Find the greatest and smallest value of $f(x, y) = y-2x$, if x, y are distinct non-negative real numbers with $\frac{x^2+y^2}{x+y}\leq 4$.

Denote by $S(k)$ the sum of the digits in the decimal representation of $k$. Prove that there are infinitely many $n\in \mathbb{Z_{+}}$ such that: ${S(2^{n}+n})<S(2^{n})$.

Let $p_n$ be the probability that $c+d$ is a perfect square when the integers $c$ and $d$ are selected independently at random from the set $\{1,2,\ldots,n\}$. Show that $\lim_{n\to\infty}p_n\sqrt n$ exists and express this limit in the form $r(\sqrt s-t)$, where $s$ and $t$ are integers and $r$ is a rational number.

There are $2^n$ airports, numbered with binary strings of length $n{}$. Any two stations whose numbers differ in exactly one digit are connected by a flight that has a price (which is the same in both directions). The sum of the prices of all $n{}$ flights leaving any station does not exceed 1. Prove that one can travel between any two airports by paying no more than 1.

Find a countable family of natural solutions to $ \frac{1}{a} +\frac{1}{b} +\frac{1}{ab}=\frac{1}{c} . $

Denote by $M(a, b, c, . . . , k)$ the least common multiple and by $D(a, b, c, . . . , k)$ the greatest common divisor of $a, b, c, . . . , k$. Prove that: a) $M(a, b)D(a, b) = ab$, b) $\frac{M(a, b, c)D(a, b)D(b, c)D(a, c)}{D(a, b, c)}= abc$.

An electric field varies according the the relationship, \[ \textbf{E} = \left( 0.57 \, \dfrac{\text{N}}{\text{C}} \right) \cdot \sin \left[ \left( 1720 \, \text{s}^{-1} \right) \cdot t \right]. \]Find the maximum displacement current through a $ 1.0 \, \text{m}^2 $ area perpendicular to $\vec{\mathbf{E}}$. Assume the permittivity of free space to be $ 8.85 \times 10^{-12} \, \text{F}/\text{m} $. Round to two significant figures. [i]Problem proposed by Kimberly Geddes[/i]

Let triangle $ABC$ have side lengths $AB=16, BC=20, AC=26.$ Let $ACDE, ABFG,$ and $BCHI$ be squares that are entirely outside of triangle $ABC$. Let $J$ be the midpoint of $EH$, $K$ be the midpoint of $DG$, and $L$ be the midpoint of $AC$. Find the area of triangle $JKL$.

If $x$, $y$, $z$ are nonnegative integers satisfying the equation below, then compute $x+y+z$. \[\left(\frac{16}{3}\right)^x\times \left(\frac{27}{25}\right)^y\times \left(\frac{5}{4}\right)^z=256.\] [i]Proposed by Jeffrey Shi[/i]

[asy] size(200); dotfactor=3; pair A=(0,0),B=(1,0),C=(2,0),D=(3,0),X=(1.2,0.7); draw(A--D); dot(A);dot(B);dot(C);dot(D); draw(arc((0.4,0.4),0.4,180,110),arrow = Arrow(TeXHead)); draw(arc((2.6,0.4),0.4,0,70),arrow = Arrow(TeXHead)); draw(B--X,dotted); draw(C--X,dotted); label("$A$",A,SW); label("$B$",B,S); label("$C$",C,S); label("$D$",D,S); label("x",X,fontsize(5pt)); //Credit to TheMaskedMagician for the diagram [/asy] Points $A , B, C$, and $D$ are distinct and lie, in the given order, on a straight line. Line segments $AB, AC$, and $AD$ have lengths $x, y$, and $z$ , respectively. If line segments $AB$ and $CD$ may be rotated about points $B$ and $C$, respectively, so that points $A$ and $D$ coincide, to form a triangle with positive area, then which of the following three inequalities must be satisfied? $\textbf{I. }x<\frac{z}{2}\qquad\textbf{II. }y<x+\frac{z}{2}\qquad\textbf{III. }y<\frac{z}{2}$ $\textbf{(A) }\textbf{I. }\text{only}\qquad\textbf{(B) }\textbf{II. }\text{only}\qquad$ $\textbf{(C) }\textbf{I. }\text{and }\textbf{II. }\text{only}\qquad\textbf{(D) }\textbf{II. }\text{and }\textbf{III. }\text{only}\qquad\textbf{(E) }\textbf{I. },\textbf{II. },\text{and }\textbf{III. }$

Prove that $$\lim_{n\to\infty}n^2\left(\int^1_0\sqrt[n]{1+x^n}\text dx-1\right)=\frac{\pi^2}{12}.$$

Let $a_1,a_2,...,a_n$ be an arithmetic progression of positive real numbers. Prove that $\tfrac {1}{\sqrt a_1+\sqrt a_2}+\tfrac {1}{\sqrt a_2+\sqrt a_3}+...+\tfrac {1}{\sqrt a_{n-1}+\sqrt a_n}=\tfrac{n-1}{\sqrt {a_1}+\sqrt{a_n}}$.

The natural numbers $x, y > 1$, are such that $x^2 + xy -y$ is the square of a natural number. Prove that $x + y + 1$ is a composite number.

Let $S=\{a_1,a_2,\ldots,a_n\}$ of nenul vectors in a plane. Show that $S{}$ can be partitioned in nenul subsets $B_1, B_2,\ldots, B_m$ with the properties: 1) each vector from $S{}$ is part of only on subset; 2) if $a_i\in B_j$ then the angle between vectors $a_i$ and $c_j$, which is the sum of all vectors from $B_j$ is not greater than $\frac{\pi}{2}$; 3) if $i\neq j$ then the angle between vectors $c_i$ and $c_j$, which is the sum of all vectors from $B_i$ and $B_j$, respectively, is greater than $\frac{\pi}{2}$. What are the possible values of $m$?

In triangle $ ABC$, points $ M,N$ lie on line $ AC$ such that $ MA\equal{}AB$ and $ NB\equal{}NC$. Also $ K,L$ lie on line $ BC$ such that $ KA\equal{}KB$ and $ LA\equal{}LC$. It is know that $ KL\equal{}\frac12{BC}$ and $ MN\equal{}AC$. Find angles of triangle $ ABC$.

Let $p$ be a prime number. Find all subsets $S\subseteq\mathbb Z/p\mathbb Z$ such that 1. if $a,b\in S$, then $ab\in S$, and 2. there exists an $r\in S$ such that for all $a\in S$, we have $r-a\in S\cup\{0\}$. [i]Proposed by Harun Khan[/i]

The equations $ 2x \plus{} 7 \equal{} 3$ and $ bx\minus{}10 \equal{} \minus{}\!2$ have the same solution for $ x$. What is the value of $ b$? $ \textbf{(A)}\minus{}\!8 \qquad \textbf{(B)}\minus{}\!4 \qquad \textbf{(C)}\minus{}\!2 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 8$

For an integer $n\ge 2$ find all $a_1,a_2,\cdots ,a_n, b_1,b_2,\cdots , b_n$ so that (a) $0\le a_1\le a_2\le \cdots \le a_n\le 1\le b_1\le b_2\le \cdots \le b_n;$ (b) $\sum_{k=1}^n (a_k+b_k)=2n;$ (c) $\sum_{k=1}^n (a_k^2+b_k^2)=n^2+3n.$

Two tangents $AT$ and $BT$ touch a circle at $A$ and $B$, respectively, and meet perpendicularly at $T$. $Q$ is on $AT$, $S$ is on $BT$, and $R$ is on the circle, so that $QRST$ is a rectangle with $QT = 8$ and $ST = 9$. Determine the radius of the circle.

Suppose that every integer is colored using one of $4$ colors. Let $m, n$ be distinct odd integers such that $m + n \ne 0$. Prove that there exist integers $a$, $ b$ of the same color such that $ a - b$ equals one of the numbers $m$, $n$, $m - n$, $m + n$.

p1. Is there any natural number n such that $n^2 + 5n + 1$ is divisible by $49$ ? Explain. p2. It is known that the parabola $y = ax^2 + bx + c$ passes through the points $(-3,4)$ and $(3,16)$, and does not cut the $x$-axis. Find all possible abscissa values ​​for the vertex point of the parabola. p3. It is known that $T.ABC$ is a regular triangular pyramid with side lengths of $2$ cm. The points $P, Q, R$, and $S$ are the centroids of triangles $ABC$, $TAB$, $TBC$ and $TCA$, respectively . Determine the volume of the triangular pyramid $P.QRS$ . p4. At an event invited $13$ special guests consisting of $ 8$ people men and $5$ women. Especially for all those special guests provided $13$ seats in a special row. If it is not expected two women sitting next to each other, determine the number of sitting positions possible for all those special guests. p5. A table of size $n$ rows and $n$ columns will be filled with numbers $ 1$ or $-1$ so that the product of all the numbers in each row and the product of all the numbers in each column is $-1$. How many different ways to fill the table?

As the graph, a pond is divided into 2n (n $\geq$ 5) parts. Two parts are called neighborhood if they have a common side or arc. Thus every part has three neighborhoods. Now there are 4n+1 frogs at the pond. If there are three or more frogs at one part, then three of the frogs of the part will jump to the three neighborhoods repsectively. Prove that for some time later, the frogs at the pond will uniformily distribute. That is, for any part either there are frogs at the part or there are frogs at the each of its neighborhoods. [img]http://www.mathlinks.ro/Forum/files/china2005_2_214.gif[/img]

Exactly three of the interior angles of a convex polygon are obtuse. What is the maximum number of sides of such a polygon? $ \textbf{(A)}\ 4 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 7 \qquad \textbf{(E)}\ 8$

Let $O$ be the center of the circumcircle, and $AD$ be the angle bisector of the acute triangle $ABC$. The perpendicular drawn from point $D$ on the line $AO$ ​​intersects the line $AC$ at the point $P$. Prove that $AP = AB$.

A square is cut into red and blue rectangles. The sum of areas of red triangles is equal to the sum of areas of the blue ones. For each blue rectangle, we write the ratio of the length of its vertical side to the length of its horizontal one and for each red rectangle, the ratio of the length of its horizontal side to the length of its vertical side. Find the smallest possible value of the sum of all the written numbers.