can is used to fill a hemispherical bowl to its brim. What is the radius of the bowl? [b]R6.[/b] In square $ABCD$, the numerical value of the length of the diagonal is three times the numerical value of the area of the square. What is the side length of the square? [b]R7.[/b] Consider triangle $ABC$ with $AB = 3$, $BC = 4$, and $AC = 5$. The altitude from $B$ to $AC$ intersects $AC$ at $H$. Compute $BH$. [b]R8.[/b] Mary shoots $5$ darts at a square with side length $2$. Let $x$ be equal to the shortest distance between any pair of her darts. What is the maximum possible value of $x$? [b]P3.[/b] Let $ABC$ be an isosceles triangle such that $AB = BC$ and all of its angles have integer degree measures. Two lines, $\ell_1$ and $\ell_2$, trisect $\angle ABC$. $\ell_1$ and $\ell_2$ intersect $AC$ at points $D$ and $E$ respectively, such that $D$ is between $A$ and $E$. What is the smallest possible integer degree measure of $\angle BDC$? [b]P4.[/b] In rectangle $ABCD$, $AB = 9$ and $BC = 8$. $W$, $X$, $Y$ , and $Z$ are on sides $AB$, $BC$, $CD$, and $DA$, respectively, such that $AW = 2WB$, $CX = 3BX$, $CY = 2DY$ , and $AZ = DZ$. If $WY$ and $XZ$ intersect at $O$, find the area of $OWBX$. [b]P5.[/b] Consider a regular $n$-gon with vertices $A_1A_2...A_n$. Find the smallest value of $n$ so that there exist positive integers $i, j, k \le n$ with $\angle A_iA_jA_k = \frac{34^o}{5}$. [b]P6.[/b] In right triangle $ABC$ with $\angle A = 90^o$ and $AB < AC$, $D$ is the foot of the altitude from $A$ to $BC$, and $M$ is the midpoint of $BC$. Given that $AM = 13$ and $AD = 5$, what is $\frac{AB}{AC}$ ? [b]P7.[/b] An ant is on the circumference of the base of a cone with radius $2$ and slant height $6$. It crawls to the vertex of the cone $X$ in an infinite series of steps. In each step, if the ant is at a point $P$, it crawls along the shortest path on the exterior of the cone to a point $Q$ on the opposite side of the cone such that $2QX = PX$. What is the total distance that the ant travels along the exterior of the cone? [b]P8.[/b] There is an infinite checkerboard with each square having side length $2$. If a circle with radius $1$ is dropped randomly on the checkerboard, what is the probability that the circle lies inside of exactly $3$ squares? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

If your average score on your first six mathematics tests was $ 84$ and your average score on your first seven mathematics tests was $ 85$, then your score on the seventh test was \[ \textbf{(A)}\ 86 \qquad \textbf{(B)}\ 88 \qquad \textbf{(C)}\ 90 \qquad \textbf{(D)}\ 91 \qquad \textbf{(E)}\ 92 \]

How many paths are there from$ A$ to the line $BC$ if the path does not go through any vertex twice and always moves to the left? [img]https://cdn.artofproblemsolving.com/attachments/e/6/a4bc3a9decc06eaeed6f7e99cb58f7b2524471.jpg[/img]

For positive integers $m,n \geq 2$, let $S_{m,n} = \{(i,j): i \in \{1,2,\ldots,m\}, j\in \{1,2,\ldots,n\}\}$ be a grid of $mn$ lattice points on the coordinate plane. Determine all pairs $(m,n)$ for which there exists a simple polygon $P$ with vertices in $S_{m,n}$ such that all points in $S_{m,n}$ are on the boundary of $P$, all interior angles of $P$ are either $90^{\circ}$ or $270^{\circ}$ and all side lengths of $P$ are $1$ or $3$.

What is the largest number of knights that can be put on a $5 \times 5$ chess board so that each knight attacks exactly two other knights? (M Gorelov)

If $$s_n = 1 + q + q^2 +... + q^n$$ and $$ S_n = 1 +\frac{1 + q}{2}+ \left( \frac{1 + q}{2}\right)^2 +... + \left( \frac{1 + q}{2}\right)^n,$$ prove that $${n + 1 \choose 1}+{n + 1 \choose 2} s_1 + {n + 1 \choose 3} s_2 + ... + {n + 1 \choose n + 1} s_n = 2^nS_n$$

In the land of Microbablia the alphabet has only two letters, ‘A’ and ‘B’. Not surprisingly, the inhabitants are obsessed with the band ABBA. Words in the local dialect with a high ABBA-factor are considered particularly lucky. To compute the ABBA-factor of a word you just count the number of occurrences of ABBA within the word (not necessarily consecutively). So for instance AABA has ABBA-factor $0$, ABBA has ABBA-factor $1$, AABBBA has ABBA-factor $6$, and ABBABBA has ABBA factor $8$. What is the greatest possible ABBA-factor for a $100$ letter word?

Let $N =\{1, 2, 3, ...\}$ be the set of all natural numbers and $f : N\to N$ be a function. Suppose $f(1) = 1$, $f(2n) = f(n)$ and $f(2n + 1) = f(2n) + 1$ for all natural numbers $n$. (i) Calculate the maximum value $M$ of $f(n)$ for $n \in N$ with $1 \le n \le 1994$. (ii) Find all $n \in N$, with 1 \le n \le 1994, such that $f(n) = M$.

Given a positive integer $h$, show that there are a finite number of triangles with integer sides $a, b, c$ and altitude relative to side $c$ equal to $h$ .

Prove that if $ x_1, x_2, \ldots, x_n $ are angles between $ 0^\circ $ and $ 180^\circ $, and $ n $ is any natural number greater than $ 1 $, then $$ \sin (x_1 + x_2 + \ldots + x_n) < \sin x_1 + \sin x_2 + \ldots + \sin x_n.$$

[b]p1.[/b] Consider the equation $$x_1x_2 + x_2x_3 + x_3x_4 + · · · + x_{n-1}x_n + x_nx_1 = 0$$ where $x_i \in \{1,-1\}$ for $i = 1, 2, . . . , n$. (a) Show that if the equation has a solution, then $n$ is even. (b) Suppose $n$ is divisible by $4$. Show that the equation has a solution. (c) Show that if the equation has a solution, then $n$ is divisible by $4$. [b]p2.[/b] (a) Find a polynomial $f(x)$ with integer coefficients and two distinct integers $a$ and $b$ such that $f(a) = b$ and $f(b) = a$. (b) Let $f(x)$ be a polynomial with integer coefficients and $a$, $b$, and $c$ be three integers. Suppose $f(a) = b$, $f(b) = c$, and $f(c) = a$. Show that $a = b = c$. [b]p3.[/b] (a) Consider the triangle with vertices $M$ $(0, 2n + 1)$, $S$ $(1, 0)$, and $U \left(0, \frac{1}{2n^2}\right)$, where $n$ is a positive integer. If $\theta = \angle MSU$, prove that $\tan \theta = 2n - 1$. (b) Find positive integers $a$ and $b$ that satisfy the following equation. $$arctan \frac18 = arctan \,\,a - arctan \,\, b$$ (c) Determine the exact value of the following infinite sum. $$arctan \frac12 + arctan \frac18 + arctan \frac{1}{18} + arctan \frac{1}{32}+ ... + arctan \frac{1}{2n^2}+ ...$$ [b]p4.[/b] (a) Prove: $(55 + 12\sqrt{21})^{1/3} +(55 - 12\sqrt{21})^{1/3}= 5$. (b) Completely factor $x^8 + x^6 + x^4 + x^2 + 1$ into polynomials with integer coefficients, and explain why your factorization is complete. [b]p5.[/b] In this problem, we simulate a hula hoop as it gyrates about your waist. We model this situation by representing the hoop with a rotating a circle of radius $2$ initially centered at $(-1, 0)$, and representing your waist with a fixed circle of radius $1$ centered at the origin. Suppose we mark the point on the hoop that initially touches the fixed circle with a black dot (see the left figure). As the circle of radius $2$ rotates, this dot will trace out a curve in the plane (see the right figure). Let $\theta$ be the angle between the positive x-axis and the ray that starts at the origin and goes through the point where the fixed circle and circle of radius $2$ touch. Determine formulas for the coordinates of the position of the dot, as functions $x(\theta)$ and $y(\theta)$. The left figure shows the situation when $\theta = 0$ and the right figure shows the situation when $\theta = 2pi/3$. [img]https://cdn.artofproblemsolving.com/attachments/8/6/d15136872118b8e14c8f382bc21b41a8c90c66.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $m,n\geq 2$ be integers. Let $f(x_1,\dots, x_n)$ be a polynomial with real coefficients such that $$f(x_1,\dots, x_n)=\left\lfloor \frac{x_1+\dots + x_n}{m} \right\rfloor\text{ for every } x_1,\dots, x_n\in \{0,1,\dots, m-1\}.$$ Prove that the total degree of $f$ is at least $n$.