Let the real numbers $a,b,c,d$ satisfy the relations $a+b+c+d=6$ and $a^2+b^2+c^2+d^2=12.$ Prove that \[36 \leq 4 \left(a^3+b^3+c^3+d^3\right) - \left(a^4+b^4+c^4+d^4 \right) \leq 48.\] [i]Proposed by Nazar Serdyuk, Ukraine[/i]

There are $13$ positive integers greater than $\sqrt{15}$ and less than $\sqrt[3]{B}$. What is the smallest integer value of $B$?

Determine the smallest positive number $a$ such that the number of all integers belonging to $(a, 2016a]$ is $2016$.

Find all polynomials $P$ with real coefficients such that for all $x, y ,z \in R$, $$P(x)+P(y)+P(z)+P(x+y+z)=P(x+y)+P(y+z)+P(z+x)$$

Find all functions $f:\mathbb{R}\backslash\{0\}\rightarrow\mathbb{R}$ for which $xf(xy) + f(-y) = xf(x)$ for all non-zero real numbers $x, y$.

Consider systems of three linear equations with unknowns $x,$ $y,$ and $z,$ \begin{align*} a_1 x + b_1 y + c_1 z = 0 \\ a_2 x + b_2 y + c_2 z = 0 \\ a_3 x + b_3 y + c_3 z = 0 \end{align*} where each of the coefficients is either $0$ or $1$ and the system has a solution other than $x = y = z = 0.$ For example, one such system is $\{1x + 1y + 0z = 0, 0x + 1y + 1z = 0, 0x + 0y + 0z = 0\}$ with a nonzero solution of $\{x, y, z\} = \{1, -1, 1\}.$ How many such systems are there? (The equations in a system need not be distinct, and two systems containing the same equations in a different order are considered different.) $\textbf{(A) } 302 \qquad \textbf{(B) } 338 \qquad \textbf{(C) } 340 \qquad \textbf{(D) } 343 \qquad \textbf{(E) } 344$

Consider the cube with the vertices at the points $(\pm 1, \pm 1, \pm 1)$. Let $V_1,...,V_{95}$ be arbitrary points within this cube. Denote $v_i = \overrightarrow{OV_i}$, where $O = (0,0,0)$ is the origin. Consider the $2^{95}$ vectors of the form $s_1v_1 + s_2v_2 +...+ s_{95}v_{95}$, where $s_i = \pm 1$. (a) If $d = 48$, prove that among these vectors there is a vector $w = (a, b, c)$ such that $a^2 + b^2 + c^2 \le 48$. (b) Find a smaller $d$ (the smaller, the better) with the same property.

$n>1$ is an integer. Let real number $x>1$ satisfy $$x^{101}-nx^{100}+nx-1=0.$$ Prove that for any real $0<a<b<1$, there exists a positive integer $m$ so that $a<\{x^m\}<b.$ [i]Proposed by Chenjie Yu[/i]

Find the number of ordered quadruplets $(a_1,a_2,a_3,a_4)$ of integers, for which $a_1\geq 1$, $a_2\geq 2$, $a_3\geq 3$, and $-10\leq a_4\leq 10$ and $a_1+a_2+a_3+a_4=2011$ .

Find all real solutions to the following system of equations. Carefully justify your answer. \[ \left\{ \begin{array}{c} \displaystyle\frac{4x^2}{1+4x^2} = y \\ \\ \displaystyle\frac{4y^2}{1+4y^2} = z \\ \\ \displaystyle\frac{4z^2}{1+4z^2} = x \end{array} \right. \]

A sequence $ (S_n), n \geq 1$ of sets of natural numbers with $ S_1 = \{1\}, S_2 = \{2\}$ and \[{ S_{n + 1} = \{k \in }\mathbb{N}|k - 1 \in S_n \text{ XOR } k \in S_{n - 1}\}. \] Determine $ S_{1024}.$

Prove that the equation $x^{19} + x^{17} = x^{16 }+ x^7 + a$ for any $a \in R$ has at least two imaginary roots

Suppose that $\xi \ne 1$ is a root of the polynomial $f(x) = x^{167} -1$. Compute $$\left|\sum_{0<a<b<167} \xi^{a^2+b^2} \right|.$$ In the above summation $a,b$ are integers

Ruby has a non-negative integer $n$. In each second, Ruby replaces the number she has with the product of all its digits. Prove that Ruby will eventually have a single-digit number or $0$. (e.g. $86\rightarrow 8\times 6=48 \rightarrow 4 \times 8 =32 \rightarrow 3 \times 2=6$) [i]Proposed by Wong Jer Ren[/i]

Let $x,y$ and $z$ be positive reals such that $x + y + z = 3$. Prove that \[ 2 \sqrt{x + \sqrt{y}} + 2 \sqrt{y + \sqrt{z}} + 2 \sqrt{z + \sqrt{x}} \le \sqrt{8 + x - y} + \sqrt{8 + y - z} + \sqrt{8 + z - x} \]

''Moskvich'' and ''Zaporozhets'' drove past the observer on the highway and the Niva moving towards them. It is known that when the Moskvich caught up with the observer, it was equidistant from the Zaporozhets and the Niva, and when the Niva caught up with the observer, it was equal. but removed from ''Moskvich'' and ''Zaporozhets''. Prove that ''Zaporozhets'' at the moment of passing by the observer was equidistant from the Niva and ''Moskvich''.

Given an integer $n\geq 2,$ let $a_{n},b_{n},c_{n}$ be integer numbers such that \[ \left( \sqrt[3]{2}-1\right) ^{n}=a_{n}+b_{n}\sqrt[3]{2}+c_{n}\sqrt[3]{4}. \] Prove that $c_{n}\equiv 1\pmod{3} $ if and only if $n\equiv 2\pmod{3}.$

Assume $f:\mathbb N_0\to\mathbb N_0$ is a function such that $f(1)>0$ and, for any nonnegative integers $m$ and $n$, $$f\left(m^2+n^2\right)=f(m)^2+f(n)^2.$$(a) Calculate $f(k)$ for $0\le k\le12$. (b) Calculate $f(n)$ for any natural number $n$.

Let \(a_0, a_1, a_2, \dots\) be an infinite sequence of positive integers with the following properties: - \(a_0\) is a given positive integer; - For each integer \(n \geq 1\), \(a_n\) is the smallest integer greater than \(a_{n-1}\) such that \(a_n + a_{n-1}\) is a perfect square. For example, if \(a_0 = 3\), then \(a_1 = 6\), \(a_2 = 10\), \(a_3 = 15\), and so on. (a) Let \(T\) be the set of numbers of the form \(a_k - a_l\), with \(k \geq l \geq 0\) integers. Prove that, regardless of the value of \(a_0\), the number of positive integers not in \(T\) is finite. (b) Calculate, as a function of \(a_0\), the number of positive integers that are not in \(T\).

Given a convex polygon $ A_1A_2 \ldots A_n$ with area $ S$ and a point $ M$ in the same plane, determine the area of polygon $ M_1M_2 \ldots M_n,$ where $ M_i$ is the image of $ M$ under rotation $ R^{\alpha}_{A_i}$ around $ A_i$ by $ \alpha_i, i \equal{} 1, 2, \ldots, n.$

Two quadratic trinomials $ f (x) $ and $ g (x) $ differ from each other only by a permutation of coefficients. Could it be that $ f (x) \geq g (x) $ for all real $ x $?

Csenge and Eszter ate a whole basket of cherries. Csenge ate a quarter of all cherries while Eszter ate four-sevenths of all cherries and forty more. How many cherries were in the basket in total?

Determine all polynomials $P$ with integer coefficients for which there exists an integer $a_n$ such that $P(a_n)=n^n$ for all positive integers $n$.

Does there exist positive $a\in\mathbb{R}$, such that \[|\cos x|+|\cos ax| >\sin x +\sin ax \] for all $x\in\mathbb{R}$? [i]N. Agakhanov[/i]

Let $ f$ be a polynomial with positive integer coefficients. Prove that if $ n$ is a positive integer, then $ f(n)$ divides $ f(f(n)\plus{}1)$ if and only if $ n\equal{}1.$