When real numbers $x,\ y$ moves in the constraint with $x^2+xy+y^2=6.$ Find the range of $x^2y+xy^2-x^2-2xy-y^2+x+y.$ 30 points

In the following addition, different letters represent different non-zero digits. What is the 5-digit number $ABCDE$? $ \begin{array}{ccccccc} A&B&C&D&E&D&B\\ &B&C&D&E&D&B\\ &&C&D&E&D&B\\ &&&D&E&D&B\\ &&&&E&D&B\\ &&&&&D&B\\ +&&&&&&B\\ \hline A&A&A&A&A&A&A \end{array} $

Carl chooses a [i]functional expression[/i]* $E$ which is a finite nonempty string formed from a set $x_1, x_2, \dots$ of variables and applications of a function $f$, together with addition, subtraction, multiplication (but not division), and fixed real constants. He then considers the equation $E = 0$, and lets $S$ denote the set of functions $f \colon \mathbb R \to \mathbb R$ such that the equation holds for any choices of real numbers $x_1, x_2, \dots$. (For example, if Carl chooses the functional equation $$ f(2f(x_1)+x_2) - 2f(x_1)-x_2 = 0, $$ then $S$ consists of one function, the identity function. (a) Let $X$ denote the set of functions with domain $\mathbb R$ and image exactly $\mathbb Z$. Show that Carl can choose his functional equation such that $S$ is nonempty but $S \subseteq X$. (b) Can Carl choose his functional equation such that $|S|=1$ and $S \subseteq X$? *These can be defined formally in the following way: the set of functional expressions is the minimal one (by inclusion) such that (i) any fixed real constant is a functional expression, (ii) for any positive integer $i$, the variable $x_i$ is a functional expression, and (iii) if $V$ and $W$ are functional expressions, then so are $f(V)$, $V+W$, $V-W$, and $V \cdot W$. [i]Proposed by Carl Schildkraut[/i]

Suppose that $m, n, k$ are positive integers satisfying $$3mk=(m+3)^n+1.$$ Prove that $k$ is odd.

A storage depot is a pyramid with height $30$ and a square base with side length $40$. Determine how many cubical $3\times 3\times 3$ boxes can be stored in this depot if the boxes are always packed so that each of their edges is parallel to either an edge of the base or the altitude of the pyramid.

Estimate the time it takes to send $60$ blocks of data over a communications channel if each block consists of $512$ "chunks" and the channel can transmit $120$ chunks per second. $ \textbf{(A)}\ 0.04 \text{ seconds}\qquad\textbf{(B)}\ 0.4 \text{ seconds}\qquad\textbf{(C)}\ 4 \text{ seconds}\qquad\textbf{(D)}\ 4 \text{ minutes}\qquad\textbf{(E)}\ 4 \text{ hours} $

Let $a, b$ and $c$ be positive integers such that $ab + 1, bc + 1$ and $ca + 1$ are all integer squares. a) Give an example of such numbers $a, b$ and $c$. b) Prove that at least one of the numbers $a, b$ and $c$ is divisible by $4$

In how many ways can one fill an $n*n$ matrix with $+1$ and $-1$ so that the product of the entries in each row and each column equals $-1$?

On the blackboard were six figures: a circle, a triangle, a square, a trapezoid, a pentagon and a hexagon, painted in six colors: blue, white, red, yellow, green and brown. Each figure had only one color and all the figures were of different colors. The next day he wondered what color each figure was. Paul replied: “The circle was red, the triangle was blue, the square was white, the trapezoid was green, the pentagon was brown, and the hexagon was yellow. Sofía answered: “The circle was yellow, the triangle was green, the square was red, the trapezoid was blue, the pentagon was brown, and the hexagon was white.” Pablo was wrong three times and Sofia twice, and it is known that the pentagon was brown. Determine if it is possible to know with certainty what the color of each of the figures was.

Prove that among the seven natural numbers forming an arithmetic progression with difference $ 30 $ , one and only one is divisible by $ 7 $ .

Determine, with proof, whether or not there exist integers $a,b,c>2010$ satisfying the equation \[a^3+2b^3+4c^3=6abc+1.\]

Let $\vartriangle ABC$ be an acute triangle, and let $A'$ and $B'$ be the feet of altitudes from $A$ to $BC$ and from $B$ to $CA$, respectively; the altitudes intersect at $H$. If $BH$ is equal to the circumradius of $\vartriangle ABC$, find $\frac{A'B}{AB}$ .

Let $ABC$ be a triangle with $AC > BC,$ let $\omega$ be the circumcircle of $\triangle ABC,$ and let $r$ be its radius. Point $P$ is chosen on $\overline{AC}$ such taht $BC=CP,$ and point $S$ is the foot of the perpendicular from $P$ to $\overline{AB}$. Ray $BP$ mets $\omega$ again at $D$. Point $Q$ is chosen on line $SP$ such that $PQ = r$ and $S,P,Q$ lie on a line in that order. Finally, let $E$ be a point satisfying $\overline{AE} \perp \overline{CQ}$ and $\overline{BE} \perp \overline{DQ}$. Prove that $E$ lies on $\omega$.

Prove that $\int_{0}^{\frac{\pi}{2}} arctg (1 - \sin^2x\cos^2x)dx = \frac{\pi^2}{4} - \pi arctg\sqrt{\frac{\sqrt{2}-1}{2}}$

Let $p>5$ be a prime number. For any integer $x$, define \[{f_p}(x) = \sum_{k=1}^{p-1} \frac{1}{(px+k)^2}\] Prove that for any pair of positive integers $x$, $y$, the numerator of $f_p(x) - f_p(y)$, when written as a fraction in lowest terms, is divisible by $p^3$.

Do there exist a rectangle that can be partitioned into a regular hexagon with side length $1$, and several right triangles with side lengths $1, \sqrt3 , 2$?

If the integers $1,2,\dots,n$ can be divided into two sets such that each of the two sets does not contain the arithmetic mean of its any two elements, what is the largest possible value of $n$? $ \textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ 10 \qquad\textbf{(E)}\ \text{None of the preceding} $

A biologist found a pond with frogs. When classifying them by their mass, he noticed the following: [i]The $50$ lightest frogs represented $30\%$ of the total mass of all the frogs in the pond, while the $44$ heaviest frogs represented $27\%$ of the total mass. [/i]As fate would have it, the frogs escaped and the biologist only has the above information. How many frogs were in the pond?

Let $ABC$ be acute, non-isosceles triangle, inscribed in the circle $(O)$. Let $D$ be perpendicular projection of $A$ onto $BC$, and $E, F$ be perpendicular projections of $D$ onto $CA,AB$ respectively. (a) Prove that $AO \perp EF$. (b) The line $AO$ intersects $DE,DF$ at $I,J$ respectively. Prove that $\vartriangle DIJ$ and $\vartriangle ABC$ are similar. (c) Prove that circumcenter of $\vartriangle DIJ$ is equidistant from $B$ and $C$

Let $0^{\circ}\leq\alpha,\beta,\gamma\leq90^{\circ}$ be angles such that \[\sin\alpha-\cos\beta=\tan\gamma\] \[\sin\beta-\cos\alpha=\cot\gamma\] Compute the sum of all possible values of $\gamma$ in degrees. [i]Proposed by Michael Ren[/i]

Let $ a_1$, $ a_2$, ..., $ a_n$ and $ b_1$, $ b_2$, ..., $ b_n$ be $ 2 \cdot n$ real numbers. Prove that the following two statements are equivalent: [b]i)[/b] For any $ n$ real numbers $ x_1$, $ x_2$, ..., $ x_n$ satisfying $ x_1 \leq x_2 \leq \ldots \leq x_ n$, we have $ \sum^{n}_{k \equal{} 1} a_k \cdot x_k \leq \sum^{n}_{k \equal{} 1} b_k \cdot x_k,$ [b]ii)[/b] We have $ \sum^{s}_{k \equal{} 1} a_k \leq \sum^{s}_{k \equal{} 1} b_k$ for every $ s\in\left\{1,2,...,n\minus{}1\right\}$ and $ \sum^{n}_{k \equal{} 1} a_k \equal{} \sum^{n}_{k \equal{} 1} b_k$.

In triangle $ABC$ the orthocenter is $H$ and the foot of the altitude from $A$ is $D$. Point $P$ satisfies $AP=HP$, and the line $PA$ is tangent to $(ABC)$. Line $PD$ intersects lines $AB, AC$ at points $X,Y$ respectively. Prove that $\angle YHX = \angle BAC$ or $\angle YHX+\angle BAC= 180^\circ$.

Find one pair of positive integers $a,b$ such that $ab(a+b)$ is not divisible by $7$, but $(a+b)^7-a^7-b^7$ is divisible by $7^7$.

Find all composite positive integers $a{}$ for which there exists a positive integer $b\geqslant a$ with the same number of divisors as $a{}$ with the following property: if $a_1<\cdots<a_n$ and $b_1<\cdots<b_n$ are the proper divisors of $a{}$ and $b{}$ respectively, then $a_i+b_i, 1\leqslant i\leqslant n$ are the proper divisors of some positive integer $c.{}$

Given a right angled triangle $ABC$ in which $\angle ACB = 90^o$. Let $D$ be an inner point of $AB$, and let $E$ be an inner point of $AC$. It is known that $\angle ADE = 90^o$, and that the length of the segment $AD$ is $8$, the length of the segment $DE$ is $15$, and the length of segment $CE$ is $3$. What is the area of triangle $ABC$?