At a movie theater tickets for adults cost $4$ dollars more than tickets for children. One afternoon the theater sold $100$ more child tickets than adult tickets for a total sales amount of $1475$ dollars. How many dollars would the theater have taken in if the same tickets were sold, but the costs of the child tickets and adult tickets were reversed?

If an arc of $ 45^\circ$ on circle $ A$ has the same length as an arc of $ 30^\circ$ on circle $ B$, then the ratio of the area of circle $ A$ to the area of circle $ B$ is $ \textbf{(A)}\ \frac {4}{9} \qquad \textbf{(B)}\ \frac {2}{3} \qquad \textbf{(C)}\ \frac {5}{6} \qquad \textbf{(D)}\ \frac {3}{2} \qquad \textbf{(E)}\ \frac {9}{4}$

Let $A$, $B$, $C$, and $D$ be points in the coordinate plane on the hyperbola $\tfrac{x^{2}}{20}-\tfrac{y^{2}}{24}=1$ such that $ABCD$ is a rhombus whose diagonals intersect at the origin. Find the greatest real number that is less than $BD^{2}$ for all such rhombi.

Let $n$ ($n \ge 1$) be an integer. Consider the equation $2\cdot \lfloor{\frac{1}{2x}}\rfloor - n + 1 = (n + 1)(1 - nx)$, where $x$ is the unknown real variable. (a) Solve the equation for $n = 8$. (b) Prove that there exists an integer $n$ for which the equation has at least $2021$ solutions. (For any real number $y$ by $\lfloor{y} \rfloor$ we denote the largest integer $m$ such that $m \le y$.)

Let $A$ and $B$ be two non-empty subsets of $X = \{1, 2, . . . , 11 \}$ with $A \cup B = X$. Let $P_A$ be the product of all elements of $A$ and let $P_B$ be the product of all elements of $B$. Find the minimum and maximum possible value of $P_A +P_B$ and find all possible equality cases. [i]Proposed by Greece[/i]

[b]a)[/b] Prove that there exists an unique sequence of real numbers $ \left( x_n \right)_{n\ge 1} $ satisfying $$ -\text{ctg} x_n=x_n\in\left( (2n+1)\pi /2,(n+1)\pi \right) , $$ for any nonnegative integer $ n. $ [b]b)[/b] Show that $ \lim_{n\to\infty } \left( \frac{x_n}{(n+1)\pi } \right)^{n^2} =e^{-1/\pi^2} . $ [i]Cătălin Zârnă[/i]

Let $f:\mathbb{Q}\times\mathbb{Q}\to\mathbb{Q}$ be a function satisfying: [list] [*] For any $x_1,x_2,y_1,y_2 \in \mathbb Q$, $$f\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right) \leq \frac{f(x_1,y_1)+f(x_2,y_2)}{2}.$$ [*] $f(0,0) \leq 0$. [*] For any $x,y \in \mathbb Q$ satisfying $x^2+y^2>100$, the inequality $f(x,y)>1$ holds.\ Prove that there is some positive rational number $b$ such that for all rationals $x,y$, $$f(x,y) \ge b\sqrt{x^2+y^2} - \frac{1}{b}.$$

Arnaldo and Bernaldo play a game where they alternate saying natural numbers, and the winner is the one who says $0$. In each turn except the first the possible moves are determined from the previous number $n$ in the following way: write \[n =\sum_{m\in O_n}2^m;\] the valid numbers are the elements $m$ of $O_n$. That way, for example, after Arnaldo says $42= 2^5 + 2^3 + 2^1$, Bernaldo must respond with $5$, $3$ or $1$. We define the sets $A,B\subset \mathbb{N}$ in the following way. We have $n\in A$ iff Arnaldo, saying $n$ in his first turn, has a winning strategy; analogously, we have $n\in B$ iff Bernaldo has a winning strategy if Arnaldo says $n$ during his first turn. This way, \[A =\{0, 2, 8, 10,\cdots\}, B = \{1, 3, 4, 5, 6, 7, 9,\cdots\}\] Define $f:\mathbb{N}\to \mathbb{N}$ by $f(n)=|A\cap \{0,1,\cdots,n-1\}|$. For example, $f(8) = 2$ and $f(11)=4$. Find \[\lim_{n\to\infty}\frac{f(n)\log(n)^{2005}}{n}\]

Kevin develops a method for shuffling a stack of $10$ cards numbered $1$ through $10$. He starts with the unshuffled pile, which is in perfect order with $1$ at the top and $10$ at the bottom. He takes the top card off the unshuffled pile and places it in what he calls the shuffled pile. Then, he flips a coin. If the coin is heads, he takes the card at the top of the unshuffled pile and places it at the top of the shuffled pile. If the coin comes up tails, he places the card at the at the bottom of the shuffled pile. He repeats this process for all the remaining cards. What is the probability that at the end of this shuffling, the top card is a prime number? Express your answer as a common fraction.

Given that $g(n) = \frac{1}{{2 + \frac{1}{{3 + \frac{1}{{... + \frac{1}{{n - 1}}}}}}}}$ and $k(n) = \frac{1}{{2 + \frac{1}{{3 + \frac{1}{{... + \frac{1}{{n - 1 + \frac{1}{n}}}}}}}}}$, for natural $n$. Prove that $\left| {g(n) - k(n)} \right| \le \frac{1}{{(n - 1)!n!}}$.

A rectangle is decomposed by $6$ vertical lines and $6$ horizontal lines in the $49$ small rectangles (see figure). The perimeter of each small rectangle is known to be a whole number of meters. In this case, will the perimeter of the large rectangle be a whole number of meters? [asy] unitsize(0.8 cm); draw((0,0)--(8.6,0)--(8.6,4.2)--(0,4.2)--cycle, linewidth(1.5*bp)); draw((0.7,0)--(0.7,4.2)); draw((1.2,0)--(1.2,4.2)); draw((4.1,0)--(4.1,4.2)); draw((5.6,0)--(5.6,4.2)); draw((6.0,0)--(6.0,4.2)); draw((7.4,0)--(7.4,4.2)); draw((0,0.4)--(8.6,0.4)); draw((0,1.0)--(8.6,1.0)); draw((0,1.5)--(8.6,1.5)); draw((0,2.5)--(8.6,2.5)); draw((0,3.1)--(8.6,3.1)); draw((0,3.5)--(8.6,3.5)); [/asy]

Find the number of positive integral solutions to the equation $\sum \limits_{i=1}^{2019} 10^{a_i}=\sum \limits_{i=1}^{2019} 10^{b_i}$, such that $a_1<a_2<\cdots <a_{2019}$ , $b_1<b_2<\cdots <b_{2019}$ and $a_{2019} < b_{2019}$ [list=1] [*] 1 [*] 2 [*] 2019 [*] None of the above [/list]

Do there exist two quadratic trinomials $ax^2 +bx+c$ and $(a+1)x^2 +(b + 1)x + (c + 1)$ with integer coeficients, both of which have two integer roots? [i]N. Agakhanov[/i]

Let $A$ and $B$ be sets of positive integers with $|A|\ge 2$ and $|B|\ge 2$. Let $S$ be a set consisting of $|A|+|B|-1$ numbers of the form $ab$ where $a\in A$ and $b\in B$. Prove that there exist pairwise distinct $x,y,z\in S$ such that $x$ is a divisor of $yz$.

For all positive integer $ n \ge 2 $, prove that $ 2^n -1 $ can't be a divisor of $ 3^n -1 $.

An equivalent of the expression $\left(\frac{x^2+1}{x}\right)\left(\frac{y^2+1}{y}\right)+\left(\frac{x^2-1}{y}\right)\left(\frac{y^2-1}{x}\right)$, $xy \not= 0$, is: $ \text{(A)}\ 1\qquad\text{(B)}\ 2xy\qquad\text{(C)}\ 2x^2y^2+2\qquad\text{(D)}\ 2xy+\frac{2}{xy}\qquad\text{(E)}\ \frac{2x}{y}+\frac{2y}{x} $

The schoolboy wrote a homework essay on the topic “How I spent my summer.” Two of his comrades from a neighboring school decided not to bother themselves with work and rewrote his essay. But while rewriting they made several mistakes - each their own. Before submitting their work, both students gave their essays to four other friends to rewrite (each gave them to two acquaintances). These four schoolchildren do the same, and so on. With each rewrite, all previous mistakes are saved and, possibly, new ones are made. It is known that on some day each new essay contained at least $10$ errors. Prove that there was a day when at least $11$ new mistakes were made in total.