[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].

Prove that a quadratic trinomial $x^2 + ax + b (a, b \in R)$ cannot attain at ten consecutive integral points values equal to powers of $2$ with non-negative integral exponent. [i](F. Petrov )[/i]

Let $S$ be the set of degree $4$ polynomials $f$ with complex number coefficients satisfying $f(1)=f(2)^2=f(3)^3$ $=$ $f(4)^4=f(5)^5=1.$ Find the mean of the fifth powers of the constant terms of all the members of $S.$

Let $a, b, c > 0$ be real numbers with $abc = 1$. Show $$1 + ab + bc + ca \ge \min \left\{ \frac{(a + b)^2}{ab} , \frac{(b+c)^2}{bc} , \frac{(c + a)^2}{ca}\right\}.$$ When does equality holds?

Let $ a_1, a_2, \ldots , a_n$ be distinct positive integers and let $ M$ be a set of $ n \minus{} 1$ positive integers not containing $ s \equal{} a_1 \plus{} a_2 \plus{} \ldots \plus{} a_n.$ A grasshopper is to jump along the real axis, starting at the point $ 0$ and making $ n$ jumps to the right with lengths $ a_1, a_2, \ldots , a_n$ in some order. Prove that the order can be chosen in such a way that the grasshopper never lands on any point in $ M.$ [i]Proposed by Dmitry Khramtsov, Russia[/i]

Let $0 < a < \frac{\pi}{2}$ and $x_1,x_2,...,x_n$ be real numbers such that $\sin x_1 + \sin x_2 +... + \sin x_n \ge n \cdot sin a $. Prove that $\sin (x_1 - a) + \sin (x_2 - a) + ... + \sin (x_n - a) \ge 0$ .

Given $t\in\mathbb C$. Complex numbers $x,y,z$ satisfy that $|x|=|y|=|z|=1$ and $\frac{t}{y}=\frac{1}{x}+\frac{1}{z}$. Calculate $$\left|\frac{2xy+2yz+3zx}{x+y+z}\right|.$$

Real numbers $a,b,c,d$ are given. Solve the system of equations (unknowns $x,y,z,u)$\[ x^{2}-yz-zu-yu=a\] \[ y^{2}-zu-ux-xz=b\] \[ z^{2}-ux-xy-yu=c\] \[ u^{2}-xy-yz-zx=d\]

Let $\{a_n\}_{n=1}^{\infty}$ be a sequence of positive integers for which \[ a_{n+2} = \left[\frac{2a_n}{a_{n+1}}\right]+\left[\frac{2a_{n+1}}{a_n}\right]. \] Prove that there exists a positive integer $m$ such that $a_m=4$ and $a_{m+1} \in\{3,4\}$. [b]Note.[/b] $[x]$ is the greatest integer not exceeding $x$.

Consider the set of all strictly decreasing sequences of $n$ natural numbers having the property that in each sequence no term divides any other term of the sequence. Let $A = (a_j)$ and $B = (b_j)$ be any two such sequences. We say that $A$ precedes $B$ if for some $k$, $a_k < b_k$ and $a_i = b_i$ for $i < k$. Find the terms of the first sequence of the set under this ordering.

Let $a_1,a_2,\ldots a_n,k$, and $M$ be positive integers such that $$\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}=k\quad\text{and}\quad a_1a_2\cdots a_n=M.$$ If $M>1$, prove that the polynomial $$P(x)=M(x+1)^k-(x+a_1)(x+a_2)\cdots (x+a_n)$$ has no positive roots.

Determine all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that for every $x,y \in \mathbb{R}$ $$f(x^{4}-y^{4})+4f(xy)^{2}=f(x^{4}+y^{4})$$

Fix two positive integers $a,k\ge2$, and let $f\in\mathbb{Z}[x]$ be a nonconstant polynomial. Suppose that for all sufficiently large positive integers $n$, there exists a rational number $x$ satisfying $f(x)=f(a^n)^k$. Prove that there exists a polynomial $g\in\mathbb{Q}[x]$ such that $f(g(x))=f(x)^k$ for all real $x$. [i]Victor Wang.[/i]

For every sequence $\{a_n\}~(n\in\mathbb N)$ we define the sequences $\{\Delta a_n\}$ and $\{\Delta^2a_n\}$ by the following formulas: \begin{align*}\Delta a_n&=a_{n+1}-a_n,\\\Delta^2a_n&=\Delta a_{n+1}-\Delta a_n.\end{align*}Further, for all $n\in\mathbb N$ for which $\Delta a_n^2\ne0$, define $$a_n'=a_n-\frac{(\Delta a_n)^2}{\Delta^2a_n}.$$ (a) For which sequences $\{a_n\}$ is the sequence $\{\Delta^2a_n\}$ constant? (b) Find all sequences $\{a_n\}$, for which the numbers $a_n'$ are defined for all $n\in\mathbb N$ and for which the sequence $\{a_n'\}$ is constant. (c) Assume that the sequence $\{a_n\}$ converges to $a=0$, and $a_n\ne a$ for all $n\in\mathbb N$ and the sequence $\{\tfrac{a_{n+1}-a}{a_n-a}\}$ converges to $\lambda\ne1$. i. Prove that $\lambda\in[-1,1)$. ii. Prove that there exists $n_0\in\mathbb N$ such that for all integers $n\ge n_0$ we have $\Delta^2a_n\ne0$. iii. Let $\lambda\ne0$. For which $k\in\mathbb Z$ is the sequence $\{\tfrac{a_n'}{a_{n+k}}\}$ not convergent? iv. Let $\lambda=0$. Prove that the sequences $\{a_n'/a_n\}$ and $\{a_n'/a_{n+1}\}$ converge to $0$. Find an example of $\{a_n\}$ for which the sequence $\{a_n'/a_{n+2}\}$ has a non-zero limit. (d) What happens with part (c) if we remove the condition $a=0$?

Find the minimum value of the expression $f(a, b, c) = (a + b)^4 + (b + c)^4 + (c + a)^4 - \frac47 (a^4 + b^4 + c^4)$, as $a, b, c$ varies over the set of all real numbers

Let $P_i(x) = x^2 + b_i x + c_i , i = 1,2, ..., n$ be pairwise distinct polynomials of degree $2$ with real coefficients so that for any $0 \le i < j \le n , i, j \in N$, the polynomial $Q_{i,j}(x) = P_i(x) + P_j(x)$ has only one real root. Find the greatest possible value of $n$.

Let $p > 3$ be a prime. Prove that if \[ \sum_{i=1 }^{p-1}{1\over i^p} = {n\over m}, \] with $\gdc(n,m) = 1$, then $p^3$ divides $n$.

A polynomial $P$ with integer coefficients has at least $13$ distinct integer roots. Prove that if an integer $n$ is not a root of $P$, then $|P(n)| \geq 7 \cdot 6!^2$, and give an example for equality.

Numbers from $1$ to $8$ are placed on the vertices of a cube, one on each of the eight vertices, so that the sum of the numbers on any three vertices of the same face is greater than $9$. Determines the minimum value that the sum of the numbers on one side can have.

Denote a set of equations in the real numbers with variables $x_1, x_2, x_3 \in \mathbb{R}$ Flensburgian if there exists an $i \in \{1, 2, 3\}$ such that every solution of the set of equations where all the variables are pairwise different, satisfies $x_i>x_j$ for all $j \neq i$. Find all positive integers $n \geq 2$, such that the following set of two equations $a^n+b=a$ and $c^{n+1}+b^2=ab$ in three real variables $a,b,c$ is Flensburgian.

We have $0\le x_i<b$ for $i=0,1,\ldots,n$ and $x_n>0,x_{n-1}>0$. If $a>b$, and $x_nx_{n-1}\ldots x_0$ represents the number $A$ base $a$ and $B$ base $b$, whilst $x_{n-1}x_{n-2}\ldots x_0$ represents the number $A'$ base $a$ and $B'$ base $b$, prove that $A'B<AB'$.

Determine all functions $f:(0,\infty)\to\mathbb{R}$ satisfying $$\left(x+\frac{1}{x}\right)f(y)=f(xy)+f\left(\frac{y}{x}\right)$$ for all $x,y>0$.

Which expression is equal to $\left | a-2-\sqrt{(a-1)^2} \right|$ for $a<0$? $\textbf{(A) } 3-2a \qquad \textbf{(B) } 1-a \qquad \textbf{(C) } 1 \qquad \textbf{(D) } a+1 \qquad \textbf{(E) } 3$

Let $[n]$ denote the set of integers $0, 1, \ldots, n-1.$ Let $\omega_n=e^{2\pi i/n}.$ Let $$f(n) = \prod_{\overset{i \in [n]}{\gcd(i,n)=1}} \prod_{\overset{j \in [n]}{\gcd(j,n)=1}} (\omega_n^i - \omega_n^j).$$ Then, $f(2024)=2^{e_1} \cdot 11^{e_2} \cdot 23^{e_3}$ for positive integers $e_1, e_2, e_3.$ Find $e_1+e_2+e_3.$

Prove that if a quadratic equation $$ ax^2 + bx + c = 0$$ with integer coefficients has a rational root, then at least one of the numbers $ a $, $ b $, $ c $ is even.