Question bank Math Olympiad Vault 2006

Do there exist positive integers $x$, $y$, and $z$ such that $x^{2006} + y^{2006} = z^{2007}$? Explain your answer.

Let $O$ be the intersection of the diagonals $AC$ and $BD$ of the convex quadrilateral $ABCD$. Let $S_{1}, S_{2}, S_{3}$, and $S_{4}$ denote the areas of the triangles $ABO$, $BCO$, $CDO$, and $DAO$.

a) Prove that $S_{1} \cdot S_{3} = S_{2} \cdot S_{4}$.

b) Does there exist a quadrilateral $ABCD$ such that $S_{1}, S_{2}, S_{3}$, and $S_{4}$ are consecutive positive integers in some order?

2006 vertices of a regular 2007-gon are red. The remaining vertex is green. Let $G$ be the total number of polygons whose one vertex is green and the others are red. Denote by $R$ the number of polygons whose all vertices are red. Which number is bigger, $R$ or $G$? Explain your answer.

A sequence of numbers $\{a_n\}$ is given by $a_1 = 1$, $a_{n+1} = 2a_n + \sqrt{3a_n^2 + 1}$ for $n \geq 1$. Prove that each term of the sequence is an integer.

A finite set of circles in the plane is called nice if it satisfies the following three conditions:
(i) No two circles intersect in more than one point;
(ii) For every point $A$ of the plane there are at most two circles passing through $A$;
(iii) Each circle from the set is tangent to exactly 5 other circles from the set.
Does there exist a nice set consisting of exactly
(a) 2006 circles?
(b) 2007 circles?

Larry can swim from Harvard to MIT (with the current of the Charles River) in 40 minutes, or back (against the current) in 45 minutes. How long does it take him to row from Harvard to MIT, if he rows the return trip in 15 minutes? (Assume that the speed of the current and Larry's swimming and rowing speeds relative to the current are all constant.) Express your answer in the format mm:ss.

Find all real solutions $(x, y)$ of the system $x^{2}+y=12=y^{2}+x$.

Let $a_{1}, a_{2}, \ldots$ be a sequence defined by $a_{1}=a_{2}=1$ and $a_{n+2}=a_{n+1}+a_{n}$ for $n \geq 1$. Find
$$
\sum_{n=1}^{\infty} \frac{a_{n}}{4^{n+1}}
$$

Tim has a working analog 12-hour clock with two hands that run continuously (instead of, say, jumping on the minute). He also has a clock that runs really slow—at half the correct rate, to be exact. At noon one day, both clocks happen to show the exact time. At any given instant, the hands on each clock form an angle between $0^{\circ}$ and $180^{\circ}$ inclusive. At how many times during that day are the angles on the two clocks equal?

Let $a$, $b$, $c$ be the roots of $x^{3}-9 x^{2}+11 x-1=0$, and let $s=\sqrt{a}+\sqrt{b}+\sqrt{c}$. Find $s^{4}-18 s^{2}-8 s$.

Let
$$
f(x) = x^{4} - 6x^{3} + 26x^{2} - 46x + 65.
$$
Let the roots of $f(x)$ be $a_{k} + i b_{k}$ for $k = 1, 2, 3, 4$. Given that the $a_{k}, b_{k}$ are all integers, find $|b_{1}| + |b_{2}| + |b_{3}| + |b_{4}|$.

Solve for all complex numbers $z$ such that $z^{4} + 4z^{2} + 6 = z$.

Compute the value of the infinite series
$$
\sum_{n=2}^{\infty} \frac{n^{4}+3 n^{2}+10 n+10}{2^{n} \cdot\left(n^{4}+4\right)}
$$

Determine the maximum value attained by
$$
\frac{x^{4}-x^{2}}{x^{6}+2 x^{3}-1}
$$
over real numbers $x\gt 1$.

A nonzero polynomial $f(x)$ with real coefficients has the property that $f(x) = f'(x) f''(x)$. What is the leading coefficient of $f(x)$?

Compute $\lim_{x \rightarrow 0} \frac{e^{x \cos x}-1-x}{\sin \left(x^{2}\right)}$.

At time $0$, an ant is at $(1,0)$ and a spider is at $(-1,0)$. The ant starts walking counterclockwise along the unit circle, and the spider starts creeping to the right along the $x$-axis. It so happens that the ant's horizontal speed is always half the spider's. What will the shortest distance ever between the ant and the spider be?

Compute $\sum_{k=1}^{\infty} \frac{k^{4}}{k!}$.

Compute
$$
\int_{0}^{1} \frac{d x}{\sqrt{x}+\sqrt[3]{x}}
$$

A triangle with vertices at $(1003,0)$, $(1004,3)$, and $(1005,1)$ in the $xy$-plane is revolved all the way around the $y$-axis. Find the volume of the solid thus obtained.