In a right triangle, ratio of the hypotenuse over perimeter of the triangle determines an interval on real numbers. Find the midpoint of this interval? $\textbf{(A)}\ \frac{2\sqrt{2} \plus{}1}{4} \qquad\textbf{(B)}\ \frac{\sqrt{2} \plus{}1}{2} \qquad\textbf{(C)}\ \frac{2\sqrt{2} \minus{}1}{4} \\ \qquad\textbf{(D)}\ \sqrt{2} \minus{}1 \qquad\textbf{(E)}\ \frac{\sqrt{2} \minus{}1}{2}$

A trapezoid has side lengths $3, 5, 7,$ and $11$. The sum of all the possible areas of the trapezoid can be written in the form of $r_1 \sqrt{n_1} + r_2 \sqrt{n_2} + r_3$, where $r_1, r_2,$ and $r_3$ are rational numbers and $n_1$ and $n_2$ are positive integers not divisible by the square of a prime. What is the greatest integer less than or equal to \[r_1 + r_2 + r_3 + n_1 + n_2?\] $ \textbf{(A)}\ 57\qquad\textbf{(B)}\ 59\qquad\textbf{(C)}\ 61\qquad\textbf{(D)}\ 63\qquad\textbf{(E)}\ 65 $

A rational number $x$ is given. Prove that there exists a sequence $x_0, x_1, x_2, \ldots$ of rational numbers with the following properties: (a) $x_0=x$; (b) for every $n\ge1$, either $x_n = 2x_{n-1}$ or $x_n = 2x_{n-1} + \textstyle\frac{1}{n}$; (c) $x_n$ is an integer for some $n$.

A cube with side length $10$ is divided into two cuboids with integral side lengths by a straight cut. Afterwards, one of these two cuboids is divided into two cuboids with integral side lengths by another straight cut. What is the smallest possible volume of the largest of the three cuboids?

Find all functions $f\colon\mathbb R^+\to\mathbb R^+$ such that \[(f(x)+f(y)+f(z))(xf(y)+yf(z)+zf(x))>(f(x)+y)(f(y)+z)(f(z)+x)\] for all $x,y,z\in\mathbb R^+$. [i]Alexander Wang[/i] [size=59](oops)[/size]

Give two statements: (1) $a,b,c$ are complex numbers, if $a^2+b^2>c^2$, then $a^2+b^2-c^2>0$. (2) $a,b,c$ are complex numbers, if $a^2+b^2-c^2>0$, then $a^2+b^2>c^2$. Then, which is true? $\text{(A)}$ (1) is correct, (2) is correct as well $\text{(B)}$ (1) is correct, (2) is incorrect $\text{(C)}$ (1) is incorrect, (2) is incorrect as well $\text{(D)}$ (1) is incorrect, (2) is correct

What time was it $2011$ minutes after midnight on January 1, 2011? $\textbf{(A)} \text{January 1 at 9:31PM}$ $\textbf{(B)} \text{January 1 at 11:51PM}$ $\textbf{(C)} \text{January 2 at 3:11AM}$ $\textbf{(D)} \text{January 2 at 9:31AM}$ $\textbf{(E)} \text{January 2 at 6:01PM}$

Prove that for any positive integer $n$, the least common multiple of the numbers $1,2,\ldots,n$ and the least common multiple of the numbers: \[\binom{n}{1},\binom{n}{2},\ldots,\binom{n}{n}\] are equal if and only if $n+1$ is a prime number. [i]Laurentiu Panaitopol[/i]

Two intersecting lines $a$ and $b$ are given in a plane. Consider all pairs of orthogonal planes $\alpha$, $\beta$ such that $a \subset \alpha$ and $b\subset \beta$. Prove that there is a circle such that every its point lies on the line $\alpha \cap \beta$ for some $\alpha$ and $\beta$.

$m, n $ are positive integers and $p$ is a prime number. Find all triples $(m, n, p)$ satisfying $(m^3+n)(n^3+m)=p^3$

If there are integers $a,b,c$ such that $a^2+b^2+c^2-ab-bc-ca$ is divisible by a prime $p$ such that $\text{gcd}(p,\frac{a^2+b^2+c^2-ab-bc-ca}{p})=1$, then prove that there are integers $x,y,z$ such that $p=x^2+y^2+z^2-xy-yz-zx$.

On an east-west shipping lane are ten ships sailing individually. The first five from the west are sailing eastwards while the other five ships are sailing westwards. They sail at the same constant speed at all times. Whenever two ships meet, each turns around and sails in the opposite direction. When all ships have returned to port, how many meetings of two ships have taken place?

Prove that for each $k$ points in the plane, no three collinear and having integral distances from each other. If we have an infinite set of points with integral distances from each other, then all points are collinear. [i](Anonymous314)[/i] PS. wording needs to be fixed , [url=http://www.mathcenter.net/forum/showthread.php?t=7288]source[/url]

Adi the Baller is shooting hoops, and makes a shot with probability $p$. He keeps shooting hoops until he misses. The value of $p$ that maximizes the chance that he makes between 35 and 69 (inclusive) buckets can be expressed as $\frac{1}{\sqrt[b]{a}}$ for a prime $a$ and positive integer $b$. Find $a+b$. Proposed by Minseok Eli Park (wolfpack)

Let $ ABC$ be an arbitrary acute angled triangle. For any point $ P$ lying within the triangle, let $ D$, $ E$, $ F$ denote the feet of the perpendiculars from $ P$ onto the sides $ AB$, $ BC$, $ CA$ respectively. Determine the set of all possible positions of the point $ P$ for which the triangle $ DEF$ is isosceles. For which position of $ P$ will the triangle $ DEF$ become equilateral?

[b]p1.[/b] What is the maximal number of pieces of two shapes, [img]https://cdn.artofproblemsolving.com/attachments/a/5/6c567cf6a04b0aa9e998dbae3803b6eeb24a35.png[/img] and [img]https://cdn.artofproblemsolving.com/attachments/8/a/7a7754d0f2517c93c5bb931fb7b5ae8f5e3217.png[/img], that can be used to tile a $7\times 7$ square? [b]p2.[/b] Six shooters participate in a shooting competition. Every participant has $5$ shots. Each shot adds from $1$ to $10$ points to shooter’s score. Every person can score totally for all five shots from $5$ to $50$ points. Each participant gets $7$ points for at least one of his shots. The scores of all participants are different. We enumerate the shooters $1$ to $6$ according to their scores, the person with maximal score obtains number $1$, the next one obtains number $2$, the person with minimal score obtains number $6$. What score does obtain the participant number $3$? The total number of all obtained points is $264$. [b]p2.[/b] There are exactly $n$ students in a high school. Girls send messages to boys. The first girl sent messages to $5$ boys, the second to $7$ boys, the third to $6$ boys, the fourth to $8$ boys, the fifth to $7$ boys, the sixth to $9$ boys, the seventh to $8$, etc. The last girl sent messages to all the boys. Prove that $n$ is divisible by $3$. [b]p4.[/b] In what minimal number of triangles can one cut a $25 \times 12$ rectangle in such a way that one can tile by these triangles a $20 \times 15$ rectangle. [b]p5.[/b] There are $2014$ stones in a pile. Two players play the following game. First, player $A$ takes some number of stones (from $1$ to $30$) from the pile, then player B takes $1$ or $2$ stones, then player $A$ takes $2$ or $3$ stones, then player $B$ takes $3$ or $4$ stones, then player A takes $4$ or $5$ stones, etc. The player who gets the last stone is the winner. If no player gets the last stone (there is at least one stone in the pile but the next move is not allowed) then the game results in a draw. Who wins the game using the right strategy? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $a_1 = \sqrt 7$ and $b_i = \lfloor a_i \rfloor$, $a_{i+1} = \dfrac{1}{b_i - \lfloor b_i \rfloor}$ for each $i\geq i$. What is the smallest integer $n$ greater than $2004$ such that $b_n$ is divisible by $4$? ($\lfloor x \rfloor$ denotes the largest integer less than or equal to $x$) $ \textbf{(A)}\ 2005 \qquad\textbf{(B)}\ 2006 \qquad\textbf{(C)}\ 2007 \qquad\textbf{(D)}\ 2008 \qquad\textbf{(E)}\ \text{None of above} $

Let $n$ be a positive integer that is not divisible by either $2$ or $5$. In the decimal expansion of $\frac{1}{n}= 0.a_1a_2a_3\cdots$ a finite number of digits after the decimal point are chosen arbitrarily to be deleted. Clearly the decimal number obtained by this procedure is also rational, so it's equal to $\frac{a}{b}$ for some integers $a,b$. Prove that $b$ is divisible by $n$.

There are 15 cities, and there is a train line between each pair operated by either the Carnegie Rail Corporation or the Mellon Transportation Company. A tourist wants to visit exactly three cities by travelling in a loop, all by travelling on one line. What is the minimum number of such 3-city loops?

Given a sequence $a_1,a_2,a_3,\ldots $ of positive integers in which every positive integer occurs exactly once. Prove that there exist integers $\ell $ and $m,\ 1<\ell <m$, such that $a_1+a_m=2a_{\ell}$.

The vertices of a triangle with sides $a\ge b\ge c$ are centers of three circles, such that no two of the circles have common interior points and none contains any other vertex of the triangle. Determine the maximum possible total area of these three circles.

Do there exist irrational numbers $a,b$ both greater than $1$, such that $\lfloor{a^m}\rfloor\neq \lfloor{b^n}\rfloor$ for all $m,n\in\mathbb{N}$ ?

Given a solid $R$ contained in a semi cylinder with the hight $1$ which has a semicircle with radius $1$ as the base. The cross section at the hight $x\ (0\leq x\leq 1)$ is the form combined with two right-angled triangles as attached figure as below. Answer the following questions. (1) Find the cross-sectional area $S(x)$ at the hight $x$. (2) Find the volume of $R$. If necessary, when you integrate, set $x=\sin t.$

In a certain language there are only two letters, $A$ and $B$. The words of this language obey the following rules: (i) The only word of length $1$ is $A$; (ii) A sequence of letters $X_1X_2...X_{n+1}$, where $X_i\in \{A,B\}$ for each $i$, forms a word of length $n+1$ if and only if it contains at least one letter $A$ and is not of the form $WA$ for a word $W$ of length $n$. Show that the number of words consisting of $1998 A$’s and $1998 B$’s and not beginning with $AA$ equals $\binom{3995}{1997}-1$

Calculate $ \lim_{n\to\infty }\left(\lim_{x\to 0} \left( -\frac{n}{x}+1+\frac{1}{x}\sum_{r=2}^{n+1}\sqrt[r!]{1+\sin rx}\right)\right) . $ [i]Constantin Rusu[/i]