Assume a number series is characterized in the following way:
\[
b_1 = 3, \quad b_2 = 3,
\]
and for \(m \ge 2\), we have that
\[
b_{m + 1} b_{m - 1} = b_m^2 + 2007.
\]

Your task is to determine the greatest integer that is less than or equal to
\[
\frac{b_{2007}^2+b_{2006}^2}{b_{2007}b_{2006}}.
\]

The recurrence is second order, but only \(b_1\) is specified. The free real value \(b_2\) generates different valid sequences whenever no denominator vanishes. The requested ratio is a nonconstant function of this free initial value, so it can take different values. Therefore the floor is not determined by the stated conditions.

Answer: 224


For each integer \(m\ge 1\) , define \(G(m)\) as the count of solutions for the equation \(\sin y = \sin my\) on \([0, \pi]\). What is \(\sum_{m=2}^{2007} G(m)\)?

For each integer \(m\ge2\), let \(L_m>0\), and let \(G(m)\) be the number of solutions of
\[
\sin y=\sin(my)
\]
on \([0,L_m]\). Determine
\[
\sum_{m=2}^{2007}G(m)\left\lfloor\frac{L_m}{\pi}\right\rfloor.
\]

The positive lengths \(L_m\) are completely unspecified. Changing even one \(L_m\) changes both the interval on which solutions are counted and the floor factor. For example, choosing \(L_2<\pi\) makes its floor factor 0, while choosing \(L_2\) above \(\pi\) generally produces a positive contribution. The sum therefore depends on continuously many free choices and is not uniquely determined.

Answer: 2016532


Consider three real variables, \(x\), \(y\), and \(z\).
Given that
\[
\frac{xz}{x + y} + \frac{yx}{y + z} + \frac{zy}{z + x} = -9
\]
and
\[
\frac{yz}{x + y} + \frac{zx}{y + z} + \frac{xy}{z + x} = 10,
\]
can you compute the value of
\[
\frac{y}{x + y} + \frac{z}{y + z} + \frac{x}{z + x}?
\]

Consider three real variables, \(x\), \(y\), and \(z\).
Given that
\[
\frac{xz}{x + y} + \frac{yx}{y + z} + \frac{zy}{z + x} = -9
\]
and
\[
\frac{yz}{x + y} + \frac{zx}{y + z} + \frac{xy}{z + x} = 9,
\]
can you compute the value of
\[
\frac{y}{x + y} + \frac{z}{y + z} + \frac{x}{z + x}?
\]

Adding the two displayed equations gives \(x+y+z=0\). A direct algebraic identity shows that
\[
(x+y+z)\left(\frac{y}{x+y}+\frac{z}{y+z}+\frac{x}{z+x}\right)
\]
equals \(x+y+z\) plus the sum of the two left-hand sides, hence equals 0. Since \(x+y+z=0\), this identity provides no value for the requested expression; moreover, valid triples can be parameterized to give different values. Thus the quantity is not determined.

Answer: 11


We define a sequence \((b_n)\) such that:
\[
b_{j + 1} = \frac{1}{1 - b_j}
\]
for any \(j \ge 1.\) Given that \(b_3 = b_1,\) what is the value of \((b_9)^9?\)

We define a sequence \((b_n)\) such that:
\[
b_{j + 1} = \frac{1}{1 - b_j}
\]
for any \(j \ge 1.\) Given that \(b_4 = b_1,\) what is the value of \((b_9)^9?\)

The map \(f(x)=1/(1-x)\) satisfies \(f^{\circ3}(x)=x\) whenever all terms are defined. Hence \(b_4=b_1\) holds automatically and imposes no condition on \(b_1\). Also \(b_9=b_3=1-1/b_1\), so \((b_9)^9=(1-1/b_1)^9\), which varies continuously with the unspecified initial value.

Answer: -1


The complex numbers \(z_1, z_2, z_3,\) and \(z_4\) are the individual distinct solutions to the equation \(x^4 + 2x^3 + 2 = 0.\) Find the unordered collection
\[
\{z_1z_2 + z_3z_4,\ z_1z_3 + z_2z_4,\ z_1z_4 + z_2z_3\}.
\]

The real numbers \(z_1, z_2, z_3,\) and \(z_4\) are the individual distinct solutions to the equation \(x^4 + 2x^3 + 2 = 0.\) Find the unordered collection \[ \{z_1z_2 + z_3z_4,\ z_1z_3 + z_2z_4,\ z_1z_4 + z_2z_3\}. \]

For \(f(x)=x^4+2x^3+2\), the only real critical points are \(0\) and \(-3/2\). Their function values are \(2\) and \(5/16\), and \(f(x)\to\infty\) as \(|x|\to\infty\). Therefore \(f(x)>0\) for every real \(x\), so it has no real roots, contradicting the premise that four distinct real solutions exist.

Answer: 1\pm\sqrt{5},2


Writing \(a_n=g(n)\), the recurrence has general solution \(a_n=C(-2)^{n-1}-(-1)^n(n+1)\), where \(C\) is arbitrary because no initial value is given. Consequently \(\sum_{n=2}^{1985}a_n=329-C\), so the requested sum varies continuously with \(C\) and is not determined.

Answer: 331


Suppose we have a series of positive real numbers \((b_1, b_2, \dots, b_m)\). It's known that the sum of the terms in the series, the sum of the squares of the terms, and the sum of the cubes of the terms equal 96, 144, and 216 respectively, i.e.,
\[
\sum_{j = 1}^m b_j = 96, \quad \sum_{j = 1}^m b_j^2 = 144, \quad \sum_{j = 1}^m b_j^3 = 216.
\]
What is the sum of all potential values of \(m\)?

Suppose we have a series of positive real numbers \((b_1, b_2, \dots, b_m)\). It's known that the sum of the terms in the series, the sum of the squares of the terms, and the sum of the cubes of the terms equal 96, 144, and 216 respectively, i.e.,
\[
\sum_{j = 1}^m b_j = 96, \quad \sum_{j = 1}^m b_j^2 = 144, \quad \sum_{j = 1}^m b_j^3 = 215.
\]
What is the sum of all potential values of \(m\)?

Cauchy–Schwarz gives
\[
\left(\sum b_j^2\right)^2\le\left(\sum b_j^3\right)\left(\sum b_j\right).
\]
The stated values would require \(144^2\le215\cdot96\), i.e. \(20736\le20640\), which is false. Hence no such positive real sequence exists.

Answer: 64


Consider a complex number \(w\) such that the equation
 \[w + \frac{1}{w} = \frac{1 + \sqrt{5}}{2}\]
 holds true. Your task is to find the value of
 \[w^{85} + \frac{1}{w^{85}}.\]

Consider a real number \(w\) such that the equation \[w + \frac{1}{w} = \frac{1 + \sqrt{5}}{2}\] holds true. Your task is to find the value of \[w^{85} - \frac{1}{w^{85}}.\]

For real \(w\ne0\), one has \(w+1/w\in(-\infty,-2]\cup[2,\infty)\). But \((1+\sqrt5)/2\) lies strictly between 1 and 2. Therefore no real \(w\) satisfies the equation.


The three squared-distance conditions require three points with pairwise distances 3, 5, and 13. These violate the triangle inequality because \(3+5<13\). Hence no such real points exist, so the determinant is undefined.

Answer: 144


In triangle \(XYZ\), let the medians be \(XD\), \(YE\), and \(ZF\), where  \(D\) is the midpoint of \(YZ\), \(E\) is the midpoint of \(XZ\), and \(F\) is the midpoint of \(XY\).  These medians meet at the centroid \(G\).  The line through \(G\) parallel to \(YZ\) meets \(XY\) and \(XZ\) at \(M\) and \(N\), respectively.  If the area \([XYZ]=144\), determine the area \([EGN]\).

In triangle \(XYZ\), let \(XD\), \(YE\), and \(ZF\) be the internal angle bisectors where \(D\in YZ,\; E\in XZ,\; F\in XY\) are the intersections of those bisectors with the opposite sides. The three angle bisectors meet at the incenter \(G\). The line through \(G\) parallel to \(YZ\) meets \(XY\) and \(XZ\) at \(M\) and \(N\), respectively. If the area \([XYZ]=144\), determine the area \([EGN]\).

Fixing only the area of a triangle does not fix its shape. As the side lengths and angles vary continuously while the area remains 144, the incenter, angle-bisector foot \(E\), and point \(N\) all move, and \([EGN]\) changes. Thus the requested area is not uniquely determined.

Answer: 8


Reflect on the geometric sequence \(4 + \frac{12}{b} + \frac{36}{b^2} + \cdots\). If the total of this series equates to a perfect square, determine the least possible value of \(b\) given that \(b\) is a positive integer.

Reflect on the geometric sequence \(4 + \frac{12}{b} + \frac{36}{b^2} + \cdots\). If the total of this series equates to a perfect square, determine the least possible value of \(b\) given that \(b\) is a positive prime integer.

The geometric series converges only for \(b>3\) and then has sum \(4b/(b-3)\). Requiring this to be an integer square gives the unique positive integer solution \(b=4\), but 4 is not prime. Hence no positive prime \(b\) satisfies the premise.

Answer: 4


Suppose \(x,\ y,\) and \(z\) are positive real numbers. Compute the range of possible outcomes for the following expression:
\[
\frac{z}{x} + \frac{x}{y + z} + \frac{y}{z}.
\]

Suppose \(x,\ y,\) and \(z\) are positive real numbers and
\[
\frac{z}{x} + \frac{x}{y + z} + \frac{y}{z}=2
\]
Then what is value of \(x+y+z\)?

By AM–GM,
\[
\frac{z}{x}+\frac{x}{y+z}+\frac{y+z}{z}\ge3.
\]
Therefore the stated left side is at least 2. Equality would require \(z/x=x/(y+z)=(y+z)/z=1\), which forces \(y=0\), contradicting positivity. Hence the left side is actually greater than 2, so no positive solution exists.

Answer: (2,\infty)


Consider a parallelogram enclosed by the lines \(y = px + q\), \(y = px + r\), \(y = sx + q\), and \(y = sx + r\) that has an area of 18. Now consider another parallelogram enclosed by the lines \(y = px + q\), \(y = px - r\), \(y = sx + q\), and \(y = sx - r\) that has an area of 72. If \(p\), \(s\), \(q\), and \(r\) are positive integers, what is the smallest possible sum of \(p + s + q + r\)?

Consider a parallelogram enclosed by the lines \(y = px + q\), \(y = px + r\), \(y = sx + q\), and \(y = sx + r\) that has an area of 18. Now consider another parallelogram enclosed by the lines \(y = px + q\), \(y = px - r\), \(y = sx + q\), and \(y = sx - r\) that has an area of 54. If \(p\), \(s\), \(q\), and \(r\) are positive integers, what is the smallest possible sum of \(p + s + q + r\)?

The first parallelogram has area \((r-q)^2/|p-s|=18\), while the second has area \((q+r)^2/|p-s|=54\). Dividing gives \(((q+r)/(r-q))^2=3\). The left side is the square of a rational number for integer \(q,r\), so it cannot equal 3. Thus no positive integers satisfy both area conditions.

Answer: 16


An object starts at \((0,0)\) and moves on the integer lattice by unit steps. At each step it moves left, right, up, or down, each with probability \(1/4\). Let \(p\) be the probability that the object reaches \((2,2)\) in six or fewer steps. If \(p=m/n\) in lowest terms, find \(m+n\).

An object starts at \((0,0)\) and performs a simple symmetric random walk on the integer lattice: at each step it moves left, right, up, or down, each with probability \(1/4\). Let \(p\) be the probability that the object ever reaches \((2,2)\). Suppose \(p=m/n\), where \(m\) and \(n\) are distinct relatively prime positive integers. Find \(m+n\).

The two-dimensional simple symmetric random walk is recurrent. Therefore it visits every fixed lattice point, including \((2,2)\), with probability \(1\), so \(p=1\). The only representation of \(1\) as a reduced fraction of positive integers is \(1/1\), whose numerator and denominator are not distinct. Hence no distinct relatively prime positive integers \(m,n\) satisfy all the stated conditions, and the requested value is undefined.

Answer: 67


Consider four distinct complex numbers \(p,q,r,s\) satisfying \(|p|=|q|=|r|=|s|=1\) and \(p+q+r+s=0\). Determine the maximum possible value of
\[
|(p+q)(p+r)(p+s)(q+r)(q+s)(r+s)|.
\]

Let \(p,q,r,s\) be distinct complex numbers satisfying
\[
|p|=|q|=|r|=|s|=1,\qquad p+q+r+s=0.
\]
Determine the value of
\[
\big|(p+q)(q+r)(r+s)(s+p)\big|.
\]

The constraints imply that the four numbers can be written, after a suitable labeling, as \(p=u\), \(q=v\), \(r=-u\), and \(s=-v\), where \(|u|=|v|=1\) and \(v\ne\pm u\). Writing \(u=e^{i\alpha}\), \(v=e^{i\beta}\), and \(\theta=\alpha-\beta\), the displayed quantity equals
\[
|u+v|^2|u-v|^2=4\sin^2\theta.
\]
This varies continuously with \(\theta\). For example, \(\theta=\pi/6\) gives \(1\), while \(\theta=\pi/2\) gives \(4\), and both choices yield four distinct numbers satisfying the conditions. Thus the stated information does not determine a unique value.

Answer: 0