Let $n$ be the answer to this problem. The polynomial $x^n+ax^2+bx+c$ has real coefficients and exactly $k$ real roots. Find the sum of the possible values of $k$.

Find all integer triples $(m,p,q)$ satisfying \[2^mp^2+1=q^5\] where $m>0$ and both $p$ and $q$ are prime numbers.

An equilateral triangle with side length $6$ has a square of side length $6$ attached to each of its edges as shown. The distance between the two farthest vertices of this figure (marked $A$ and $B$ in the figure) can be written as $m + \sqrt{n}$ where $m$ and $n$ are positive integers. Find $m + n$. [asy] draw((0,0)--(1,0)--(1/2,sqrt(3)/2)--cycle); draw((1,0)--(1+sqrt(3)/2,1/2)--(1/2+sqrt(3)/2,1/2+sqrt(3)/2)--(1/2,sqrt(3)/2)); draw((0,0)--(-sqrt(3)/2,1/2)--(-sqrt(3)/2+1/2,1/2+sqrt(3)/2)--(1/2,sqrt(3)/2)); dot((-sqrt(3)/2+1/2,1/2+sqrt(3)/2)); label("A", (-sqrt(3)/2+1/2,1/2+sqrt(3)/2), N); draw((1,0)--(1,-1)--(0,-1)--(0,0)); dot((1,-1)); label("B", (1,-1), SE); [/asy]

Let $p$ be a prime number, and let $n$ be a positive integer. Find the number of quadruples $(a_1, a_2, a_3, a_4)$ with $a_i\in \{0, 1, \ldots, p^n - 1\}$ for $i = 1, 2, 3, 4$, such that \[p^n \mid (a_1a_2 + a_3a_4 + 1).\]

Let $A,B$ be adjacent vertices of a regular $n$-gon ($n\ge5$) with center $O$. A triangle $XYZ$, which is congruent to and initially coincides with $OAB$, moves in the plane in such a way that $Y$ and $Z$ each trace out the whole boundary of the polygon, with $X$ remaining inside the polygon. Find the locus of $X$.

Barb the baker creates a new temperature system for baking bread, Breadus, which is linearly based on Fahrenheit. Bread rises at $110$ F$^\circ$, which is $0$ on the Breadus scale. Bread bakes at $350$ F$^\circ$, which is $100$ on the Breadus scale. Bread is done when it’s internal temperature is $200$ F$^\circ.$ What is this temperature on the Breadus scale? $\textbf{(A) }33\qquad\textbf{(B) }34.5\qquad\textbf{(C) }36\qquad\textbf{(D) }37.5\qquad\textbf{(E) }39$

Define two functions $f(t)=\frac 12\left(t+\frac{1}{t}\right),\ g(t)=t^2-2\ln t$. When real number $t$ moves in the range of $t>0$, denote by $C$ the curve by which the point $(f(t),\ g(t))$ draws on the $xy$-plane. Let $a>1$, find the area of the part bounded by the line $x=\frac 12\left(a+\frac{1}{a}\right)$ and the curve $C$.

In $\triangle ABC$, $AB=AC=28$ and $BC=20$. Points $D,E,$ and $F$ are on sides $\overline{AB}$, $\overline{BC}$, and $\overline{AC}$, respectively, such that $\overline{DE}$ and $\overline{EF}$ are parallel to $\overline{AC}$ and $\overline{AB}$, respectively. What is the perimeter of parallelogram $ADEF$? [asy] size(180); pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); real r=5/7; pair A=(10,sqrt(28^2-100)),B=origin,C=(20,0),D=(A.x*r,A.y*r); pair bottom=(C.x+(D.x-A.x),C.y+(D.y-A.y)); pair E=extension(D,bottom,B,C); pair top=(E.x+D.x,E.y+D.y); pair F=extension(E,top,A,C); draw(A--B--C--cycle^^D--E--F); dot(A^^B^^C^^D^^E^^F); label("$A$",A,NW); label("$B$",B,SW); label("$C$",C,SE); label("$D$",D,W); label("$E$",E,S); label("$F$",F,dir(0)); [/asy] $\textbf{(A) }48\qquad \textbf{(B) }52\qquad \textbf{(C) }56\qquad \textbf{(D) }60\qquad \textbf{(E) }72\qquad$

$\omega$ is incircle of $\triangle ABC$ touch $AC$ in $S$. Point $Q$ lies on $\omega$ and midpoints of $AQ$ and $QC$ lies on $\omega$ . Prove that $QS$ bisects $\angle AQC$

Jose, Thuy, and Kareem each start with the number $10$. Jose subtracts $1$ from the number $10$, doubles his answer, and then adds $2$. Thuy doubles the number $10$, subtracts $1$ from her answer, and then adds $2$. Kareem subtracts $1$ from the number $10$, adds $2$ to his number, and then doubles the result. Who gets the largest final answer? $\text{(A)}\ \text{Jose} \qquad \text{(B)}\ \text{Thuy} \qquad \text{(C)}\ \text{Kareem} \qquad \text{(D)}\ \text{Jose and Thuy} \qquad \text{(E)}\ \text{Thuy and Kareem}$

Suppose that $x,$ $y,$ and $z$ are three positive numbers that satisfy the equations $xyz=1,$ $x+\frac{1}{z}=5,$ and $y+\frac{1}{x}=29.$ Then $z+\frac{1}{y}=\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

Triangle $AOB$ is rotated in plane around point $O$ for $90^{\circ}$ and it maps in triangle $A_1OB_1$ ($A$ maps to $A_1$, $B$ maps to $B_1$). Prove that median of triangle $OAB_1$ of side $AB_1$ is orthogonal to $A_1B$

Two sets $M=\{x,xy,\lg(xy)\},N=\{0,|x|,y\}$, if $M=N$, then $(x+\frac{1}{y})+(x^2+\frac{1}{y^2})+\cdots+(x^{2001}+\frac{1}{y^{2001}})=$________.

Given a sequence $(x_n )$ such that $\lim_{n\to \infty} x_n - x_{n-2}=0,$ prove that $$\lim_{n\to \infty} \frac{x_n -x_{n-1}}{n}=0.$$

In $\triangle {ABC}$, tangents of the circumcircle $\odot {O}$ at $B, C$ and at $A, B$ intersects at $X, Y$ respectively. $AX$ cuts $BC$ at ${D}$ and $CY$ cuts $AB$ at ${F}$. Ray $DF$ cuts arc $AB$ of the circumcircle at ${P}$. $Q, R$ are on segments $AB, AC$ such that $P, Q, R$ are collinear and $QR \parallel BO$. If $PQ^2=PR \cdot QR$, find $\angle ACB$.

In a rhombus, $ ABCD$, line segments are drawn within the rhombus, parallel to diagonal $ BD$, and terminated in the sides of the rhombus. A graph is drawn showing the length of a segment as a function of its distance from vertex $ A$. The graph is: $ \textbf{(A)}\ \text{A straight line passing through the origin.} \\ \textbf{(B)}\ \text{A straight line cutting across the upper right quadrant.} \\ \textbf{(C)}\ \text{Two line segments forming an upright V.} \\ \textbf{(D)}\ \text{Two line segments forming an inverted V.} \\ \textbf{(E)}\ \text{None of these.}$

Let $F(x,y)$ and $G(x,y)$ be relatively prime homogeneous polynomials of degree at least one having integer coefficients. Prove that there exists a number $c$ depending only on the degrees and the maximum of the absolute values of the coefficients of $F$ and $G$ such that $F(x,y)\neq G(x,y)$ for any integers $x$ and $y$ that are relatively prime and satisfy $\max \{ |x|,|y|\} > c$. [K. Gyory]

Do there exist positive integers $k$ and $n$ such that for any finite graph $G$ with diameter $k+1$ there exists a set $S$ of at most $n$ vertices such that for any $v\in V(G)\setminus S$, there exists a vertex $u\in S$ of distance at most $k$ from $v$? [i]David Yang.[/i]

The base of the pyramid is a convex quadrangle. Is there necessarily a section of this pyramid that does not intersect the base and is an inscribed quadrangle? (M. Volchkevich)

Points $A_1, B_1, C_1$ inside an acute-angled triangle $ABC$ are selected on the altitudes from $A, B, C$ respectively so that the sum of the areas of triangles $ABC_1, BCA_1$, and $CAB_1$ is equal to the area of triangle $ABC$. Prove that the circumcircle of triangle $A_1B_1C_1$ passes through the orthocenter $H$ of triangle $ABC$.

Two players play the following game. They alternately write divisors of $100!$ on the blackboard, not repeating any of the numbers written before. The player after whose move the greatest common divisor of the written numbers equals $1,$ loses the game. Which player has a winning strategy?

For a positive integer $a$, define the sequence ($x_n$) by $x_1 = x_2 = 1$ and $x_{n+2 }= (a^4 +4a^2 +2)x_{n+1} -x_n -2a^2$ , for n $\ge 1$. Show that $x_n$ is a perfect square and that for $n > 2$ its square root equals the first entry in the matrix $\begin{pmatrix} a^2+1 & a \\ a & 1 \end{pmatrix}^{n-2}$

Given $3n$ points in the plane, no three collinear, is it always possible to form $n$ triangles (with vertices at the points), so that no point in the plane lies in more than one triangle?

If $ m$ men can do a job in $ d$ days, then $ m\plus{}r$ men can do the job in: $\textbf{(A)}\ d+r\text{ days} \qquad \textbf{(B)}\ d-r\text{ days} \qquad \textbf{(C)}\ \dfrac{md}{m+r}\text{ days} \qquad \textbf{(D)}\ \dfrac{d}{m+r}\text{ days} \qquad \textbf{(E)}\ \text{None of these}$

[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]