Is there sequence $(a_n)$ of natural numbers, such that differences $\{a_{n+1}-a_n\}$ take every natural value and only one time and differences $\{a_{n+2}-a_n\}$ take every natural value greater $2015$ and only one time ? [i]A. Golovanov[/i]

Let $x$ be the smallest positive integer that cannot be expressed in the form $\frac{2^a - 2^b}{2^c - 2^d}$, where $a$, $b$, $c$, $d$ are non-negative integers. Prove that $x$ is odd.

Dene the function $f : R \to R$ by $$f(x) =\begin{cases} \dfrac{1}{x^2+\sqrt{x^4+2x}}\,\,\, \text{if} \,\,\,x \notin (- \sqrt[3]{2}, 0] \\ \,\,\, 0 \,\,\,, \,\,\, \text{otherwise} \end{cases}$$ The sum of all real numbers $x$ for which $f^{10}(x) = 1$ can be written as $\frac{a+b\sqrt{c}}{d}$ , where $a, b,c, d$ are integers, $d$ is positive, $c$ is square-free, and gcd$(a,b, d) = 1$. Find $1000a + 100b + 10c + d.$ (Here, $f^n(x)$ is the function $f(x)$ iterated $n$ times. For example, $f^3(x) = f(f(f(x)))$.)

Suppose we have a square $ABCD$ and a point $S$ in the interior of this square. Under homothety with centre $S$ and ratio of magnification $k > 1$, this square becomes another square $A'B'C'D'$. Prove that the sum of the areas of the two quadrilaterals $A'ABB'$ and $C'CDD'$ are equal to the sum of the areas of the two quadrilaterals $B'BCC'$ and $D'DAA'$. [asy] unitsize(3 cm); pair[] A, B, C, D; pair S; A[1] = (0,1); B[1] = (0,0); C[1] = (1,0); D[1] = (1,1); S = (0.3,0.6); A[0] = interp(S,A[1],2/3); B[0] = interp(S,B[1],2/3); C[0] = interp(S,C[1],2/3); D[0] = interp(S,D[1],2/3); draw(A[0]--B[0]--C[0]--D[0]--cycle); draw(A[1]--B[1]--C[1]--D[1]--cycle); draw(A[1]--S, dashed); draw(B[1]--S, dashed); draw(C[1]--S, dashed); draw(D[1]--S, dashed); dot("$A$", A[0], N); dot("$B$", B[0], SE); dot("$C$", C[0], SW); dot("$D$", D[0], SE); dot("$A'$", A[1], NW); dot("$B'$", B[1], SW); dot("$C'$", C[1], SE); dot("$D'$", D[1], NE); dot("$S$", S, dir(270)); [/asy]

Two positive numbers $x$ and $y$ are given. Show that $\left(1 +\frac{x}{y} \right)^3 + \left(1 +\frac{y}{x}\right)^3 \ge 16$.

Compute the sum of all prime numbers $p$ with $p \ge 5$ such that $p$ divides $(p + 3)^{p-3} + (p + 5)^{p-5}$. .

A clueless ant makes the following route: starting at point $ A $ goes $ 1$ cm north, then $ 2$ cm east, then $ 3$ cm south, then $ 4$ cm west, immediately $ 5$ cm north, continues $ 6$ cm east, and so on, finally $ 41$ cm north and ends in point $ B $. Calculate the distance between $ A $ and $ B $ (in a straight line).

Let $n$ and $k$ be positive integers. Find all monic polynomials $f\in \mathbb{Z}[X]$, of degree $n$, such that $f(a)$ divides $f(2a^k)$ for $a\in \mathbb{Z}$ with $f(a)\neq 0$.

(i): Prove that if $n$ is a positive integer, then $$\binom{2n}{n}=\frac{(2n)!}{(n!)^2}$$ is a positive integer that is divisible by all prime numbers $p$ with $n<p\le 2n$, and that $$\binom{2n}{n}<2^{2n}.$$ (ii): For $x$ a positive real number, let $\pi(x)$ denote the number of prime numbers $p \le x$. [Thus, $\pi(10) = 4$ since there are $4$ primes, viz., $2$, $3$, $5$, and $7$, not exceeding $10$.]Prove that if $n \ge 3$ is an integer, then (a)$$\pi(2n) < \pi(n) + {{2n}\over{\log_2(n)}};$$(b)$$\pi(2^n) < {{2^{n+1}\log_2(n-1)}\over{n}};$$(c) Deduce that, for all real numbers $x \ge 8$,$$\pi(x) < {{4x \log_2(\log_2(x))}\over{\log_2(x)}}.$$

Let $ABC$ be a triangle in which $AB < AC, D$ is the foot of the altitude from $A, H$ is the orthocenter, $O$ is the circumcenter, $M$ is the midpoint of the side $BC, A'$ is the reflection of $A$ across $O$, and $S$ is the intersection of the tangents at $B$ and $C$ to the circumcircle. The tangent at $A'$ to the circumcircle intersects $SC$ and $SB$ at $X$ and $Y$ , respectively. If $M,S,X,Y$ are concyclic, prove that lines $OD$ and $SA'$ are parallel.

A block of size $ a\times b\times c$ is composed of $ 1\times 1\times 2$ domino blocks. Assuming that each of the three possible directions of domino blocks occurs equally many times, what are the possible values of $ a$, $ b$, $ c$?

Show that the polynomial $x^{8} +98 x^{4}+1$ can be expressed as the product of two nonconstant polynomials with integer coefficients.

Prove that there exist a subset $A$ of $\{1,2,\cdots,2^n\}$ with $n$ elements, such that for any two different non-empty subset of $A$, the sum of elements of one subset doesn't divide another's.

Square $ABCD$ has side length $5$. Draw E on $BC$ and $F$ on $AD$ such that $BE < AF$. Next, flip $ABCD$ across $EF$ to a square $A'B'C'D'$ such that $C'$ lies in the interior of $ABCD$ and $C$ lies in the interior of $A'B'C'D'$. Suppose that $CC' = 4$ and $DD' = 2$. Compute $AA'$.

Let \[T_0=2, T_1=3, T_2=6,\] and for $n\ge 3$, \[T_n=(n+4)T_{n-1}-4nT_{n-2}+(4n-8)T_{n-3}.\] The first few terms are \[2, 3, 6, 14, 40, 152, 784, 5158, 40576, 363392.\] Find a formula for $T_n$ of the form \[T_n=A_n+B_n,\] where $\{A_n\}$ and $\{B_n\}$ are well known sequences.

Find the largest real number $\alpha$ such that, for all non-negative real numbers $x$, $y$ and $z$, the following inequality holds: \[ (x+y+z)^3 + \alpha (x^2z + y^2x + z^2y) \geq \alpha (x^2y + y^2z + z^2x). \]

Let $ABC$ be a triangle, $P$ and $Q$ the intersections of the parallel line to $BC$ that passes through $A$ with the external angle bisectors of angles $B$ and $C$, respectively. The perpendicular to $BP$ at $P$ and the perpendicular to $CQ$ at $Q$ meet at $R$. Let $I$ be the incenter of $ABC$. Show that $AI = AR$.

Which number we need to substract from numerator and add to denominator of $\frac{\overline{28a3}}{7276}$ such that we get fraction equal to $\frac{2}{7}$

[b]p1.[/b] A $10 \times 10$ array of squares is given. In each square, a student writes the product of the row number and the column number of the square (the upper left hand corner of this array is shown below). Determine the sum of the $100$ integers written in the array. [img]https://cdn.artofproblemsolving.com/attachments/5/9/527fdf90529221f6d06af169de1728da296538.png[/img] [b]p2.[/b] The equilateral triangle $DEF$ is inscribed in the equilateral triangle $ABC$ so that $ED$ is perpendicular to $BC$. If the area of $ABC$ equals one square unit, determine the area of $DEF$. [img]https://cdn.artofproblemsolving.com/attachments/c/0/6e1a303a45fa89576e26bc8fd30ce6564aaad1.png[/img] [b]p3.[/b] Consider a symmetric triangular set of points as shown (every point lies a distance of one unit from each of its neighbors). A collection of $m$ lines has the property that for every point in the arrangement, there is at least one line in the collection that passes through that point. Prove or disprove that $m \ge 10$. [img]https://cdn.artofproblemsolving.com/attachments/0/9/540f2781312f86672df1578bfe4f68b51d3b2c.png[/img] [b]p4.[/b] Let $P$ be a convex polygon drawn on graph paper (defined as the grid of all lines with equations $x = a$ and $y = b$, with $a$ and $b$ integers). We know that all the vertices of $P$ are at the intersections of grid lines and none of its sides is parallel to a grid line. Let $H$ be the sum of the lengths of the horizontal segments of the grid which are contained in the interior of $P$, and let $V$ be the sum of the lengths of the vertical segments of the grid in the interior of $P$. Prove that $H = V$ . [b]p5.[/b] Peter, Paul, and Mary play the following game. Given a fixed positive integer $k$ which is at most $2013$, they randomly choose a subset $A$ of $\{1, 2,..., 2013\}$ with $k$ elements. The winner is Peter, Paul, or Mary, respectively, if the sum of the numbers in $A$ leaves a remainder of $0$, $1$, or $2$ when divided by $3$. Determine the values of $k$ for which this game is fair (i.e., such that the three possible outcomes are equally likely). PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $n \geq 3$ be a positive integer. Find the maximum number of diagonals in a regular $n$-gon one can select, so that any two of them do not intersect in the interior or they are perpendicular to each other.

Six standard fair six-sided dice are rolled and arranged in a row at random. Compute the expected number of dice showing the same number as the sixth die in the row.

A large flat plate of glass is suspended $\sqrt{2/3}$ units above a large flat plate of wood. (The glass is infinitely thin and causes no funny refractive effects.) A point source of light is suspended $\sqrt{6}$ units above the glass plate. An object rests on the glass plate of the following description. Its base is an isosceles trapezoid $ABCD$ with $AB \parallel DC$, $AB = AD = BC = 1$, and $DC = 2$. The point source of light is directly above the midpoint of $CD$. The object’s upper face is a triangle $EF G$ with $EF = 2$, $EG = F G =\sqrt3$. $G$ and $AB$ lie on opposite sides of the rectangle $EF CD$. The other sides of the object are $EA = ED = 1$, $F B = F C = 1$, and $GD = GC = 2$. Compute the area of the shadow that the object casts on the wood plate.

Prove that there are different points $A_0 \,\, ,A_1 \,\, , \cdots A_{2550}$ on the $XY$ plane corresponding to the following properties simultaneously. (i) Any three points are not on the same line. (ii) If $ d(A_i,A_j)$ represents the distance between $A_i\,\, , A_j $ then $$ \sum_{0 \leq i < j \leq 2550}\{d(A_i,A_j)\} < 10^{-2008}$$ Note : $ \{x \}$ represents the decimal part of x e.g. $ \{ 3.16\} = 0.16$. [i] (passer-by)[/i]

Let $a_{1}=1$, $a_{2}=2$, $a_{3}$, $a_{4}$, $\cdots$ be the sequence of positive integers of the form $2^{\alpha}3^{\beta}$, where $\alpha$ and $\beta$ are nonnegative integers. Prove that every positive integer is expressible in the form \[a_{i_{1}}+a_{i_{2}}+\cdots+a_{i_{n}},\] where no summand is a multiple of any other.

We are given the square $ABCD$. On sides $AB$ and $CD$ we are given points $ K$ and $L$ respectively, and on segment $KL$ we are given point $M$ . Prove that the second intersection point (i.e. the one other than $M$) of the intersection points of circles circumscribed around triangles $AKM$ and $MLC$ lies on the diagonal $AC$. (V . N . Dubrovskiy)