Suppose that $S$ is a series of real numbers between $2$ and $8$ inclusive, and that for any two elements $y > x$ in $S,$ $$ 98y - 102x - xy \ge 4. $$ What is the maximum possible size for the set $S?$ $$ \mathrm a. ~ 12\qquad \mathrm b.~14\qquad \mathrm c. ~16 \qquad \mathrm d. ~18 \qquad \mathrm e. 20 $$

The least value of the function $ ax^2\plus{}bx\plus{}c$ with $a>0$ is: $\textbf{(A)}\ -\dfrac{b}{a} \qquad \textbf{(B)}\ -\dfrac{b}{2a} \qquad \textbf{(C)}\ b^2-4ac \qquad \textbf{(D)}\ \dfrac{4ac-b^2}{4a}\qquad \textbf{(E)}\ \text{None of these}$

The lengths of the sides $a, b$ and $c$ of a right triangle satisfy the relations $a <b <c$, and $\alpha$ is the measure of the smallest angle of the triangle. For which real values $k$ the equation $ax^2 + bx + kc = 0$ has real solutions for any measure of the angle $\alpha$ not exceeding $18^o$

Each positive integer is coloured red or blue. A function $f$ from the set of positive integers to itself has the following two properties: (a) if $x\le y$, then $f(x)\le f(y)$; and (b) if $x,y$ and $z$ are (not necessarily distinct) positive integers of the same colour and $x+y=z$, then $f(x)+f(y)=f(z)$. Prove that there exists a positive number $a$ such that $f(x)\le ax$ for all positive integers $x$. [i](United Kingdom) Ben Elliott[/i]

[b]Black hole thermodynamics [/b] The goal of this problem is to explore some interesting properties of Black Holes. The following equation was obtained by L. Smarr in 1973: \[ M^2 = \frac{1}{16\pi} A + \frac{4\pi}{A}\left(J^2 + \frac{1}{4}Q^4\right)+\frac{1}{2}Q^2\] where $M$, $J$, $Q$ and $A$ are the mass, angular momentum, charge and area of the event horizon of a black hole. To make contact with thermodynamics we write for the entropy of the Black Hole, \[S = \frac{1}{4}k_B A\] where $k_B$ is the Boltzmann constant. [list=1] [*] Work in natural units $G = \hbar = c = 1$ and show that the equation for the entropy is dimensionally correct. [/*] [*] Take $k_B = 1/8\pi$ (by choosing units) and derive an expression for $S(M,J,Q)$. Is this expression unique? (Hint: What is the entropy of the Schwarzschild Black Hole which corresponds to $J=Q=0$?) \item We suppose the mass-energy $M$ (since $c=1$) plays the role of internal energy. Show that $T,\Omega,\Phi$ defined via, \[ dM = T dS + \Omega dJ + \Phi dQ\] are given by, \begin{eqnarray*} & T = \frac{1}{M} \left[1- \frac{1}{16S^2}\left(J^2 + \frac{1}{4}Q^4\right)\right] \\ & \Omega = \frac{J}{8MS}\\ & \Phi = \frac{Q}{2M}\left[1+\frac{Q^2}{8S}\right]. \end{eqnarray*} This is the analog of the first law of thermodynamics. [/*] [*]Look at the expression for $M(S,J,Q)$ closely and derive the analog of the Gibbs-Duhem Relation familiar from Thermodynamics. [/*] [*] Show that, \[ S \to \frac{1}{4}M^2 - \frac{1}{8}Q^2 \] as $T \to 0$. What does this say about the third law of thermodynamics? Give reasons to support your answer. \item An alternative statement to the third law is that "it is impossible to reach absolute-zero in a finite number of steps". What can we conclude from part (e)? [/*] [/list]

Let $O$ and $I$ be the circumcenter and incenter of triangle $ABC$. The perpendicular from $I$ to $OI$ meets $AB$ and the external bisector of angle $C$ at points $X$ and $Y$ respectively. In what ratio does $I$ divide the segment $XY$?

Let $d$ be a diameter of a circle $k$, and let $A$ be an arbitrary point on this diameter $d$ in the interior of $k$. Further, let $P$ be a point in the exterior of $k$. The circle with diameter $PA$ meets the circle $k$ at the points $M$ and $N$. Find all points $B$ on the diameter $d$ in the interior of $k$ such that \[\measuredangle MPA = \measuredangle BPN \quad \text{and} \quad PA \leq PB.\] (i. e. give an explicit description of these points without using the points $M$ and $N$).

In the country of East Westmore, statisticians estimate there is a baby born every 8 hours and a death every day. To the nearest hundred, how many people are added to the population of East Westmore each year? $\textbf{(A)}\hspace{.05in}600 \qquad \textbf{(B)}\hspace{.05in}700 \qquad \textbf{(C)}\hspace{.05in}800 \qquad \textbf{(D)}\hspace{.05in}900 \qquad \textbf{(E)}\hspace{.05in}1000 $

Two positive integers are written on the blackboard. Mary records in her notebook the square of the smaller number and replaces the larger number on the blackboard by the difference of the two numbers. With the new pair of numbers, she repeats the process, and continues until one of the numbers on the blackboard becomes zero. What will be the sum of the numbers in Mary's notebook at that point? (4)

On a cube, 27 points are marked in the following manner: one point in each corner, one point on the middle of each edge, one point on the middle of each face, and one in the middle the cube. The number of lines containing three out of these points is A. 33 B. 42 C. 49 D. 72 E. 81

A basketball team's players were successful on $50\%$ of their two-point shots and $40\%$ of their three-point shots, which resulted in $54$ points. They attempted $50\%$ more two-point shots than three-point shots. How many three-point shots did they attempt? $ \textbf{(A) }10\qquad\textbf{(B) }15\qquad\textbf{(C) }20\qquad\textbf{(D) }25\qquad\textbf{(E) }30 $

Jonathan finds all ordered triples $(a, b, c)$ of positive integers such that $abc = 720$. For each ordered triple, he writes their sum $a + b + c$ on the board. (Numbers may appear more than once.) What is the sum of all the numbers written on the board?

A [i]mirrored polynomial[/i] is a polynomial $f$ of degree $100$ with real coefficients such that the $x^{50}$ coefficient of $f$ is $1$, and $f(x) = x^{100} f(1/x)$ holds for all real nonzero $x$. Find the smallest real constant $C$ such that any mirrored polynomial $f$ satisfying $f(1) \ge C$ has a complex root $z$ obeying $|z| = 1$.

Let $ABC$ be a triangle with $\angle ACB > 90^{\circ}$, and let $D$ be a point on $BC$ such that $AD$ is perpendicular to $BC$. Consider the circumference $\Gamma$ with with diameter $BC$. A line $\ell$ passes through $D$ and is tangent to $\Gamma$ at $P$, cuts $AC$ at $M$ (such that $M$ is in between $A$ and $C$), and cuts the side $AB$ at $N$. Prove that $M$ is the midpoint of $DP$ if and only if $N$ is the midpoint of $AB$.

Two years ago Tom was $25\%$ shorter than Mary. Since then Tom has grown $20\%$ taller, and Mary has grown $4$ inches taller. Now Mary is $20\%$ taller than Tom. How many inches tall is Tom now?

The shorthand of a clock has the length 3, the longhand has the length 4. Determine the distance between the endpoints of the hands at the time, where their distance increases the most.

Starting at a vertex $x_0$, we walk over the edges of a connected graph $G$ according to the following rules: 1. We never walk the same edge twice in the same direction. 2. Once we reach a vertex $x \ne x_0$, never visited before, we mark the edge by which we come to $x$. We can use this marked edge to leave vertex $x$ only if we already have traversed, in both directions, all other edges incident to $x$. Show that, regardless of the path followed, we will always be stuck at $x_0$ and that all other edges will have been traveled in both directions.

Let $n$ be a positive integer greater than $2$. Solve in nonnegative real numbers the following system of equations \[x_{k}+x_{k+1}=x_{k+2}^{2}\quad , \quad k=1,2,\cdots,n\] where $x_{n+1}=x_{1}$ and $x_{n+2}=x_{2}$.

Show that the set of positive integers which cannot be represented as a sum of distinct perfect squares is finite.

Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that $$(y + 1)f(x) + f(xf(y) + f(x + y))= y$$ for all $x, y \in \mathbb{R}$.

Show that the equation $x^2 +y^5 =z^3$ has infinitely many solutions in integers $x, y, z$ for which $xyz \neq 0$.

Each cell of a $10 \times 10$ board is painted red, blue or white, with exactly twenty of them red. No two adjacent cells are painted in the same colour. A domino consists of two adjacent cells, and it is said to be good if one cell is blue and the other is white. (a) Prove that it is always possible to cut out $30$ good dominoes from such a board. (b) Give an example of such a board from which it is possible to cut out $40$ good dominoes. (c) Give an example of such a board from which it is not possible to cut out more than $30$ good dominoes.

The set of all real solutions of the inequality \[ |x \minus{} 1| \plus{} |x \plus{} 2| < 3\] is $ \textbf{(A)}\ x \in ( \minus{} 3,2) \qquad \textbf{(B)}\ x \in ( \minus{} 1,2) \qquad \textbf{(C)}\ x \in ( \minus{} 2,1) \qquad$ $ \textbf{(D)}\ x \in \left( \minus{} \frac32,\frac72\right) \qquad \textbf{(E)}\ \O \text{ (empty})$ Note: I updated the notation on this problem.

Determine whether or not the length of symmedian is not greater than the length of the angle bisector drawn from the same vertex?

Two integers between $1$ and $100$ inclusive are chosen such that their difference is $7$ and their product is a multiple of $5$. In how many ways can this choice be made?