If $x \ge 0$, then $\sqrt{x \sqrt{x \sqrt{x}}} = $ $ \textbf{(A)}\ x\sqrt{x}\qquad\textbf{(B)}\ x\sqrt[4]{x}\qquad\textbf{(C)}\ \sqrt[8]{x}\qquad\textbf{(D)}\ \sqrt[8]{x^{3}}\qquad\textbf{(E)}\ \sqrt[8]{x^{7}} $

a) Does there exist a function $f: N \to N$ such that $f(f(n))=f(n+1) - f(n)$ for all $n \in N$? b) Does there exist a function $f: N \to N$ such that $f(f(n))=f(n+2) - f(n)$ for all $n \in N$? I. Voronovich

Prove that if from any $2007$ consecutive terms of an infinite arithmetic progression of integers starting with $2$, one can choose a term relatively prime to all the $2006$ other terms, then there is also a term amongst any $2008$ consecutive terms relatively prime to the rest.

Find all positive integers $k$ for which there exists a nonlinear function $f:\mathbb{Z} \rightarrow\mathbb{Z}$ such that the equation $$f(a)+f(b)+f(c)=\frac{f(a-b)+f(b-c)+f(c-a)}{k}$$ holds for any integers $a,b,c$ satisfying $a+b+c=0$ (not necessarily distinct). [i]Evan Chen[/i]

a) Factorize $xy - x - y + 1$. b) Prove that if integers $a$ and $b$ satisfy $ |a + b| > |1 + ab|$, then $ab = 0$.

Find $ \lim_{a\to\infty} \frac {1}{a^2}\int_0^a \log (1 \plus{} e^x)\ dx.$

$\textbf{N1.}$ Find all nonnegative integers $a,b,c$ such that \begin{align*} a^2+b^2+c^2-ab-bc-ca = a+b+c \end{align*} [i]Proposed by usjl[/i]

Let the class of functions $f_n$ be defined such that $f_1(x)=|x^3-x^2|$ and $f_{k+1}(x)=|f_k(x)-x^3|$ for all $k\ge1$. Denote by $S_n$ the sum of all $y$-values of $f_n(x)$'s "sharp" points in the First Quadrant. (A "sharp" point is a point for which the derivative is not defined.) Find the ratio of odd to even terms, $$\lim_{k\to\infty}\frac{S_{2k+1}}{S_{2k}}$$

How many ways can the integers from $-7$ to $7$ inclusive be arranged in a sequence such that the absolute value of the numbers in the sequence does not decrease?

[b]p1.[/b] Three positive integers sum to $16$. What is the least possible value of the sum of their squares? [b]p2.[/b] Ben is thinking of an odd positive integer less than $1000$. Ben subtracts $ 1$ from his number and divides by $2$, resulting in another number. If his number is still odd, Ben repeats this procedure until he gets an even number. Given that the number he ends on is $2$, how many possible values are there for Ben’s original number? [b]p3.[/b] Triangle $ABC$ is isosceles, with $AB = BC = 18$ and has circumcircle $\omega$. Tangents to $\omega$ at $ A$ and $ B$ intersect at point $D$. If $AD = 27$, what is the length of $AC$? [b]p4.[/b] How many non-decreasing sequences of five natural numbers have first term $ 1$, last term $ 11$, and have no three terms equal? [b]p5.[/b] Adam is bored, and has written the string “EMCC” on a piece of paper. For fun, he decides to erase every letter “C”, and replace it with another instance of “EMCC”. For example, after one step, he will have the string “EMEMCCEMCC”. How long will his string be after $8$ of these steps? [b]p6.[/b] Eric has two coins, which land heads $40\%$ and $60\%$ of the time respectively. He chooses a coin randomly and flips it four times. Given that the first three flips contained two heads and one tail, what is the probability that the last flip was heads? [b]p7.[/b] In a five person rock-paper-scissors tournament, each player plays against every other player exactly once, with each game continuing until one player wins. After each game, the winner gets $ 1$ point, while the loser gets no points. Given that each player has a $50\%$ chance of defeating any other player, what is the probability that no two players end up with the same amount of points? [b]p8.[/b] Let $\vartriangle ABC$ have $\angle A = \angle B = 75^o$. Points $D, E$, and $F$ are on sides $BC$, $CA$, and $AB$, respectively, so that $EF$ is parallel to $BC$, $EF \perp DE$, and $DE = EF$. Find the ratio of $\vartriangle DEF$’s area to $\vartriangle ABC$’s area. [b]p9.[/b] Suppose $a, b, c$ are positive integers such that $a+b =\sqrt{c^2 + 336}$ and $a-b =\sqrt{c^2 - 336}$. Find $a+b+c$. [b]p10.[/b] How many times on a $12$-hour analog clock are there, such that when the minute and hour hands are swapped, the result is still a valid time? (Note that the minute and hour hands move continuously, and don’t always necessarily point to exact minute/hour marks.) [b]p11.[/b] Adam owns a square $S$ with side length $42$. First, he places rectangle $A$, which is $6$ times as long as it is wide, inside the square, so that all four vertices of $A$ lie on sides of $S$, but none of the sides of $ A$ are parallel to any side of $S$. He then places another rectangle $B$, which is $ 7$ times as long as it is wide, inside rectangle $A$, so that all four vertices of $ B$ lie on sides of $ A$, and again none of the sides of $B$ are parallel to any side of $A$. Find the length of the shortest side of rectangle $ B$. [b]p12.[/b] Find the value of $\sqrt{3 \sqrt{3^3 \sqrt{3^5 \sqrt{...}}}}$, where the exponents are the odd natural numbers, in increasing order. [b]p13.[/b] Jamesu and Fhomas challenge each other to a game of Square Dance, played on a $9 \times 9$ square grid. On Jamesu’s turn, he colors in a $2\times 2$ square of uncolored cells pink. On Fhomas’s turn, he colors in a $1 \times 1$ square of uncolored cells purple. Once Jamesu can no longer make a move, Fhomas gets to color in the rest of the cells purple. If Jamesu goes first, what the maximum number of cells that Fhomas can color purple, assuming both players play optimally in trying to maximize the number of squares of their color? [b]p14.[/b] Triangle $ABC$ is inscribed in circle $\omega$. The tangents to $\omega$ from $B$ and $C$ meet at $D$, and segments $AD$ and $BC$ intersect at $E$. If $\angle BAC = 60^o$ and the area of $\vartriangle BDE$ is twice the area of $\vartriangle CDE$, what is $\frac{AB}{AC}$ ? [b]p15.[/b] Fhomas and Jamesu are now having a number duel. First, Fhomas chooses a natural number $n$. Then, starting with Jamesu, each of them take turns making the following moves: if $n$ is composite, the player can pick any prime divisor $p$ of $n$, and replace $n$ by $n - p$, if $n$ is prime, the player can replace n by $n - 1$. The player who is faced with $ 1$, and hence unable to make a move, loses. How many different numbers $2 \le n \le 2019$ can Fhomas choose such that he has a winning strategy, assuming Jamesu plays optimally? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ ABC$ be an acute angled triangle whose circumcenter is $ O$. The three diameters of the circumcircle that pass through $ A$, $ B$, and $ C$, meet the opposite sides $ BC$, $ CA$, and $ AB$ at the points $ A_1$, $ B_1$ and $ C_1$, respectively. The circumradius of $ ABC$ is of length $ 2P$, where $ P$ is a prime number. The lengths of $ OA_1$, $ OB_1$, $ OC_1$ are integers. What are the lengths of the sides of the triangle?

Doug can paint a room in $ 5$ hours. Dave can paint the same room in $ 7$ hours. Doug and Dave paint the room together and take a one-hour break for lunch. Let $ t$ be the total time, in hours, required for them to complete the job working together, including lunch. Which of the following equations is satisfied by $ t$? $ \textbf{(A)}\ \left(\frac{1}{5}\plus{}\frac{1}{7}\right)(t\plus{}1)\equal{}1 \qquad \textbf{(B)}\ \left(\frac{1}{5}\plus{}\frac{1}{7}\right)t\plus{}1\equal{}1 \qquad \textbf{(C)}\ \left(\frac{1}{5}\plus{}\frac{1}{7}\right)t\equal{}1 \\ \textbf{(D)}\ \left(\frac{1}{5}\plus{}\frac{1}{7}\right)(t\minus{}1)\equal{}1 \qquad \textbf{(E)}\ (5\plus{}7)t\equal{}1$

One wants to distribute $n$ same sized cakes between $k$ people equally by cutting every cake at most once. If the number of positive divisors of $n$ is denoted as $d(n)$, show that the number of different values of $k$ which makes such distribution possible is $n+d(n)$

Let $f:\mathbb{R}^{\geq 0}\to\mathbb{R}$ be a function such that $f(x)-3x$ and $f(x)-x^3$ are ascendant functions. Prove that $f(x)-x^2-x$ is an ascendant function, too. (We call the function $g(x)$ ascendant, when for every $x\leq{y}$ we have $g(x)\leq{g(y)}$.)

Do there exist a bounded function $f: \mathbb{R}\to\mathbb{R}$ such that $f(1)>0$ and $f(x)$ satisfies an inequality $f^2(x+y)\ge f^2(x)+2f(xy)+f^2(y)$?

Find the complex numbers $ z_1,z_2,z_3 $ of same absolute value having the property that: $$ 1=z_1z_2z_3=z_1+z_2+z_3. $$