Does every positive polynomial in two real variables attain its lower bound in the plane?

How many pairs of postive integers $(a,b)$ with $a\geq b$ are there such that $a^2+b^2$ divides both $a^3+b$ and $a+b^3$? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ \text{Infinitely many} $

Let the polynomial $P(x)=x^3+19x^2+94x+a$ where $a\in\mathbb{N}$. If $p$ a prime number, prove that no more than three numbers of the numbers $P(0), P(1),\ldots, P(p-1)$ are divisible by $p$.

Determine the smallest constant $N$ so that the following may hold true: Geostan has deployed secret agents in Combostan. All pairs of agents can communicate, either directly or through other agents. The distance between two agents is the smallest number of agents in a communication chain between the two agents. Andreas and Edvard are among these agents, and Combostan has given Noah the task of determining the distance between Andreas and Edvard. Noah has a list of numbers, one for each agent. The number of an agent describes the longest of the two distances from the agent to Andreas and Edvard. However, Noah does not know which number corresponds to which agent, or which agents have direct contact. Given this information, he can write down $N$ numbers and prove that the distance between Andreas and Edvard is one of these $N$ numbers. The number $N$ is independent of the agents’ communication network.

Find all functions $f:\mathbb R\to\mathbb R$ satisfying the equality $$ f(f(x)+f(y))=(x+y)f(x+y) $$ for all real $x$ and $y$. [i](B. Serankou)[/i]

On each of the $2014^2$ squares of a $2014 \times 2014$-board a light bulb is put. Light bulbs can be either on or off. In the starting situation a number of the light bulbs is on. A move consists of choosing a row or column in which at least $1007$ light bulbs are on and changing the state of all $2014$ light bulbs in this row or column (from on to off or from off to on). Find the smallest non-negative integer $k$ such that from each starting situation there is a finite sequence of moves to a situation in which at most $k$ light bulbs are on.

Find the number of integers $n$ with $1 \le n \le 2017$ so that $(n-2)(n-0)(n-1)(n-7)$ is an integer multiple of $1001$.

Let $P,Q$ be the polynomials: $$x^3-4x^2+39x-46, x^3+3x^2+4x-3,$$ respectively. 1. Prove that each of $P, Q$ has an unique real root. Let them be $\alpha,\beta$, respectively. 2. Prove that $\{ \alpha\}>\{ \beta\} ^2$, where $\{ x\}=x-\lfloor x\rfloor$ is the fractional part of $x$.

The tangents at $B$ and $C$ to the circumcircle of the acute-angled triangle $ABC$ meet at $X$. Let $M$ be the midpoint of $BC$. Prove that [i](a)[/i] $\angle BAM = \angle CAX$, and [i](b)[/i] $\frac{AM}{AX} = \cos\angle BAC.$

Find all $f:\mathbb{R}\rightarrow \mathbb{R}$ such that the equation $$f(x^2+yf(x))=(1-x)f(y-x)$$ holds for all $x,y\in\mathbb{R}$.

Suppose hops, skips and jumps are specific units of length. If $b$ hops equals $c$ skips, $d$ jumps equals $e$ hops, and $f$jumps equals $g$ meters, then one meter equals how many skips? $ \textbf{(A)}\ \frac{bdg}{cef}\qquad\textbf{(B)}\ \frac{cdf}{beg}\qquad\textbf{(C)}\ \frac{cdg}{bef}\qquad\textbf{(D)}\ \frac{cef}{bdg}\qquad\textbf{(E)}\ \frac{ceg}{bdf} $

$\boxed{A5}$Let $n\in{N},n>2$,and suppose $a_1,a_2,...,a_{2n}$ is a permutation of the numbers $1,2,...,2n$ such that $a_1<a_3<...<a_{2n-1}$ and $a_2>a_4>...>a_{2n}.$Prove that \[(a_1-a_2)^2+(a_3-a_4)^2+...+(a_{2n-1}-a_{2n})^2>n^3\]

Find all the real solutions to $$n=\sum_{i=1}^n x_i=\sum_{1\le i<j\le n} x_ix_j$$ [i]Proposed by Carlos Dominguez, Valle[/i]

Several pounamu stones weigh altogether $10$ tons and none of them weigh more than $1$ tonne. A truck can carry a load which weight is at most $3$ tons. What is the smallest number of trucks such that bringing all stones from the quarry will be guaranteed?

Let $ABC$ be an acute triangle with circumcenter $O$ and circumradius $R$. $AO$ meets the circumcircle of $BOC$ at $A'$, $BO$ meets the circumcircle of $COA$ at $B'$ and $CO$ meets the circumcircle of $AOB$ at $C'$. Prove that \[OA'\cdot OB'\cdot OC'\geq 8R^{3}.\] Sorry if this has been posted before since this is a very classical problem, but I failed to find it with the search-function.

For a positive integer $n$, let $\langle n \rangle$ denote the perfect square integer closest to $n$. For example, $\langle 74 \rangle = 81$, $\langle 18 \rangle = 16$. If $N$ is the smallest positive integer such that $$ \langle 91 \rangle \cdot \langle 120 \rangle \cdot \langle 143 \rangle \cdot \langle 180 \rangle \cdot \langle N \rangle = 91 \cdot 120 \cdot 143 \cdot 180 \cdot N $$ find the sum of the squares of the digits of $N$.

Consider a triangle $\vartriangle ABC$ with $BC = 10$. An excircle is a circle that is tangent to one side of the triangle as well as the extensions of the other two sides; suppose that the excircle opposite vertex $B$ has center $I_2$ and exradius $r_2 = 11$, and suppose that the excircle opposite vertex $C$ has center $I_3$ and exradius $r_3 = 13$. Compute $I_2I_3$.

Let $ABC$ be a triangle with circumcircle $\Gamma$, and points $E$ and $F$ are chosen from sides $CA$, $AB$, respectively. Let the circumcircle of triangle $AEF$ and $\Gamma$ intersect again at point $X$. Let the circumcircles of triangle $ABE$ and $ACF$ intersect again at point $K$. Line $AK$ intersect with $\Gamma$ again at point $M$ other than $A$, and $N$ be the reflection point of $M$ with respect to line $BC$. Let $XN$ intersect with $\Gamma$ again at point $S$ other that $X$. Prove that $SM$ is parallel to $BC$. [i] Proposed by Ming Hsiao[/i]

Determine all integers $n \geq 2$ for which the number $11111$ in base $n$ is a perfect square.

How many subsets (including the empty-set) of $\{1, 2..., 6\}$ do not have three consecutive integers?

Let $\clubsuit\left(x\right)$ denote the sum of the digits of the positive integer $x$. For example, $\clubsuit\left(8\right)=8$ and $\clubsuit\left(123\right)=1+2+3=6$. For how many two-digit values of $x$ is $\clubsuit\left(\clubsuit\left(x\right)\right)=3$?

In the binomial expansion of $(\sqrt{x}+2)^{2n+1}$, sum of coefficients that power of $x$ is an integer is________.

In an acute-angled triangle $ABC$, $M$ is the midpoint of side $BC$, and $D, E$ and $F$ the feet of the altitudes from $A, B$ and $C$, respectively. Let $H$ be the orthocenter of $\Delta ABC$, $S$ the midpoint of $AH$, and $G$ the intersection of $FE$ and $AH$. If $N$ is the intersection of the median $AM$ and the circumcircle of $\Delta BCH$, prove that $\angle HMA = \angle GNS$. [i]Proposed by Marko Djikic[/i]

Five people are standing in a straight line, and the distance between any two people is a unique positive integer number of units. Find the least possible distance between the leftmost and rightmost people in the line in units.

For how many integer values of $ k$ do the graphs of $ x^2 \plus{} y^2 \equal{} k^2$ and $ xy \equal{} k$ [u]not[/u] intersect? $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 8$