[b]p1.[/b] Fluffy the Dog is an extremely fluffy dog. Because of his extreme fluffiness, children always love petting Fluffy anywhere. Given that Fluffy likes being petted $1/4$ of the time, out of $120$ random people who each pet Fluffy once, what is the expected number of times Fluffy will enjoy being petted? [b]p2.[/b] Andy thinks of four numbers $27$, $81$, $36$, and $41$ and whispers the numbers to his classmate Cynthia. For each number she hears, Cynthia writes down every factor of that number on the whiteboard. What is the sum of all the different numbers that are on the whiteboard? (Don't include the same number in your sum more than once) [b]p3.[/b] Charles wants to increase the area his square garden in his backyard. He increases the length of his garden by $2$ and increases the width of his garden by $3$. If the new area of his garden is $182$, then what was the original area of his garden? [b]p4.[/b] Antonio is trying to arrange his flute ensemble into an array. However, when he arranges his players into rows of $6$, there are $2$ flute players left over. When he arranges his players into rows of $13$, there are $10$ flute players left over. What is the smallest possible number of flute players in his ensemble such that this number has three prime factors? [b]p5.[/b] On the AMC $9$ (Acton Math Competition $9$), $5$ points are given for a correct answer, $2$ points are given for a blank answer and $0$ points are given for an incorrect answer. How many possible scores are there on the AMC $9$, a $15$ problem contest? [b]p6.[/b] Charlie Puth produced three albums this year in the form of CD's. One CD was circular, the second CD was in the shape of a square, and the final one was in the shape of a regular hexagon. When his producer circumscribed a circle around each shape, he noticed that each time, the circumscribed circle had a radius of $10$. The total area occupied by $1$ of each of the different types of CDs can be expressed in the form $a + b\pi + c\sqrt{d}$ where $d$ is not divisible by the square of any prime. Find $a + b + c + d$. [b]p7.[/b] You are picking blueberries and strawberries to bring home. Each bushel of blueberries earns you $10$ dollars and each bushel of strawberries earns you $8$ dollars. However your cart can only fit $24$ bushels total and has a weight limit of $100$ lbs. If a bushel of blueberries weighs $8$ lbs and each bushel of strawberries weighs $6$ lbs, what is your maximum profit. (You can only pick an integer number of bushels) [b]p8.[/b] The number $$\sqrt{2218 + 144\sqrt{35} + 176\sqrt{55} + 198\sqrt{77}}$$ can be expressed in the form $a\sqrt5 + b\sqrt7 + c\sqrt{11}$ for positive integers $a, b, c$. Find $abc$. [b]p9.[/b] Let $(x, y)$ be a point such that no circle passes through the three points $(9,15)$, $(12, 20)$, $(x, y)$, and no circle passes through the points $(0, 17)$, $(16, 19)$, $(x, y)$. Given that $x - y = -\frac{p}{q}$ for relatively prime positive integers $p$, $q$, Find $p + q$. [b]p10.[/b] How many ways can Alfred, Betty, Catherine, David, Emily and Fred sit around a $6$ person table if no more than three consecutive people can be in alphabetical order (clockwise)? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Sequences $x_n$ and $y_n$ satisfy the simultaneous relationships $x_k = x_{k+1} + y_{k+1}$ and $x_k > y_k$ for all $k \ge 1$. Furthermore, either $y_k = y_{k+1}$ or $y_k = x_{k+1}$. If $x_1 = 3 + \sqrt2$, $x_3 = 5 -\sqrt2$, and $y_1 = y_5$, evaluate $$(y_1)^2 + (y_2)^2 + (y_3)^2 + . . .$$

Denote by $\mathbb{N}$ the positive integers. Let $f:\mathbb{N} \rightarrow \mathbb{N}$ be a function such that, for any $w,x,y,z \in \mathbb{N}$, \[ f(f(f(z)))f(wxf(yf(z)))=z^{2}f(xf(y))f(w). \] Show that $f(n!) \ge n!$ for every positive integer $n$. [i]Pakawut Jiradilok[/i]

Find all real numbers $q$, such that for all real $p \geq 0$, the equation $x^2-2px+q^2+q-2=0$ has at least one real root in $(-1;0)$.

[b]p1.[/b] Find all integer solutions of the equation $a^2 - b^2 = 2002$. [b]p2.[/b] Prove that the disks drawn on the sides of a convex quadrilateral as on diameters cover this quadrilateral. [b]p3.[/b] $30$ students from one school came to Mathematical Olympiad. In how many different ways is it possible to place them in four rooms? [b]p4.[/b] A $12$ liter container is filled with gasoline. How to split it in two equal parts using two empty $5$ and $8$ liter containers? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all cubic polynomials $x^3 +ax^2 +bx+c$ admitting the rational numbers $a$, $b$ and $c$ as roots.

You are given $n\ge 4$ positive real numbers. It turned out that all $\frac{n(n-1)}{2}$ of their pairwise products form an arithmetic progression in some order. Show that all given numbers are equal. [i](Proposed by Anton Trygub)[/i]

Do there exist two real monic polynomials $P(x)$ and $Q(x)$ of degree 3,such that the roots of $P(Q(X))$ are nine pairwise distinct nonnegative integers that add up to $72$? (In a monic polynomial of degree 3, the coefficient of $x^{3}$ is $1$.)

Let be two polynoms $ P,Q\in\mathbb{R} [X] $ having the property that $$ \left| \{ n\in\{ 0\}\cup\mathbb{N} | P(n)\le Q(n) \} \right| =\left| \{ n\in\{ 0\}\cup\mathbb{N} | P(n)\ge Q(n) \} \right| =\infty .$$ Show that $ P=Q. $ [i]Laurențiu Panaitopol[/i]

Does there exist a polynomial \( P(x,y) \) in two variables with real coefficients, such that the following two conditions hold: 1) \( P(x,y) = P(x, x-y) = P(y-x, y) \) for any real numbers \( x \) and \( y \); 2) There does not exist a polynomial \( Q(z) \) in one variable with real coefficients such that \( P(x,y) = Q(x^2 - xy + y^2) \) for any real numbers \( x \) and \( y \)?

$\boxed{A3}$The sequence $a_1,a_2,a_3,...$ is defined by $a_1=a_2=1,a_{2n+1}=2a_{2n}-a_n$ and $a_{2n+2}=2a_{2n+1}$ for $n\in{N}.$Prove that if $n>3$ and $n-3$ is divisible by $8$ then $a_n$ is divisible by $5$

Let $f(0, 0) = 5^{2003}, f(0, n) = 0$ for every integer $n \neq 0$ and \[\begin{array}{c}\ f(m, n) = f(m-1, n) - 2 \cdot \Bigg\lfloor \frac{f(m-1, n)}{2}\Bigg\rfloor + \Bigg\lfloor\frac{f(m-1, n-1)}{2}\Bigg\rfloor + \Bigg\lfloor\frac{f(m-1, n+1)}{2}\Bigg\rfloor \end{array}\] for every natural number $m > 0$ and for every integer $n$. Prove that there exists a positive integer $M$ such that $f(M, n) = 1$ for all integers $n$ such that $|n| \leq \frac{(5^{2003}-1)}{2}$ and $f(M, n) = 0$ for all integers n such that $|n| > \frac{5^{2003}-1}{2}.$

A farmer divides his herd of $n$ cows among his four sons so that one son gets one-half the herd, a second son, one-fourth, a third son, one-fifth, and the fourth son, 7 cows. Then $n$ is: $ \textbf{(A)}\ 80 \qquad\textbf{(B)}\ 100\qquad\textbf{(C)}\ 140\qquad\textbf{(D)}\ 180\qquad\textbf{(E)}\ 240 $

Find all polynomials $p(x)$ with real coeffcients such that \[p(a + b - 2c) + p(b + c - 2a) + p(c + a - 2b) = 3p(a - b) + 3p(b - c) + 3p(c - a)\] for all $a, b, c\in\mathbb{R}$. [i](2nd Benelux Mathematical Olympiad 2010, Problem 2)[/i]

Prove the following inequalities: (1) $ x\minus{}\sin x\leq \tan x\minus{}x\ \ \left(0\leq x<\frac{\pi}{2}\right)$ (2) $ \int_0^x \cos (\tan t\minus{}t)\ dt\leq \sin (\sin x)\plus{}\frac 12 \left(x\minus{}\frac{\sin 2x}{2}\right)\ \left(0\leq x\leq \frac{\pi}{3}\right)$

Prove that $$ |\cos n\beta - \cos n\alpha| \leq n^2 |\cos \beta - \cos\alpha|,$$ where $n$ is a natural number . Check for what values of $ n $, $ \alpha $, $ \beta $ equality holds.

Given an integer $n \geq 3$, determine if there are $n$ integers $b_1, b_2, \dots , b_n$, distinct two-by-two (that is, $b_i \neq b_j$ for all $i \neq j$) and a polynomial $P(x)$ with coefficients integers, such that $P(b_1) = b_2, P(b_2) = b_3, \dots , P(b_{n-1}) = b_n$ and $P(b_n) = b_1$.

A sequence of natural numbers $a_1,a_2,...$ satisfies $a_1 = 1, a_{n+2} = 2a_{n+1} - a_n +2$ for $n \in N$. Prove that for every natural $n$ there exists a natural $m$ such that $a_na_{n+1} = a_m$.

Let $ g: \mathbb{C} \rightarrow \mathbb{C}$, $ \omega \in \mathbb{C}$, $ a \in \mathbb{C}$, $ \omega^3 \equal{} 1$, and $ \omega \ne 1$. Show that there is one and only one function $ f: \mathbb{C} \rightarrow \mathbb{C}$ such that \[ f(z) \plus{} f(\omega z \plus{} a) \equal{} g(z),z\in \mathbb{C} \]

Let $f$ be a polynomial of degree $98$, such that $f (k) =\frac{1}{k}$ for $k=1,2,3,\ldots,99$. Determine $f(100)$.

[i]Version 1[/i]. Let $n$ be a positive integer, and set $N=2^{n}$. Determine the smallest real number $a_{n}$ such that, for all real $x$, \[ \sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant a_{n}(x-1)^{2}+x . \] [i]Version 2[/i]. For every positive integer $N$, determine the smallest real number $b_{N}$ such that, for all real $x$, \[ \sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant b_{N}(x-1)^{2}+x . \]

Find all pairs of complex numbers $(z,w) \in \mathbb{C}^2$ such that the relation \[|z^{2n}+z^nw^n+w^{2n} | = 2^{2n}+2^n+1 \] holds for all positive integers $n$.

Chandler the Octopus along with his friends Maisy the Bear and Jeff the Frog are solving LMT problems. It takes Maisy $3$ minutes to solve a problem, Chandler $4$ minutes to solve a problem and Jeff $5$ minutes to solve a problem. They start at $12:00$ pm, and Chandler has a dentist appointment from $12:10$ pm to $12:30$, after which he comes back and continues solving LMT problems. The time it will take for them to finish solving $50$ LMT problems, in hours, is $m/n$ ,where $m$ and $n$ are relatively prime positive integers. Find $m +n$. [b]Note:[/b] they may collaborate on problems. [i]Proposed by Aditya Rao[/i]

Given $2025$ pairwise distinct positive integer numbers \(a_1, a_2, \ldots, a_{2025}\), find the maximum possible number of equal numbers among the fractions of the form \[ \frac{a_i^2 + a_j^2}{a_i + a_j} \] [i]Proposed by Mykhailo Shtandenko[/i]

Find all real numbers $x$ such that $-1 < x \le 2 $ and $$\sqrt{2 - x}+\sqrt{2 + 2x} =\sqrt{\frac{x^4 + 1}{x^2 + 1}}+ \frac{x + 3}{x + 1}.$$ .