Given an integer $n\ge 2$, prove that \[\lfloor \sqrt n \rfloor + \lfloor \sqrt[3]n\rfloor + \cdots +\lfloor \sqrt[n]n\rfloor = \lfloor \log_2n\rfloor + \lfloor \log_3n\rfloor + \cdots +\lfloor \log_nn\rfloor\]. [hide="Edit"] Thanks to shivangjindal for pointing out the mistake (and sorry for the late edit)[/hide]

Let $L$ be the number of times the letter $L$ appeared on the Speed Round, $M$ be the number of times the letter $M$ appeared on the Speed Round, and $T$ be the number of times the letter $T$ appeared on the Speed Round. Find the value of $LMT$.

If $\alpha$, $\beta$, and $\gamma$ are the roots of $x^3 - x - 1 = 0$, compute $\frac{1+\alpha}{1-\alpha} + \frac{1+\beta}{1-\beta} + \frac{1+\gamma}{1-\gamma}$.

The numbers from 1 to 1996 are written down ------ 12345678910111213.... How many zeros are written? A. 489 B. 699 C. 796 D. 996 E. None of these

You have a set of five weights, together with a balance that allows you to compare the weight of two things. The weights are known to be $10$, $20$,$30$,$40$ and $50$ grams, but are otherwise identical except for their labels. The $10$ and $50$ gram weights are clearly labelled, but the labels have been erased on the remaining weights. Using the balance exactly once, is it possible to determine what one of the three unlabelled weights is? If so, explain how, and if not, explain why not.

Find all functions $f: \mathbb{Z^+} \to \mathbb{Z^+}$ such that for all integers $a, b, c$ we have $$ af(bc)+bf(ac)+cf(ab)=(a+b+c)f(ab+bc+ac). $$ [i]Note. The set $\mathbb{Z^+}$ refers to the set of positive integers.[/i] [i]Proposed by Mojtaba Zare, Iran[/i]

Let $x,y,z$ be 3 different real numbers not equal to $0$ that satisfiying $x^2-xy=y^2-yz=z^2-zx$. Find all the values of $\frac{x}{z}+\frac{y}{x}+\frac{z}{y}$ and $(x+y+z)^3+9xyz$.

p1. Among the numbers $\frac15$ and $\frac14$ there are infinitely many fractional numbers. Find $999$ decimal numbers between $\frac15$ and $\frac14$ so that the difference between the next fractional number with the previous fraction constant. (i.e. If $x_1, x_2, x_3, x_4,..., x_{999}$ is a fraction that meant, then $x_2 - x_1= x_3 - x_3= ...= x_n - x_{n-1}=...=x_{999}-x_{998}$) p2. The pattern in the image below is: "Next image obtained by adding an image of a black equilateral triangle connecting midpoints of the sides of each white triangle that is left in the previous image." The pattern is continuous to infinity. [img]https://cdn.artofproblemsolving.com/attachments/e/f/81a6b4d20607c7508169c00391541248b8f31e.png[/img] It is known that the area of ​​the triangle in Figure $ 1$ is $ 1$ unit area. Find the total area of ​​the area formed by the black triangles in figure $5$. Also find the total area of the area formed by the black triangles in the $20$th figure. p3. For each pair of natural numbers $a$ and $b$, we define $a*b = ab + a - b$. The natural number $x$ is said to be the [i]constituent [/i] of the natural number $n$ if there is a natural number $y$ that satisfies $x*y = n$. For example, $2$ is a constituent of $6$ because there is a natural number 4 so that $2*4 = 2\cdot 4 + 2 - 4 = 8 + 2 - 4 = 6$. Find all constituent of $2005$. p4. Three people want to eat at a restaurant. To find who pays them to make a game. Each tossing one coin at a time. If the result is all heads or all tails, then they toss again. If not, then "odd person" (i.e. the person whose coin appears different from the two other's coins) who pay. Determine the number of all possible outcomes, if the game ends in tossing: a. First. b. Second. c. Third. d. Tenth. p5. Given the equation $x^2 + 3y^2 = n$, where $x$ and $y$ are integers. If $n < 20$ what number is $n$, and which is the respective pair $(x,y)$ ? Show that it is impossible to solve $x^2 + 3y^2 = 8$ in integers.

Find the complex numbers $ a,b $ having the properties that $ |a|=|b|=1=\bar{a} +\bar{b} -ab. $

Determine all pairs $(f,g)$ of functions from the set of real numbers to itself that satisfy \[g(f(x+y)) = f(x) + (2x + y)g(y)\] for all real numbers $x$ and $y$. [i]Proposed by Japan[/i]

Solve in $ \mathbb{R}^3 $ the system: $$ \left\{ \begin{matrix} 3^x+4^x=5^y \\8^y+15^y=17^z \\ 20^z+21^z=29^x \end{matrix} \right. $$ [i]Cristinel Mortici[/i]

Let $ABC$ be an equilateral triangle and $M$ be a point inside the triangle. We denote by $A'$, $B'$, $C'$ the projections of the point $M$ on the sides $BC$, $CA$ and $AB$ respectively. Prove that the lines $AA'$, $BB'$ and $CC'$ are concurrent if and only if $M$ belongs to an altitude of the triangle.

Solve the equation \[2^{x^{2}+x}+\log_{2}x = 2^{x+1}\]

Find a real function $f : [0,\infty)\to \mathbb R$ such that $f(2x+1) = 3f(x)+5$, for all $x$ in $[0,\infty)$.

$F_{n}$ denotes the Fibonacci Sequence where $F_{1}=0, F_{2}=1, F_{n}=F_{n-1}+F_{n-2},\ \forall \ n \geq 3$ Find$$\sum\limits_{n=3}^{\infty}\frac{18+999F_n}{F_{n-1}\times F_{n+1}}$$ [list=1] [*] 2016 [*] 2017 [*] 2018 [*] None of these [/list]

Find all continuous functions $f : \mathbb R \to \mathbb R$ such that for all $x \in \mathbb R$ and for all $n \in \mathbb N^{\ast}$ we have \[ n^2 \int_{x}^{x + \frac{1}{n}} f(t) \, dt = n f(x) + \frac12 . \] [i]Mihai Piticari[/i]

[u]Set 7[/u] [b]G19.[/b] How many ordered triples $(x, y, z)$ with $1 \le x, y, z \le 50$ are there such that both $x + y + z$ and $xy + yz + zx$ are divisible by$ 6$? [b]G20.[/b] Triangle $ABC$ has orthocenter $H$ and circumcenter $O$. If $D$ is the foot of the perpendicular from $A$ to $BC$, then $AH = 8$ and $HD = 3$. If $\angle AOH = 90^o$, find $BC^2$. [b]G21.[/b] Nate flips a fair coin until he gets two heads in a row, immediately followed by a tails. The probability that he flips the coin exactly $12$ times is $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [u]Set 8[/u] [b]G22.[/b] Let $f$ be a function defined by $f(1) = 1$ and $$f(n) = \frac{1}{p}f\left(\frac{n}{p}\right)f(p) + 2p - 2,$$ where $p$ is the least prime dividing $n$, for all integers $n \ge 2$. Find $f(2022)$. [b]G23.[/b] Jessica has $15$ balls numbered $1$ through $15$. With her left hand, she scoops up $2$ of the balls. With her right hand, she scoops up $2$ of the remaining balls. The probability that the sum of the balls in her left hand is equal to the sum of the balls in her right hand can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [b]G24.[/b] Let $ABCD$ be a cyclic quadrilateral such that its diagonal $BD = 17$ is the diameter of its circumcircle. Given $AB = 8$, $BC = CD$, and that a line $\ell$ through A intersects the incircle of $ABD$ at two points $P$ and $Q$, the maximum area of $CP Q$ can be expressed as a fraction $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find $m + n$. [u]Set 9[/u] [i]This set consists of three estimation problems, with scoring schemes described.[/i] [b]G25.[/b] Estimate $N$, the total number of participants (in person and online) at MOAA this year. An estimate of $e$ gets a total of max $ \left( 0, \lfloor 150 \left( 1- \frac{|N-e|}{N}\right) \rfloor -120 \right)$ points. [b]G26.[/b] If $A$ is the the total number of in person participants at MOAA this year, and $B$ is the total number of online participants at MOAA this year, estimate $N$, the product $AB$. An estimate of $e$ gets a total of max $(0, 30 - \lceil \log10(8|N - e| + 1)\rceil )$ points. [b]G27.[/b] Estimate $N$, the total number of letters in all the teams that signed up for MOAA this year, both in person and online. An estimate of e gets a total of max $(0, 30 - \lceil 7 log5(|N - E|)\rceil )$ points. PS. You should use hide for answers. Sets 1-3 have been posted [url=https://artofproblemsolving.com/community/c3h3131303p28367061]here [/url] and 4-6 [url=https://artofproblemsolving.com/community/c3h3131305p28367080]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Some positive integers are initially written on a board, where each $2$ of them are different. Each time we can do the following moves: (1) If there are 2 numbers (written in the board) in the form $n, n+1$ we can erase them and write down $n-2$ (2) If there are 2 numbers (written in the board) in the form $n, n+4$ we can erase them and write down $n-1$ After some moves, there might appear negative numbers. Find the maximum value of the integer $c$ such that: Independetly of the starting numbers, each number which appears in any move is greater or equal to $c$

[b]p1.[/b] Compute the greatest integer less than or equal to $$\frac{10 + 12 + 14 + 16 + 18 + 20}{21}$$ [b]p2.[/b] Let$ A = 1$.$B = 2$, $C = 3$, $...$, $Z = 26$. Find $A + B +M + C$. [b]p3.[/b] In Mr. M's farm, there are $10$ cows, $8$ chickens, and $4$ spiders. How many legs are there (including Mr. M's legs)? [b]p4.[/b] The area of an equilateral triangle with perimeter $18$ inches can be expressed in the form $a\sqrt{b}{c}$ , where $a$ and $c$ are relatively prime and $b$ is not divisible by the square of any prime. Find $a + b + c$. [b]p5.[/b] Let $f$ be a linear function so $f(x) = ax + b$ for some $a$ and $b$. If $f(1) = 2017$ and $f(2) = 2018$, what is $f(2019)$? [b]p6.[/b] How many integers $m$ satisfy $4 < m^2 \le 216$? [b]p7.[/b] Allen and Michael Phelps compete at the Olympics for swimming. Allen swims $\frac98$ the distance Phelps swims, but Allen swims in $\frac59$ of Phelps's time. If Phelps swims at a rate of $3$ kilometers per hour, what is Allen's rate of swimming? The answer can be expressed as $m/n$ for relatively prime positive integers $m, n$. Find $m + n$. [b]p8.[/b] Let $X$ be the number of distinct arrangements of the letters in "POONAM," $Y$ be the number of distinct arrangements of the letters in "ALLEN" and $Z$ be the number of distinct arrangements of the letters in "NITHIN." Evaluate $\frac{X+Z}{Y}$ : [b]p9.[/b] Two overlapping circles, both of radius $9$ cm, have centers that are $9$ cm apart. The combined area of the two circles can be expressed as $\frac{a\pi+b\sqrt{c}+d}{e}$ where $c$ is not divisible by the square of any prime and the fraction is simplified. Find $a + b + c + d + e$. [b]p10.[/b] In the Boxborough-Acton Regional High School (BARHS), $99$ people take Korean, $55$ people take Maori, and $27$ people take Pig Latin. $4$ people take both Korean and Maori, $6$ people take both Korean and Pig Latin, and $5$ people take both Maori and Pig Latin. $1$ especially ambitious person takes all three languages, and and $100$ people do not take a language. If BARHS does not oer any other languages, how many students attend BARHS? [b]p11.[/b] Let $H$ be a regular hexagon of side length $2$. Let $M$ be the circumcircle of $H$ and $N$ be the inscribed circle of $H$. Let $m, n$ be the area of $M$ and $N$ respectively. The quantity $m - n$ is in the form $\pi a$, where $a$ is an integer. Find $a$. [b]p12.[/b] How many ordered quadruples of positive integers $(p, q, r, s)$ are there such that $p + q + r + s \le 12$? [b]p13.[/b] Let $K = 2^{\left(1+ \frac{1}{3^2} \right)\left(1+ \frac{1}{3^4} \right)\left(1+ \frac{1}{3^8}\right)\left(1+ \frac{1}{3^{16}} \right)...}$. What is $K^8$? [b]p14.[/b] Neetin, Neeton, Neethan, Neethine, and Neekhil are playing basketball. Neetin starts out with the ball. How many ways can they pass 5 times so that Neethan ends up with the ball? [b]p15.[/b] In an octahedron with side lengths $3$, inscribe a sphere. Then inscribe a second sphere tangent to the first sphere and to $4$ faces of the octahedron. The radius of the second sphere can be expressed in the form $\frac{\sqrt{a}-\sqrt{b}}{c}$ , where the square of any prime factor of $c$ does not evenly divide into $b$. Compute $a + b + c$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Calculate the sum $$S=\sqrt{1+\frac{8\cdot 1^2-1}{1^2\cdot 3^2}}+\sqrt{1+\frac{8\cdot 2^2-1}{3^2\cdot 5^2}}+...+ \sqrt{1+\frac{8\cdot 1003^2-1}{2005^2\cdot 2007^2}}$$

Write the equation of the line tangent to the graph of the function $y = x^4-x^2 + x$ to at least at two points.

Determine all triples $(x, y, z)$ of nonnegative real numbers that verify the following system of equations: $$x^2 - y = (z -1)^2 $$ $$y^2 - z = (x -1)^2$$ $$z^2 - x = (y - 1)^2$$

Prove that every bijective function $ f: \mathbb{Z}\rightarrow\mathbb{Z}$ can be written in the way $ f\equal{}u\plus{}v$ where $ u,v: \mathbb{Z}\rightarrow\mathbb{Z}$ are bijective functions.

Four runners are preparing to begin a $1$-mile race from the same starting line. When the race starts, runners Alice, Bob, and Charlie all travel at constant speeds of $8$ mph, $4$ mph, and $2$ mph, respectively. The fourth runner, Dave, is initially half as slow as Charlie, but Dave has a superpower where he suddenly doubles his running speed every time a runner finishes the race. How many hours does it take for Dave to finish the race?

Let $f_1 = (a_1, a_2, \dots , a_n) , n > 2$, be a sequence of integers. From $f_1$ one constructs a sequence $f_k$ of sequences as follows: if $f_k = (c_1, c_2, \dots, cn)$, then $f_{k+1} = (c_{i_{1}}, c_{i_{2}}, c_{i_{3}} + 1, c_{i_{4}} + 1, . . . , c_{i_{n}} + 1)$, where $(c_{i_{1}}, c_{i_{2}},\dots , c_{i_{n}})$ is a permutation of $(c_1, c_2, \dots, c_n)$. Give a necessary and sufficient condition for $f_1$ under which it is possible for $f_k$ to be a constant sequence $(b_1, b_2,\dots , b_n), b_1 = b_2 =\cdots = b_n$, for some $k.$