Ten points in the plane are given, with no three collinear. Four distinct segments joining pairs of these points are chosen at random, all such segments being equally likely. The probability that some three of the segments form a triangle whose vertices are among the ten given points is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

What is the smallest positive integer $t$ such that there exist integers $x_1,x_2,\ldots,x_t$ with \[x^3_1+x^3_2+\,\ldots\,+x^3_t=2002^{2002}\,?\]

For a positive integer $n$, let $A_n$ be the set of quadruplets $(a,b,c,d)$ of integers, satisfying the following properties simultaneously: [list] [*] $0\le a\le c\le n,$ [*] $0\le b\le d\le n,$ [*] $c+d>n,$ and [*] $bc=ad+1.$ [/list] Moreover, define $$\alpha_n=\sum_{(a,b,c,d)\in A_n}\frac{1}{ab+cd}.$$ Find all real numbers $t$ such that $\alpha_n>t$ for every positive integer $n$.

Let $a$ and $b$ be positive integers such that $\frac{5a^4 + a^2}{b^4 + 3b^2 + 4}$ is an integer. Show that $a$ is not prime.

Find all prime numbers $p$ and $q$ such that $3p^2q+2pq^2=483$

$(GBR 1)$ The polynomial $P(x) = a_0x^k + a_1x^{k-1} + \cdots + a_k$, where $a_0,\cdots, a_k$ are integers, is said to be divisible by an integer $m$ if $P(x)$ is a multiple of $m$ for every integral value of $x$. Show that if $P(x)$ is divisible by $m$, then $a_0 \cdot k!$ is a multiple of $m$. Also prove that if $a, k,m$ are positive integers such that $ak!$ is a multiple of $m$, then a polynomial $P(x)$ with leading term $ax^k$can be found that is divisible by $m.$

Let $n$ be a nonzero integer. Prove that $n^4-7n^2+1$ can never be a perfect square.

Let be a natural number $ n, $ and $ n $ real numbers $ a_1,a_2,\ldots ,a_n. $ Prove that there exists a real number $ a $ such that $ a+a_1,a+a_2,\ldots ,a+a_n $ are all irrational.

Find all positive integers $x, y$ and primes $p$, such that $x^5+y^4=pxy$.

For a real number $a$, denote by $(a]$ the smallest integer that is not less than $a$. Find all real values of $x$ for which holds the equality $$(\sin x]^2 + (\cos x]^2 =|tg x| +|ctg x|.$$

For each positive integer $n$, a new number $t_n$ is formed from the numbers $2^n$ and $5^n$ which consists of the digits from $2^n$ followed by the digits from $5^n$. For example, $t_4$ is $16625$. How many digits does the number $t_{2008}$ have?

Find all integer solutions of the equation \[\frac {x^{7} \minus{} 1}{x \minus{} 1} \equal{} y^{5} \minus{} 1.\]

A jar contains 4 blue marbles, 3 green marbles, and 5 red marbles. If Helen reaches in the jar and selects a marble at random, then the probability that she selects a red marble can be expressed as $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

For even positive integer $n$ we put all numbers $1, 2, \cdots , n^2$ into the squares of an $n \times n$ chessboard (each number appears once and only once). Let $S_1$ be the sum of the numbers put in the black squares and $S_2$ be the sum of the numbers put in the white squares. Find all $n$ such that it is possible to have $\frac{S_1}{S_2}=\frac{39}{64}$.

We say that a square-free positive integer $n$ is [i]almost prime[/i] if \[n \mid x^{d_1}+x^{d_2}+\dots+x^{d_k}-kx\] for all integers $x$, where $1=d_1<d_2<\dots<d_k=n$ are all the positive divisors of $n$. Suppose that $r$ is a Fermat prime (i.e. it is a prime of the form $2^{2^m}+1$ for an integer $m \ge 0$), $p$ is a prime divisor of an almost prime integer $n$, and $p \equiv 1 \pmod{r}$. Show that, with the above notation, $d_i \equiv 1 \pmod{r}$ for all $1 \le i \le k$. (An integer $n$ is called [i]square-free[/i] if it is not divisible by $d^2$ for any integer $d>1$.)

[b]p1.[/b] What is $7\%$ of one half of $11\%$ of $20000$ ? [b]p2.[/b] Three circles centered at $A, B$, and $C$ are tangent to each other. Given that $AB = 8$, $AC = 10$, and $BC = 12$, find the radius of circle $ A$. [b]p3. [/b]How many positive integer values of $x$ less than $2012$ are there such that there exists an integer $y$ for which $\frac{1}{x} +\frac{2}{2y+1} =\frac{1}{y}$ ? [b]p4. [/b]The positive difference between $ 8$ and twice $x$ is equal to $11$ more than $x$. What are all possible values of $x$? [b]p5.[/b] A region in the coordinate plane is bounded by the equations $x = 0$, $x = 6$, $y = 0$, and $y = 8$. A line through $(3, 4)$ with slope $4$ cuts the region in half. Another line going through the same point cuts the region into fourths, each with the same area. What is the slope of this line? [b]p6.[/b] A polygon is composed of only angles of degrees $138$ and $150$, with at least one angle of each degree. How many sides does the polygon have? [b]p7.[/b] $M, A, T, H$, and $L$ are all not necessarily distinct digits, with $M \ne 0$ and $L \ne 0$. Given that the sum $MATH +LMT$, where each letter represents a digit, equals $2012$, what is the average of all possible values of the three-digit integer $LMT$? [b]p8. [/b]A square with side length $\sqrt{10}$ and two squares with side length $\sqrt{7}$ share the same center. The smaller squares are rotated so that all of their vertices are touching the sides of the larger square at distinct points. What is the distance between two such points that are on the same side of the larger square? [b]p9.[/b] Consider the sequence $2012, 12012, 20120, 20121, ...$. This sequence is the increasing sequence of all integers that contain “$2012$”. What is the $30$th term in this sequence? [b]p10.[/b] What is the coefficient of the $x^5$ term in the simplified expansion of $(x +\sqrt{x} +\sqrt[3]{x})^{10}$ ? PS. You had better use hide for answers.

Prove that for each prime number $p$ and positive integer $n$, $p^n$ divides \[\binom{p^n}{p}-p^{n-1}. \]

[b]p1. [/b]Ed, Fred and George are playing on a see-saw that is slightly off center. When Ed sits on the left side and George, who weighs $100$ pounds, on the right side, they are perfectly balanced. Similarly, if Fred, who weighs $400$ pounds, sits on the left and Ed sits on the right, they are also perfectly balanced. Assuming the see-saw has negligible weight, what is the weight of Ed, in pounds? [b]p2.[/b] How many digits does the product $2^{42}\cdot 5^{38}$ have? [b]p3.[/b] Square $ABCD$ has equilateral triangles drawn external to each side, as pictured. If each triangle is folded upwards to meet at a point $E$, then a square pyramid can be made. If the center of square $ABCD$ is $O$, what is the measure of $\angle OEA$? [img]https://cdn.artofproblemsolving.com/attachments/9/a/39c0096ace5b942a9d3be1eafe7aa7481fbb9f.png[/img] [b]p4.[/b] How many solutions $(x, y)$ in the positive integers are there to $3x + 7y = 1337$ ? [b]p5.[/b] A trapezoid with height $12$ has legs of length $20$ and $15$ and a larger base of length $42$. What are the possible lengths of the other base? [b]p6.[/b] Let $f(x) = 6x + 7$ and $g(x) = 7x + 6$. Find the value of a such that $g^{-1}(f^{-1}(g(f(a)))) = 1$. [b]p7.[/b] Billy and Cindy want to meet at their favorite restaurant, and they have made plans to do so sometime between $1:00$ and $2:00$ this Sunday. Unfortunately, they didn’t decide on an exact time, so they both decide to arrive at a random time between $1:00$ and $2:00$. Silly Billy is impatient, though, and if he has to wait for Cindy, he will leave after $15$ minutes. Cindy, on the other hand, will happily wait for Billy from whenever she arrives until $2:00$. What is the probability that Billy and Cindy will be able to dine together? [b]p8.[/b] As pictured, lines are drawn from the vertices of a unit square to an opposite trisection point. If each triangle has legs with ratio $3 : 1$, what is the area of the shaded region? [img]https://cdn.artofproblemsolving.com/attachments/e/9/35a6340018edcddfcd7e085f8f6e56686a8e07.png[/img] [b]p9.[/b] For any positive integer $n$, let $f_1(n)$ denote the sum of the squares of the digits of $n$. For $k \ge 2$, let $f_k(n) = f_{k-1}(f_1(n))$. Then, $f_1(5) = 25$ and $f_3(5) = f_2(25) = 85$. Find $f_{2012}(15)$. [b]p10.[/b] Given that $2012022012$ has $ 8$ distinct prime factors, find its largest prime factor. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[u]Round 1[/u] [b]1.1.[/b] Suppose a certain menu has $3$ sandwiches and $5$ drinks. How many ways are there to pick a meal so that you have exactly a drink and a sandwich? [b]1.2.[/b] If $a + b = 4$ and $a + 3b = 222222$, find $10a + b$. [b]1.3.[/b] Compute $$\left\lfloor \frac{2019 \cdot 2017}{2018} \right\rfloor $$ where $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$. [u]Round 2[/u] [b]2.1.[/b] Andrew has $10$ water bottles, each of which can hold at most $10$ cups of water. Three bottles are thirty percent filled, five are twenty-four percent filled, and the rest are empty. What is the average amount of water, in cups, contained in the ten water bottles? [b]2.2.[/b] How many positive integers divide $195$ evenly? [b]2.3.[/b] Square $A$ has side length $\ell$ and area $128$. Square $B$ has side length $\ell/2$. Find the length of the diagonal of Square $B$. [u]Round 3[/u] [b]3.1.[/b] A right triangle with area $96$ is inscribed in a circle. If all the side lengths are positive integers, what is the area of the circle? Express your answer in terms of $\pi$. [b]3.2.[/b] A circular spinner has four regions labeled $3, 5, 6, 10$. The region labeled $3$ is $1/3$ of the spinner, $5$ is $1/6$ of the spinner, $6$ is $1/10$ of the spinner, and the region labeled $10$ is $2/5$ of the spinner. If the spinner is spun once randomly, what is the expected value of the number on which it lands? [b]3.3.[/b] Find the integer k such that $k^3 = 8353070389$ [u]Round 4[/u] [b]4.1.[/b] How many ways are there to arrange the letters in the word [b]zugzwang [/b] such that the two z’s are not consecutive? [b]4.2.[/b] If $O$ is the circumcenter of $\vartriangle ABC$, $AD$ is the altitude from $A$ to $BC$, $\angle CAB = 66^o$ and $\angle ABC = 44^o$, then what is the measure of $\angle OAD$ ? [b]4.3.[/b] If $x > 0$ satisfies $x^3 +\frac{1}{x^3} = 18$, find $x^5 +\frac{1}{x^5}$ [u]Round 5[/u] [b]5.1.[/b] Let $C$ be the answer to Question $3$. Neethen decides to run for school president! To be entered onto the ballot, however, Neethen needs $C + 1$ signatures. Since no one else will support him, Neethen gets the remaining $C$ other signatures through bribery. The situation can be modeled by $k \cdot N = 495$, where $k$ is the number of dollars he gives each person, and $N$ is the number of signatures he will get. How many dollars does Neethen have to bribe each person with to get exactly C signatures? [b]5.2.[/b] Let $A$ be the answer to Question $1$. With $3A - 1$ total votes, Neethen still comes short in the election, losing to Serena by just $1$ vote. Darn! Neethen sneaks into the ballot room, knowing that if he destroys just two ballots that voted for Serena, he will win the election. How many ways can Neethen choose two ballots to destroy? [b]5.3.[/b] Let $B$ be the answer to Question $2$. Oh no! Neethen is caught rigging the election by the principal! For his punishment, Neethen needs to run the perimeter of his school three times. The school is modeled by a square of side length $k$ furlongs, where $k$ is an integer. If Neethen runs $B$ feet in total, what is $k + 1$? (Note: one furlong is $1/8$ of a mile). [u]Round 6[/u] [b]6.1.[/b] Find the unique real positive solution to the equation $x =\sqrt{6 + 2\sqrt6 + 2x}- \sqrt{6 - 2\sqrt6 - 2x} -\sqrt6$. [b]6.2.[/b] Consider triangle ABC with $AB = 13$ and $AC = 14$. Point $D$ lies on $BC$, and the lengths of the perpendiculars from $D$ to $AB$ and $AC$ are both $\frac{56}{9}$. Find the largest possible length of $BD$. [b]6.3.[/b] Let $f(x, y) = \frac{m}{n}$, where $m$ is the smallest positive integer such that $x$ and $y$ divide $m$, and $n$ is the largest positive integer such that $n$ divides both $x$ and $y$. If $S = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$, what is the median of the distinct values that $f(a, b)$ can take, where $a, b \in S$? [u]Round 7[/u] [b]7.1.[/b] The polynomial $y = x^4 - 22x^2 - 48x - 23$ can be written in the form $$y = (x - \sqrt{a} - \sqrt{b} - \sqrt{c})(x - \sqrt{a} +\sqrt{b} +\sqrt{c})(x +\sqrt{a} -\sqrt{b} +\sqrt{c})(x +\sqrt{a} +\sqrt{b} -\sqrt{c})$$ for positive integers $a, b, c$ with $a \le b \le c$. Find $(a + b)\cdot c$. [b]7.2.[/b] Varun is grounded for getting an $F$ in every class. However, because his parents don’t like him, rather than making him stay at home they toss him onto a number line at the number $3$. A wall is placed at $0$ and a door to freedom is placed at $10$. To escape the number line, Varun must reach 10, at which point he walks through the door to freedom. Every $5$ minutes a bell rings, and Varun may walk to a different number, and he may not walk to a different number except when the bell rings. Being an $F$ student, rather than walking straight to the door to freedom, whenever the bell rings Varun just randomly chooses an adjacent integer with equal chance and walks towards it. Whenever he is at $0$ he walks to $ 1$ with a $100$ percent chance. What is the expected number of times Varun will visit $0$ before he escapes through the door to freedom? [b]7.3.[/b] Let $\{a_1, a_2, a_3, a_4, a_5, a_6\}$ be a set of positive integers such that every element divides $36$ under the condition that $a_1 < a_2 <... < a_6$. Find the probability that one of these chosen sets also satisfies the condition that every $a_i| a_j$ if $i|j$. [u]Round 8[/u] [b]8.[/b] How many numbers between $1$ and $100, 000$ can be expressed as the product of at most $3$ distinct primes? Your answer will be scored according to the following formula, where $X$ is the correct answer and $I$ is your input. $$max \left\{ 0, \left\lceil min \left\{13 - \frac{|I-X|}{0.1 |I|}, 13 - \frac{|I-X|}{0.1 |I-2X|} \right\} \right\rceil \right\}$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

How many positive integers of $n$ digits exist such that each digit is $1, 2$, or $3$? How many of these contain all three of the digits $1, 2$, and $3$ at least once?

It is given that the number $4^{11}+1$ is divisible by some prime greater than $1000$. Determine this prime. [i] Proposed by David Altizio [/i]

Consider a sequence of non-negative integers g$_1,g_2,g_3,...$ each consisting of three digits (numbers smaller than $100$ are also written with three digits; the number $27$, for example, is written as $027$). Each number consists of the preceding by taking the product of the three digits that make up the preceding. The resulting sequence is of course dependent on the choice of $g_1$ (e.g. $g_1 = 359$ leads to $g_2= 135$, $g_3= 015$, $g_4 = 000$).Prove that independent of the choice of $g_1$: (a) $g_{n+1}\le g_n$ (b) $g_{10}= 000$.

$(BUL 1)$ Prove that the equation $\sqrt{x^3 + y^3 + z^3}=1969$ has no integral solutions.

Prove that exists integer $n > 2003$ that in sequence $\binom{n}{0}$, $\binom{n}{1}$, $\binom{n}{2}$, ..., $\binom{n}{2003}$ each element is a divisor of all elements which are after him.

Prove that for any positive integers $x, y, z$ with $xy-z^2 = 1$ one can find non-negative integers $a, b, c, d$ such that $x = a^2 + b^2, y = c^2 + d^2, z = ac + bd$. Set $z = (2q)!$ to deduce that for any prime number $p = 4q + 1$, $p$ can be represented as the sum of squares of two integers.