Determine the maximum number of three-term arithmetic progressions which can be chosen from a sequence of $n$ real numbers \[a_1<a_2<\cdots<a_n.\]

Find all funtions $f : Z_{>0} \to Z_{>0}$ that satisfy $f(f(f(n))) + f(f(n)) + f(n) = 3n$ for all $n \in Z_{>0}$ .

Can there be drawn on a circle of radius $1$ a number of $1975$ distinct points, so that the distance (measured on the chord) between any two points (from the considered points) is a rational number?

Let $a\geq 4$ be a positive integer. Determine the smallest value of $n\geq 5$, $n\neq a$, such that $a$ can be represented in the form$$a=\frac{x_1^2+x_2^2+\cdots + x_n^ 2}{x_1x_2\cdots x_n}$$for a suitable choice of the $n$ positive integers $x_1,x_2,\ldots ,x_n$.

Prove that for all positive real numbers $a$, $b$ the following inequality holds: \begin{align*} \sqrt{\frac{a^2+b^2}{2}}+\frac{2ab}{a+b}\ge \frac{a+b}{2}+ \sqrt{ab} \end{align*} When does equality hold?

Find all functions $ f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying \[ f(f(x \plus{} y)) \equal{} f(x \plus{} y) \plus{} f(x)f(y) \minus{} xy\] for all real numbers $x$ and $y$.

Find all triples $(x,y,z)\in \mathbb{R}^{3}$ such that $x,y,z>3$ and $\frac{(x+2)^2}{y+z-2}+\frac{(y+4)^2}{z+x-4}+\frac{(z+6)^2}{x+y-6}=36$

Let $ a$, $ b$, $ c$ be real numbers such that for every two of the equations \[ x^2\plus{}ax\plus{}b\equal{}0, \quad x^2\plus{}bx\plus{}c\equal{}0, \quad x^2\plus{}cx\plus{}a\equal{}0\] there is exactly one real number satisfying both of them. Determine all possible values of $ a^2\plus{}b^2\plus{}c^2$.

Determine all functions $f : \mathbb R^+ \to \mathbb R$ that satisfy the equation $$f(xy) = f(x)f(y)f(x+y)$$ for all positive real numbers $x$ and $y$.

The degrees of polynomials $P$ and $Q$ with real coefficients do not exceed $n$. These polynomials satisfy the identity \[ P(x) x^{n + 1} + Q(x) (x+1)^{n + 1} = 1. \] Determine all possible values of $Q \left( - \frac{1}{2} \right)$.

Let $a,b,c,d,e,f$ be real numbers such that the polynomial \[ p(x)=x^8-4x^7+7x^6+ax^5+bx^4+cx^3+dx^2+ex+f \] factorises into eight linear factors $x-x_i$, with $x_i>0$ for $i=1,2,\ldots,8$. Determine all possible values of $f$.

[hide=Intro]The answers to each of the ten questions in this section are integers containing only the digits $ 1$ through $ 8$, inclusive. Each answer can be written into the grid on the answer sheet, starting from the cell with the problem number, and continuing across or down until the entire answer has been written. Answers may cross dark lines. If the answers are correctly filled in, it will be uniquely possible to write an integer from $ 1$ to $ 8$ in every cell of the grid, so that each number will appear exactly once in every row, every column, and every marked $2$ by $4$ box. You will get $7$ points for every correctly filled answer, and a $15$ point bonus for filling in every gridcell. It will help to work back and forth between the grid and the problems, although every problem is uniquely solvable on its own. Please write clearly within the boxes. No points will be given for a cell without a number, with multiple numbers, or with illegible handwriting.[/hide] [img]https://cdn.artofproblemsolving.com/attachments/9/b/f4db097a9e3c2602b8608be64f06498bd9d58c.png[/img] [b]1 ACROSS:[/b] Jack puts $ 10$ red marbles, $ 8$ green marbles and 4 blue marbles in a bag. If he takes out $11$ marbles, what is the expected number of green marbles taken out? [b]2 DOWN:[/b] What is the closest integer to $6\sqrt{35}$ ? [b]3 ACROSS: [/b]Alan writes the numbers $ 1$ to $64$ in binary on a piece of paper without leading zeroes. How many more times will he have written the digit $ 1$ than the digit $0$? [b]4 ACROSS:[/b] Integers a and b are chosen such that $-50 < a, b \le 50$. How many ordered pairs $(a, b)$ satisfy the below equation? $$(a + b + 2)(a + 2b + 1) = b$$ [b]5 DOWN: [/b]Zach writes the numbers $ 1$ through $64$ in binary on a piece of paper without leading zeroes. How many times will he have written the two-digit sequence “$10$”? [b]6 ACROSS:[/b] If you are in a car that travels at $60$ miles per hour, $\$1$ is worth $121$ yen, there are $8$ pints in a gallon, your car gets $10$ miles per gallon, a cup of coffee is worth $\$2$, there are 2 cups in a pint, a gallon of gas costs $\$1.50$, 1 mile is about $1.6$ kilometers, and you are going to a coffee shop 32 kilometers away for a gallon of coffee, how much, in yen, will it cost? [b]7 DOWN:[/b] Clive randomly orders the letters of “MIXING THE LETTERS, MAN”. If $\frac{p}{m^nq}$ is the probability that he gets “LMT IS AN EXTREME THING” where p and q are odd integers, and $m$ is a prime number, then what is $m + n$? [b]8 ACROSS:[/b] Joe is playing darts. A dartboard has scores of $10, 7$, and $4$ on it. If Joe can throw $12$ darts, how many possible scores can he end up with? [b]9 ACROSS:[/b] What is the maximum number of bounded regions that $6$ overlapping ellipses can cut the plane into? [b]10 DOWN:[/b] Let $ABC$ be an equilateral triangle, such that $A$ and $B$ both lie on a unit circle with center $O$. What is the maximum distance between $O$ and $C$? Write your answer be in the form $\frac{a\sqrt{b}}{c}$ where $b$ is not divisible by the square of any prime, and $a$ and $c$ share no common factor. What is $abc$ ? PS. You had better use hide for answers.

[b]p1.[/b] Aidan walks into a skyscraper’s first floor lobby and takes the elevator up $50$ floors. After exiting the elevator, he takes the stairs up another $10$ floors, then takes the elevator down $30$ floors. Find the floor number Aidan is currently on. [b]p2.[/b] Jeff flips a fair coin twice and Kaylee rolls a standard $6$-sided die. The probability that Jeff flips $2$ heads and Kaylee rolls a $4$ is $P$. Find $\frac{1}{P}$ . [b]p3.[/b] Given that $a\odot b = a + \frac{a}{b}$ , find $(4\odot 2)\odot 3$. [b]p4.[/b] The following star is created by gluing together twelve equilateral triangles each of side length $3$. Find the outer perimeter of the star. [img]https://cdn.artofproblemsolving.com/attachments/e/6/ad63edbf93c5b7d4c7e5d68da2b4632099d180.png[/img] [b]p5.[/b] In Lexington High School’sMath Team, there are $40$ students: $20$ of whom do science bowl and $22$ of whom who do LexMACS. What is the least possible number of students who do both science bowl and LexMACS? [b]p6.[/b] What is the least positive integer multiple of $3$ whose digits consist of only $0$s and $1$s? The number does not need to have both digits. [b]p7.[/b] Two fair $6$-sided dice are rolled. The probability that the product of the numbers rolled is at least $30$ can be written as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Find $a +b$. [b]p8.[/b] At the LHSMath Team Store, $5$ hoodies and $1$ jacket cost $\$13$, and $5$ jackets and $1$ hoodie cost $\$17$. Find how much $15$ jackets and $15$ hoodies cost, in dollars. [b]p9.[/b] Eric wants to eat ice cream. He can choose between $3$ options of spherical ice cream scoops. The first option consists of $4$ scoops each with a radius of $3$ inches, which costs a total of $\$3$. The second option consists of a scoop with radius $4$ inches, which costs a total of $\$2$. The third option consists of $5$ scoops each with diameter $2$ inches, which costs a total of $\$1$. The greatest possible ratio of volume to cost of ice cream Eric can buy is nπ cubic inches per dollar. Find $3n$. [b]p10.[/b] Jen claims that she has lived during at least part of each of five decades. What is the least possible age that Jen could be? (Assume that age is always rounded down to the nearest integer.) [b]p11.[/b] A positive integer $n$ is called Maisylike if and only if $n$ has fewer factors than $n -1$. Find the sum of the values of $n$ that are Maisylike, between $2$ and $10$, inclusive. [b]p12.[/b] When Ginny goes to the nearby boba shop, there is a $30\%$ chance that the employee gets her drink order wrong. If the drink she receives is not the one she ordered, there is a $60\%$ chance that she will drink it anyways. Given that Ginny drank a milk tea today, the probability she ordered it can be written as $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers. Find the value of $a +b$. [b]p13.[/b] Alex selects an integer $m$ between $1$ and $100$, inclusive. He notices there are the same number of multiples of $5$ as multiples of $7$ betweenm and $m+9$, inclusive. Find how many numbers Alex could have picked. [b]p14.[/b] In LMTown there are only rainy and sunny days. If it rains one day there’s a $30\%$ chance that it will rain the next day. If it’s sunny one day there’s a $90\%$ chance it will be sunny the next day. Over n days, as n approaches infinity, the percentage of rainy days approaches $k\%$. Find $10k$. [b]p15.[/b] A bag of letters contains $3$ L’s, $3$ M’s, and $3$ T’s. Aidan picks three letters at random from the bag with replacement, and Andrew picks three letters at random fromthe bag without replacement. Given that the probability that both Aidan and Andrew pick one each of L, M, and T can be written as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers, find $m+n$. [b]p16.[/b] Circle $\omega$ is inscribed in a square with side length $2$. In each corner tangent to $2$ of the square’s sides and externally tangent to $\omega$ is another circle. The radius of each of the smaller $4$ circles can be written as $(a -\sqrt{b})$ where $a$ and $b$ are positive integers. Find $a +b$. [img]https://cdn.artofproblemsolving.com/attachments/d/a/c76a780ac857f745067a8d6c4433efdace2dbb.png[/img] [b]p17.[/b] In the nonexistent land of Lexingtopia, there are $10$ days in the year, and the Lexingtopian Math Society has $5$ members. The probability that no two of the LexingtopianMath Society’s members share the same birthday can be written as $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers. Find $a +b$. [b]p18.[/b] Let $D(n)$ be the number of diagonals in a regular $n$-gon. Find $$\sum^{26}_{n=3} D(n).$$ [b]p19.[/b] Given a square $A_0B_0C_0D_0$ as shown below with side length $1$, for all nonnegative integers $n$, construct points $A_{n+1}$, $B_{n+1}$, $C_{n+1}$, and $D_{n+1}$ on $A_nB_n$, $B_nC_n$, $C_nD_n$, and $D_nA_n$, respectively, such that $$\frac{A_n A_{n+1}}{A_{n+1}B_n}=\frac{B_nB_{n+1}}{B_{n+1}C_n} =\frac{C_nC_{n+1}}{C_{n+1}D_n}=\frac{D_nD_{n+1}}{D_{n+1}A_n} =\frac34.$$ [img]https://cdn.artofproblemsolving.com/attachments/6/a/56a435787db0efba7ab38e8401cf7b06cd059a.png[/img] The sum of the series $$\sum^{\infty}_{i=0} [A_iB_iC_iD_i ] = [A_0B_0C_0D_0]+[A_1B_1C_1D_1]+[A_2B_2C_2D_2]...$$ where $[P]$ denotes the area of polygon $P$ can be written as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Find $a +b$. [b]p20.[/b] Let $m$ and $n$ be two real numbers such that $$\frac{2}{n}+m = 9$$ $$\frac{2}{m}+n = 1$$ Find the sum of all possible values of $m$ plus the sumof all possible values of $n$. [b]p21.[/b] Let $\sigma (x)$ denote the sum of the positive divisors of $x$. Find the smallest prime $p$ such that $$\sigma (p!) \ge 20 \cdot \sigma ([p -1]!).$$ [b]p22.[/b] Let $\vartriangle ABC$ be an isosceles triangle with $AB = AC$. Let $M$ be the midpoint of side $\overline{AB}$. Suppose there exists a point X on the circle passing through points $A$, $M$, and $C$ such that $BMCX$ is a parallelogram and $M$ and $X$ are on opposite sides of line $BC$. Let segments $\overline{AX}$ and $\overline{BC}$ intersect at a point $Y$ . Given that $BY = 8$, find $AY^2$. [b]p23.[/b] Kevin chooses $2$ integers between $1$ and $100$, inclusive. Every minute, Corey can choose a set of numbers and Kevin will tell him how many of the $2$ chosen integers are in the set. How many minutes does Corey need until he is certain of Kevin’s $2$ chosen numbers? [b]p24.[/b] Evaluate $$1^{-1} \cdot 2^{-1} +2^{-1} \cdot 3^{-1} +3^{-1} \cdot 4^{-1} +...+(2015)^{-1} \cdot (2016)^{-1} \,\,\, (mod \,\,\,2017).$$ [b]p25.[/b] In scalene $\vartriangle ABC$, construct point $D$ on the opposite side of $AC$ as $B$ such that $\angle ABD = \angle DBC = \angle BC A$ and $AD =DC$. Let $I$ be the incenter of $\vartriangle ABC$. Given that $BC = 64$ and $AD = 225$, find$ BI$ . [img]https://cdn.artofproblemsolving.com/attachments/b/1/5852dd3eaace79c9da0fd518cfdcd5dc13aecf.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let be a twice-differentiable function $ f:(0,\infty )\longrightarrow\mathbb{R} $ that admits a polynomial function of degree $ 1 $ or $ 2, $ namely, $ \alpha :(0,\infty )\longrightarrow\mathbb{R} $ as its asymptote. Prove the following propositions: [b]a)[/b] $ f''>0\implies f-\alpha >0 $ [b]b)[/b] $ \text{supp} f''=(0,\infty )\wedge f-\alpha >0\implies f''=0 $

Prove that all numbers of the sequence \[ \frac{107811}{3}, \quad \frac{110778111}{3}, \frac{111077781111}{3}, \quad \ldots \] are exact cubes.

Solve the system of simultaneous equations \[\sqrt x - \frac 1y - 2w + 3z = 1,\]\[x + \frac{1}{y^2} - 4w^2 - 9z^2 = 3,\]\[x \sqrt x - \frac{1}{y^3} - 8w^3 + 27z^3 = -5,\]\[x^2 + \frac{1}{y^4} - 16w^4 - 81z^4 = 15.\]

Problem: Find all polynomials satisfying the equation $ f(x^2) = (f(x))^2 $ for all real numbers x. I'm not exactly sure where to start though it doesn't look too difficult. Thanks!

Radii of five concentric circles $\omega_0,\omega_1,\omega_2,\omega_3,\omega_4$ form a geometric progression with common ratio $q$ in this order. What is the maximal value of $q$ for which it's possible to draw a broken line $A_0A_1A_2A_3A_4$ consisting of four equal segments such that $A_i$ lies on $\omega_i$ for every $i=\overline{0,4}$? [hide=thanks ]Thanks to the user Vlados021 for translating the problem.[/hide]

Find the minimum value of $4(x^2 + y^2 + z^2 + w^2) + (xy - 7)^2 + (yz - 7)^2 + (zw - 7)^2 + (wx - 7)^2$ as $x, y, z$, and $w$ range over all real numbers.

$f(x) = x^3+3x^2 - 1$. Find the number of real solutions of $f(f(x)) = 0$.

[u]Set 6[/u] [b]B16.[/b] Let $\ell_r$ denote the line $x + ry + r^2 = 420$. Jeffrey draws the lines $\ell_a$ and $\ell_b$ and calculates their single intersection point. [b]B17.[/b] Let set $L$ consist of lines of the form $3x + 2ay = 60a + 48$ across all real constants a. For every line $\ell$ in $L$, the point on $\ell$ closest to the origin is in set $T$ . The area enclosed by the locus of all the points in $T$ can be expressed in the form nπ for some positive integer $n$. Compute $n$. [b]B18.[/b] What is remainder when the $2020$-digit number $202020 ... 20$ is divided by $275$? [u]Set 7[/u] [b]B19.[/b] Consider right triangle $\vartriangle ABC$ where $\angle ABC = 90^o$, $\angle ACB = 30^o$, and $AC = 10$. Suppose a beam of light is shot out from point $A$. It bounces off side $BC$ and then bounces off side $AC$, and then hits point $B$ and stops moving. If the beam of light travelled a distance of $d$, then compute $d^2$. [b]B20.[/b] Let $S$ be the set of all three digit numbers whose digits sum to $12$. What is the sum of all the elements in $S$? [b]B21.[/b] Consider all ordered pairs $(m, n)$ where $m$ is a positive integer and $n$ is an integer that satisfy $$m! = 3n^2 + 6n + 15,$$ where $m! = m \times (m - 1) \times ... \times 1$. Determine the product of all possible values of $n$. [u]Set 8[/u] [b]B22.[/b] Compute the number of ordered pairs of integers $(m, n)$ satisfying $1000 > m > n > 0$ and $6 \cdot lcm(m - n, m + n) = 5 \cdot lcm(m, n)$. [b]B23.[/b] Andrew is flipping a coin ten times. After every flip, he records the result (heads or tails). He notices that after every flip, the number of heads he had flipped was always at least the number of tails he had flipped. In how many ways could Andrew have flipped the coin? [b]B24.[/b] Consider a triangle $ABC$ with $AB = 7$, $BC = 8$, and $CA = 9$. Let $D$ lie on $\overline{AB}$ and $E$ lie on $\overline{AC}$ such that $BCED$ is a cyclic quadrilateral and $D, O, E$ are collinear, where $O$ is the circumcenter of $ABC$. The area of $\vartriangle ADE$ can be expressed as $\frac{m\sqrt{n}}{p}$, where $m$ and $p$ are relatively prime positive integers, and $n$ is a positive integer not divisible by the square of any prime. What is $m + n + p$? [u]Set 9[/u] [i]This set consists of three estimation problems, with scoring schemes described.[/i] [b]B25.[/b] Submit one of the following ten numbers: $$3 \,\,\,\, 6\,\,\,\, 9\,\,\,\, 12\,\,\,\, 15\,\,\,\, 18\,\,\,\, 21\,\,\,\, 24\,\,\,\, 27\,\,\,\, 30.$$ The number of points you will receive for this question is equal to the number you selected divided by the total number of teams that selected that number, then rounded up to the nearest integer. For example, if you and four other teams select the number $27$, you would receive $\left\lceil \frac{27}{5}\right\rceil = 6$ points. [b]B26.[/b] Submit any integer from $1$ to $1,000,000$, inclusive. The standard deviation $\sigma$ of all responses $x_i$ to this question is computed by first taking the arithmetic mean $\mu$ of all responses, then taking the square root of average of $(x_i -\mu)^2$ over all $i$. More, precisely, if there are $N$ responses, then $$\sigma =\sqrt{\frac{1}{N} \sum^N_{i=1} (x_i -\mu)^2}.$$ For this problem, your goal is to estimate the standard deviation of all responses. An estimate of $e$ gives $\max \{ \left\lfloor 130 ( min \{ \frac{\sigma }{e},\frac{e}{\sigma }\}^{3}\right\rfloor -100,0 \}$ points. [b]B27.[/b] For a positive integer $n$, let $f(n)$ denote the number of distinct nonzero exponents in the prime factorization of $n$. For example, $f(36) = f(2^2 \times 3^2) = 1$ and $f(72) = f(2^3 \times 3^2) = 2$. Estimate $N = f(2) + f(3) +.. + f(10000)$. An estimate of $e$ gives $\max \{30 - \lfloor 7 log_{10}(|N - e|)\rfloor , 0\}$ points. PS. You had better use hide for answers. First sets have been posted [url=https://artofproblemsolving.com/community/c4h2777391p24371239]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all real parameters $a$ for which the equation $x^8 +ax^4 +1 = 0$ has four real roots forming an arithmetic progression.

Let $a,b,c,d$ be positive real numbers such that $abcd=1$. Prove the inequality \[\frac{1}{\sqrt{a+2b+3c+10}}+\frac{1}{\sqrt{b+2c+3d+10}}+\frac{1}{\sqrt{c+2d+3a+10}}+\frac{1}{\sqrt{d+2a+3b+10}} \le 1.\]

Find all functions $ f: \mathbb{Q}^{\plus{}} \mapsto \mathbb{Q}^{\plus{}}$ such that: \[ f(x) \plus{} f(y) \plus{} 2xy f(xy) \equal{} \frac {f(xy)}{f(x\plus{}y)}.\]

Let $x$, $y$, and $z$ be positive real numbers such that $x + y + z = 1$. Prove that $\frac{(x+y)^3}{z} + \frac{(y+z)^3}{x} + \frac{(z+x)^3}{y} + 9xyz \ge 9(xy + yz + zx)$. When does equality hold?