Let $k\ge2$ be an integer. The function $f:\mathbb N\to\mathbb N$ is defined by $$f(n)=n+\left\lfloor\sqrt[k]{n+\sqrt[k]n}\right\rfloor.$$Determine the set of values taken by the function $f$.

For positive integers $x$, let$$f(x)=\begin{cases} \frac{f\left(\frac{x}{2}\right)}{2} &\mbox{if }x\mbox{ is even,} \\ 2^{-x} &\mbox{if }x\mbox{ is odd.} \end{cases}$$Find $f(1)+f(2)+f(3)+\dots$.

For each positive integer $n$, let $a_n = 0$ (or $1$) if the number of $1$’s in the binary representation of $n$ is even (or odd), respectively. Show that there do not exist positive integers $k$ and $m$ such that $$a_{k+j}=a_{k+m+j} =a_{k+2m+j}$$ for $0 \leq j \leq m-1.$

[u]Round 1[/u] [b]p1.[/b] Three snails – Alice, Bobby, and Cindy – were racing down a road. Whenever one snail passed another, it waved at the snail it passed. During the race, Alice waved $3$ times and was waved at twice. Bobby waved $4$ times and was waved at $3$ times. Cindy waved $5$ times. How many times was she waved at? [b]p2.[/b] Sherlock and Mycroft are playing Battleship on a $4\times 4$ grid. Mycroft hides a single $3\times 1$ cruiser somewhere on the board. Sherlock can pick squares on the grid and fire upon them. What is the smallest number of shots Sherlock has to fire to guarantee at least one hit on the cruiser? [b]p3.[/b] Thirty girls – $13$ of them in red dresses and $17$ in blue dresses – were dancing in a circle, hand-in-hand. Afterwards, each girl was asked if the girl to her right was in a blue dress. Only the girls who had both neighbors in red dresses or both in blue dresses told the truth. How many girls could have answered “Yes”? [b]p4.[/b] Herman and Alex play a game on a $5\times 5$ board. On his turn, a player can claim any open square as his territory. Once all the squares are claimed, the winner is the player whose territory has the longer border. Herman goes first. If both play their best, who will win, or will the game end in a draw? [img]https://cdn.artofproblemsolving.com/attachments/5/7/113d54f2217a39bac622899d3d3eb51ec34f1f.png[/img] [b]p5.[/b] Is it possible to find $2014$ distinct positive integers whose sum is divisible by each of them? [u]Round 2[/u] [b]p6.[/b] Hermione and Ron play a game that starts with 129 hats arranged in a circle. They take turns magically transforming the hats into animals. On each turn, a player picks a hat and chooses whether to change it into a badger or into a raven. A player loses if after his or her turn there are two animals of the same species right next to each other. Hermione goes first. Who loses? [b]p7.[/b] Three warring states control the corner provinces of the island whose map is shown below. [img]https://cdn.artofproblemsolving.com/attachments/e/a/4e2f436be1dcd3f899aa34145356f8c66cda82.png[/img] As a result of war, each of the remaining $18$ provinces was occupied by one of the states. None of the states was able to occupy any province on the coast opposite their corner. The states would like to sign a peace treaty. To do this, they each must send ambassadors to a place where three provinces, one controlled by each state, come together. Prove that they can always find such a place to meet. For example, if the provinces are occupied as shown here, the squares mark possible meeting spots. [img]https://cdn.artofproblemsolving.com/attachments/e/b/81de9187951822120fc26024c1c1fbe2138737.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $\theta$ be a real number such that $1 + \sin 2\theta -\left(\frac12 \sin 2\theta\right)^2= 0$. Compute the maximum value of $(1 + \sin \theta )(1 + \cos \theta)$.

Let $N$ be an integer greater than $1$ and let $T_n$ be the number of non empty subsets $S$ of $\{1,2,.....,n\}$ with the property that the average of the elements of $S$ is an integer.Prove that $T_n - n$ is always even.

Find the number of positive integers $n$ less than $10000$ such that there are more $4$’s in the digits of $n + 1$ than in the digits of $n$.

Let $P, Q \in \mathbb{R}[x]$ be relatively prime nonconstant polynomials. Show that there can be at most three real numbers $\lambda$ such that $P + \lambda Q$ is the square of a polynomial. [i]Alison Miller[/i]

Two thousand points are given on a circle. Label one of the points 1. From this point, count 2 points in the clockwise direction and label this point 2. From the point labeled 2, count 3 points in the clockwise direction and label this point 3. (See figure.) Continue this process until the labels $1, 2, 3, \dots, 1993$ are all used. Some of the points on the circle will have more than one label and some points will not have a label. What is the smallest integer that labels the same point as 1993? [asy] int x=101, y=3*floor(x/4); draw(Arc(origin, 1, 360*(y-3)/x, 360*(y+4)/x)); int i; for(i=y-2; i<y+4; i=i+1) { dot(dir(360*i/x)); } label("3", dir(360*(y-2)/x), dir(360*(y-2)/x)); label("2", dir(360*(y+1)/x), dir(360*(y+1)/x)); label("1", dir(360*(y+3)/x), dir(360*(y+3)/x));[/asy]

Let $S_n = n^2 + 20n + 12$, $n$ a positive integer. What is the sum of all possible values of $n$ for which $S_n$ is a perfect square?

Anna and Boris have the same copy of $5\times5$ table filled with $25$ distinct numbers. After choosing the maximal number in the table, Anna erases the row and the column that contain this number. Then she continue the same operations with a smaller table till it is possible.Boris basically does the same; however, each time choosing the minimal number in a table. Can it happen that the total sum of the numbers chosen by Boris a) is greater than the total sum of the numbers chosen by Anna? [i](6 points)[/i] b) is greater than the total sum of any $5$ numbers of initial table given that no two of the numbers are in the same row or in the same column? [i](2 points)[/i]

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].