A baseball league consists of two four-team divisions. Each team plays every other team in its division $N$ games. Each team plays every team in the other division $M$ games with $N>2M$ and $M>4$. Each team plays a 76 game schedule. How many games does a team play within its own division? $ \textbf{(A) } 36 \qquad \textbf{(B) } 48 \qquad \textbf{(C) } 54 \qquad \textbf{(D) } 60 \qquad \textbf{(E) } 72 $

Find all triples $(a, b, c)$ of natural numbers such that the sets $$\{ gcd (a, b), gcd(b, c), gcd(c, a), lcm (a, b), lcm (b, c), lcm (c, a)\}$$ and $$\{2, 3, 5, 30, 60\}$$ are the same. Remark: For example, the sets $\{1, 2013\}$ and $\{1, 1, 2013\}$ are equal.

Express $\sqrt{2 + \sqrt{3}}$ in the form $\frac{a + \sqrt{b}}{\sqrt{c}}$, where $a$ is a positive integer and $b$ and $c$ are square-free positive integers.

Recall that a palindrome is a number that reads the same forward and backward. Find the greatest integer less than $1000$ that is a palindrome both when written in base ten and when written in base eight, such as $292=444_{\text{eight}}$.

For a real number $a$, $[a]$ denotes the largest integer not exceeding $a$. Find all positive real numbers $x$ satisfying the equation $$x\cdot [x]+2022=[x^2]$$

Let $a$ and $b$ be distinct real numbers for which \[\dfrac ab+\dfrac{a+10b}{b+10a}=2.\] Find $\dfrac ab$. $\textbf{(A) }0.6\qquad\textbf{(B) }0.7\qquad\textbf{(C) }0.8\qquad\textbf{(D) }0.9\qquad\textbf{(E) }1$

In ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, $F$ is its left focal point, $A$ is its right vertex, $B$ is its upper vertex. If the eccentricity of the ellipse is $\frac{\sqrt5-1}{2}$, then $\angle ABF=$________.

Let $p,q$ be positive integers. Consider a rectangle $ABCD$ with lengths of sides $p$ and $q$ that consists of $pq$ unital squares. How many of these squares are crossed by diagonal $AC$?

A circle $\mathfrak{D}$ is drawn through the vertices $A$ and $B$ of $\triangle ABC$. If $\mathfrak{D}$ intersects $AC$ at a point $M$ and $BC$ at $P$ and $MP$ contains the incenter of $\triangle ABC$, then the length $MP$ is (in standard notation, where $t=\frac{1}{a+b+c}$): (A)$at(b+c)$ (B)$ct(b+a)$ (C)$bct$ (D)$abt$

[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.

Prove that for every natural number $k$ ($k \geq 2$) there exists an irrational number $r$ such that for every natural number $m$, \[[r^m] \equiv -1 \pmod k .\] [i]Remark.[/i] An easier variant: Find $r$ as a root of a polynomial of second degree with integer coefficients. [i]Proposed by Yugoslavia.[/i]

Find all the pairs of integers $ (m, n)$ such that $ \sqrt {n +\sqrt {2016}} +\sqrt {m-\sqrt {2016}} \in \mathbb {Q}.$

$A, B, C$ can walk at $5$ km/hr. They have a car that can accomodate any two of them whch travels at $50$ km/hr. Can they reach a point $62$ km away in less than $3$ hrs?

For each interger $n\geq 4$, we consider the $m$ subsets $A_1, A_2,\dots, A_m$ of $\{1, 2, 3,\dots, n\}$, such that $A_1$ has exactly one element, $A_2$ has exactly two elements,...., $A_m$ has exactly $m$ elements and none of these subsets is contained in any other set. Find the maximum value of $m$.

Let $n$ be positive integer.A square $A$ of side length $n$ is divided by $n^2$ unit squares. All unit squares are painted in $n$ distinct colors such that each color appears exactly $n$ times. Prove that there exists a positive integer $N$ , such that for any $n>N$ the following is true: There exists a square $B$ of side length $\sqrt{n}$ and side parallel to the sides of $A$ such that $B$ contains completely cells of $4$ distinct colors.

Pinocchio claims that he can divide an isoceles triangle into three triangles, any two of which can be put together to form a new isosceles triangle. Is Pinocchio lying? (A Shapovalov)

El Chapulín observed that the number $2014$ has an unusual property. By placing its eight positive divisors in increasing order, the fifth divisor is equal to three times the third minus $4$. A number of eight divisors with this unusual property is called the [i]red[/i] number . How many [i]red[/i] numbers smaller than $2014$ exist?

Juku has the first $100$ volumes of the Harrie Totter book series at his home. For every$ i$ and $j$, where $1 \le i < j \le 100$, call the pair $(i, j)$ reversed if volume No. $j$ is before volume No, $i$ on Juku’s shelf. Juku wants to arrange all volumes of the series to one row on his shelf in such a way that there does not exist numbers $i, j, k$, where $1 \le i < j < k \le 100$, such that pairs $(i, j)$ and $(j, k)$ are both reversed. Find the largest number of reversed pairs that can occur under this condition

Let ABCD be trapezium that is inscribed in circle (O) with larger edge BC. P is a point lying outer segment BC. PA cut (O) at N(that means PA isn't tangent of (O)), the circle with diameter PD intersect (O) at E, DE meet BC at N. Prove that MN always pass through a fixed point.

Alice plays the following game of solitaire on a $20 \times 20$ chessboard. She begins by placing $100$ pennies, $100$ nickels, $100$ dimes, and $100$ quarters on the board so that each of the $400$ squares contains exactly one coin. She then chooses $59$ of these coins and removes them from the board. After that, she removes coins, one at a time, subject to the following rules: - A penny may be removed only if there are four squares of the board adjacent to its square (up, down, left, and right) that are vacant (do not contain coins). Squares “off the board” do not count towards this four: for example, a non-corner square bordering the edge of the board has three adjacent squares, so a penny in such a square cannot be removed under this rule, even if all three adjacent squares are vacant. - A nickel may be removed only if there are at least three vacant squares adjacent to its square. (And again, “off the board” squares do not count.) - A dime may be removed only if there are at least two vacant squares adjacent to its square (“off the board” squares do not count). - A quarter may be removed only if there is at least one vacant square adjacent to its square (“off the board” squares do not count). Alice wins if she eventually succeeds in removing all the coins. Prove that it is impossiblefor her to win.

Let $ \Delta ABC$ be a triangle, with $ m(\angle A)=90^{\circ}$ and $ m(\angle B)=30^{\circ}.$ If $M$ is the middle of $[AB],$ $N$ is the middle of $[BC],$ and $P\in[BC],\ Q\in[MN],$ such that \[\frac{PB}{PC}=4\cdot\frac{QM}{QN}+3,\] prove that $ \Delta APQ$ is an equilateral triangle. [b]Author: MARIN BANCOȘ[/b] [b]Regional Mathematical Contest GRIGORE MOISIL, Romania, Baia Mare, 24.03.2012, 7th grade[/b]

For what pairs of positive real numbers $(a,b)$ does the improper integral $(1)$ converge? \begin{align}\int_{b}^{\infty}\left(\sqrt{\sqrt{x+a}-\sqrt{x}}-\sqrt{\sqrt{x}-\sqrt{x-b}}\right)\,\mathrm{d}x \end{align}

Let $x,y,z \in\mathbb{Q}$,such that $(x+y+z)^3=9(x^2y+y^2z+z^2x).$ Prove that $x=y=z$

$n$ people are in the plane, so that the closest person is unique and each one shoot this closest person with a squirt gun. If $n$ is odd, prove that there exists at least one person that nobody shot. If $n$ is even, will there always be a person who escape? Justify that.

Let $D,E,F$ be points on the sides $BC,CA,AB$ of a triangle $ABC$, respectively. Suppose that the inradii of the triangles $AEF,BFD,CDE$ are all equal to $r_1$. If $r_2$ and $r$ are the inradii of triangles $DEF$ and $ABC$ respectively, prove that $r_1 +r_2 =r$.