Prove that for all positive real numbers $a$ and $ b$: $$\frac{(a + b)^3}{4} \ge a^2b + ab^2$$

[b]p1.[/b] Let $a$ be an integer. How many fractions $\frac{a}{100}$ are greater than $\frac17$ and less than $\frac13$ ?. [b]p2.[/b] Justin Bieber invited Justin Timberlake and Justin Shan to eat sushi. There were $5$ different kinds of fish, $3$ different rice colors, and $11$ different sauces. Justin Shan insisted on a spicy sauce. If the probability of a sushi combination that pleased Justin Shan is $6/11$, then how many non-spicy sauces were there? [b]p3.[/b] A palindrome is any number that reads the same forward and backward (for example, $99$ and $50505$ are palindromes but $2020$ is not). Find the sum of all three-digit palindromes whose tens digit is $5$. [b]p4.[/b] Isaac is given an online quiz for his chemistry class in which he gets multiple tries. The quiz has $64$ multiple choice questions with $4$ choices each. For each of his previous attempts, the computer displays Isaac's answer to that question and whether it was correct or not. Given that Isaac is too lazy to actually read the questions, the maximum number of times he needs to attempt the quiz to guarantee a $100\%$ can be expressed as $2^{2^k}$. Find $k$. [b]p5.[/b] Consider a three-way Venn Diagram composed of three circles of radius $1$. The area of the entire Venn Diagram is of the form $\frac{a}{b}\pi +\sqrt{c}$ for positive integers $a$, $b$, $c$ where $a$, $b$ are relatively prime. Find $a+b+c$. (Each of the circles passes through the center of the other two circles) [b]p6.[/b] The sum of two four-digit numbers is $11044$. None of the digits are repeated and none of the digits are $0$s. Eight of the digits from $1-9$ are represented in these two numbers. Which one is not? [b]p7.[/b] Al wants to buy cookies. He can buy cookies in packs of $13$, $15$, or $17$. What is the maximum number of cookies he can not buy if he must buy a whole number of packs of each size? [b]p8.[/b] Let $\vartriangle ABC$ be a right triangle with base $AB = 2$ and hypotenuse $AC = 4$ and let $AD$ be a median of $\vartriangle ABC$. Now, let $BE$ be an altitude in $\vartriangle ABD$ and let $DF$ be an altitude in $\vartriangle ADC$. The quantity $(BE)^2 - (DF)^2$ can be expressed as a common fraction $\frac{a}{b}$ in lowest terms. Find $a + b$. [b]p9.[/b] Let $P(x)$ be a monic cubic polynomial with roots $r$, $s$, $t$, where $t$ is real. Suppose that $r + s + 2t = 8$, $2rs + rt + st = 12$ and $rst = 9$. Find $|P(2)|$. [b]p10.[/b] Let S be the set $\{1, 2,..., 21\}$. How many $11$-element subsets $T$ of $S$ are there such that there does not exist two distinct elements of $T$ such that one divides the other? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Define the functions $f, F : \mathbb N \to \mathbb N$, by \[f(n)=\left[ \frac{3-\sqrt 5}{2} n \right] , F(k) =\min \{n \in \mathbb N|f^k(n) > 0 \},\] where $f^k = f \circ \cdots \circ f$ is $f$ iterated $n$ times. Prove that $F(k + 2) = 3F(k + 1) - F(k)$ for all $k \in \mathbb N.$

Let $P \in \mathbb{R}[x]$. Suppose that the multiset of real roots (where roots are counted with multiplicity) of $P(x)-x$ and $P^3(x)-x$ are distinct. Prove that for all $n\in \mathbb{N}$, $P^n(x)-x$ has at least $\sigma(n)-2$ distinct real roots. (Here $P^n(x):=P(P^{n-1}(x))$ with $P^1(x) = P(x)$, and $\sigma(n)$ is the sum of all positive divisors of $n$). [i]Proposed by Malay Mahajan[/i]

Let $a_0,a_1,a_2,\dots $ be a sequence of real numbers such that $a_0=0, a_1=1,$ and for every $n\geq 2$ there exists $1 \leq k \leq n$ satisfying \[ a_n=\frac{a_{n-1}+\dots + a_{n-k}}{k}. \]Find the maximum possible value of $a_{2018}-a_{2017}$.

Ephramis taking his final exams. He has $7$ exams and his school holds finals over $3$ days. For a certain arrangement of finals, let $f$ be the maximum number of finals Ephram takes on any given day. Find the expected value of $f$ .

Find all non-constant polynoms $ f\in\mathbb{Q} [X] $ that don't have any real roots in the interval $ [0,1] $ and for which there exists a function $ \xi :[0,1]\longrightarrow\mathbb{Q} [X]\times\mathbb{Q} [X], \xi (x):=\left( g_x,h_x \right) $ such that $ h_x(x)\neq 0 $ and $ \int_0^x \frac{dt}{f(t)} =\frac{g_x(x)}{h_x(x)} , $ for all $ x\in [0,1] . $

Let $f(x)$ be a quadratic polynomial with two real roots in the interval $[-1,1]$. Prove that if the maximum value of $|f(x)|$ in the interval $[-1,1]$ is equal to $1$, then the maximum value of $|f'(x)|$ in the interval $[-1,1]$ is not less than $1$.

Let $p=4k+3$ be a prime. Prove that if \[\dfrac {1} {0^2+1}+\dfrac{1}{1^2+1}+\cdots+\dfrac{1}{(p-1)^2+1}=\dfrac{m} {n}\] (where the fraction $\dfrac {m} {n}$ is in reduced terms), then $p \mid 2m-n$. [i]Proposed by A. Golovanov[/i]

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 $Q \left( - \frac{1}{2} \right)$.

Find all functions $f:\mathbb{R}\to\mathbb{R}$ so that $(x+y)(f(x)-f(y))=(x-y)f(x+y)$ for all $x,y$ that belongs to $\mathbb{R}$.

Find the functions $f: \mathbb{R} \to \mathbb{R}$ that satisfy $$(f(x)-y)f(x+f(y))=f(x^2)-yf(y),$$ for all real numbers $x$ and $y.$

The equation \[ x^{10}+(13x-1)^{10}=0 \] has 10 complex roots $r_1, \overline{r_1}, r_2, \overline{r_2}, r_3, \overline{r_3}, r_4, \overline{r_4}, r_5, \overline{r_5},$ where the bar denotes complex conjugation. Find the value of \[ \frac 1{r_1\overline{r_1}}+\frac 1{r_2\overline{r_2}}+\frac 1{r_3\overline{r_3}}+\frac 1{r_4\overline{r_4}}+\frac 1{r_5\overline{r_5}}. \]

Consider (3-variable) polynomials \[P_n(x,y,z)=(x-y)^{2n}(y-z)^{2n}+(y-z)^{2n}(z-x)^{2n}+(z-x)^{2n}(x-y)^{2n}\] and \[Q_n(x,y,z)=[(x-y)^{2n}+(y-z)^{2n}+(z-x)^{2n}]^{2n}.\] Determine all positive integers $n$ such that the quotient $Q_n(x,y,z)/P_n(x,y,z)$ is a (3-variable) polynomial with rational coefficients.

Let $k$, $\ell $ and $m$ be positive integers. Let $ABCDEF$ be a hexagon that has a center of symmetry whose angles are all $120^\circ$ and let its sidelengths be $AB=k$, $BC=\ell$ and $CD=m$. Let $f(k,\ell,m)$ denote the number of ways we can partition hexagon $ABCDEF$ into rhombi with unit sides and an angle of $120^\circ$. Prove that by fixing $\ell$ and $m$, there exists polynomial $g_{\ell,m}$ such that $f(k,\ell,m)=g_{\ell,m}(k)$ for every positive integer $k$, and find the degree of $g_{\ell,m}$ in terms of $\ell$ and $m$. [i]Submitted by Zoltán Gyenes, Budapest[/i]

The monic quadratic trinomial $f(x)$ has $2$ different roots. Could it be that the equation $f(f(x)) = 0$ has $3$ different root, and the equation $f(f(f(x))) = 0$ has $7$ different roots?

Let $p(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$ be a monic polynomial of degree $n>2$, with real coefficients and all its roots real and different from zero. Prove that for all $k=0,1,2,\cdots,n-2$, at least one of the coefficients $a_k,a_{k+1}$ is different from zero.

How many polynomial functions $f$ of degree $\ge 1$ satisfy \[ f(x^2)=[f(x)]^2=f(f(x)) \ ? \] $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ \text{finitely many but more than 2} \\ \qquad\textbf{(E)}\ \text{infinitely many} $

Find all quadruplets $(x_1, x_2, x_3, x_4)$ of real numbers such that the next six equalities apply: $$\begin{cases} x_1 + x_2 = x^2_3 + x^2_4 + 6x_3x_4\\ x_1 + x_3 = x^2_2 + x^2_4 + 6x_2x_4\\ x_1 + x_4 = x^2_2 + x^2_3 + 6x_2x_3\\ x_2 + x_3 = x^2_1 + x^2_4 + 6x_1x_4\\ x_2 + x_4 = x^2_1 + x^2_3 + 6x_1x_3 \\ x_3 + x_4 = x^2_1 + x^2_2 + 6x_1x_2 \end{cases}$$

Jason and Valerie agree to meet for game night, which runs from $4:00$ PM to $5:00$ PM. Jason and Valerie each choose a random time from $4:00$ PM to $5:00$ PM to show up. If Jason arrives first, he will wait $20$ minutes for Valerie before leaving. If Valerie arrives first, she will wait $10$ minutes for Jason before leaving. What is the probability that Jason and Valerie successfully meet each other for game night?

Let $m$ be the largest real solution to the equation \[\frac{3}{x-3}+\frac{5}{x-5}+\frac{17}{x-17}+\frac{19}{x-19}= x^2-11x-4.\] There are positive integers $a,b,c$ such that $m = a + \sqrt{b+\sqrt{c}}$. Find $a+b+c$.

The number $2023$ is written $2023$ times on a blackboard. On one move, you can choose two numbers $x, y$ on the blackboard, delete them and write $\frac{x+y} {4}$ instead. Prove that when one number remains, it is greater than $1$.

Let $a_1,a_2,...,a_{2020}$ be positive real numbers. Prove that: $$\max{(a^2_1-a_2,a^2_2-a_3,...,a^2_{2020}-a_1)}\ge\max{(a^2_1-a_1,a^2_2-a_2,...,a^2_{2020}-a_{2020})}$$

Let $x_0,x_1,\dots, x_{n-1}$ be real numbers such that $0<|x_0|<|x_1|<\dots<|x_{n-1}|$. We will write the sum of the elements of each one of the $2^n$ subsets of $\{x_0,x_1,\dots,x_{n-1}\}$ in a paper. Prove that the $2^n$ written numbers are consecutive elements of a arithmetic progression if and only if the ratios $$|\frac{x_i}{x_j}|, 0\leq j<i\leq n-1$$ are equal(s) to the ratio(s) obtained with the numbers $2^0,2^1,\dots,2^{n-1}$. Note: The sum of the elements of the empty set is $0$.

Starting with a list of three numbers, the “[i]Make-My-Day[/i]” procedure creates a new list by replacing each number by the sum of the other two. For example, from $\{1, 3, 8\}$ “[i]Make-My-Day[/i]” gives $\{11, 9, 4\}$ and a new “[i]MakeMy-Day[/i]” leads to $\{13, 15, 20\}$. If we begin with $\{20, 1, 8\}$, what is the maximum difference between two numbers on the list after $2018$ consecutive “[i]Make-My-Day[/i]”s?