Question bank Math Olympiad Vault 2008

Alice rolls two octahedral dice with the numbers $2,3,4,5,6,7,8,9$. What's the probability the two dice sum to $11$?

Let $a_{0} = \frac{6}{7}$, and
$$
a_{n+1} = \begin{cases}2 a_{n} & \text{ if } a_{n} \lt \frac{1}{2} \\ 2 a_{n} - 1 & \text{ if } a_{n} \geq \frac{1}{2}\end{cases}
$$
Find $a_{2008}$.

Consider a circular cone with vertex $V$, and let $ABC$ be a triangle inscribed in the base of the cone, such that $AB$ is a diameter and $AC = BC$. Let $L$ be a point on $BV$ such that the volume of the cone is 4 times the volume of the tetrahedron $ABCL$. Find the value of $BL / LV$.

Find the number of subsets $S$ of $\{1,2, \ldots, 63\}$ the sum of whose elements is $2008$.

Find the sum of all positive integers $n$ such that $n$ divides $n^{2}+n+2$.

Let $f(r) = \sum_{j=2}^{2008} \frac{1}{j^{r}} = \frac{1}{2^{r}} + \frac{1}{3^{r}} + \cdots + \frac{1}{2008^{r}}$. Find $\sum_{k=2}^{\infty} f(k)$.

Al has a rectangle of integer side lengths $a$ and $b$, and area $1000$. What is the smallest perimeter it could have?

Solve the following system of equations for $w$.
$$
\begin{aligned}
& 2w + x + y + z = 1 \\
& w + 2x + y + z = 2 \\
& w + x + 2y + z = 2 \\
& w + x + y + 2z = 1
\end{aligned}
$$

Suppose we have an (infinite) cone $\mathcal{C}$ with apex $A$ and a plane $\pi$. The intersection of $\pi$ and $\mathcal{C}$ is an ellipse $\mathcal{E}$ with major axis $BC$, such that $B$ is closer to $A$ than $C$, and $BC = 4$, $AC = 5$, $AB = 3$. Suppose we inscribe a sphere in each part of $\mathcal{C}$ cut up by $\mathcal{E}$ with both spheres tangent to $\mathcal{E}$. What is the ratio of the radii of the spheres (smaller to larger)?

Find the number of distinct primes dividing $1 \cdot 2 \cdot 3 \cdots 9 \cdot 10$.

Let $P(x)$ be a polynomial with degree $2008$ and leading coefficient $1$ such that
$$
P(0)=2007,\ P(1)=2006,\ P(2)=2005,\ \ldots,\ P(2007)=0.
$$
Determine the value of $P(2008)$. You may use factorials in your answer.

Evaluate the infinite sum $\sum_{n=1}^{\infty} \frac{n}{n^{4}+4}$.

You have a $2 \times 3$ grid filled with integers between $1$ and $9$. The numbers in each row and column are distinct, the first row sums to $23$, and the columns sum to $14$, $16$, and $17$ respectively.

| | $14$ | $16$ | $17$ |
| :---: | :---: | :---: | :---: |
| $23$ | $a$ | $b$ | $c$ |
| | $x$ | $y$ | $z$ |

What is $x+2y+3z$?

In a game show, Bob is faced with 7 doors, 2 of which hide prizes. After he chooses a door, the host opens three other doors, of which one is hiding a prize. Bob chooses to switch to another door. What is the probability that his new door is hiding a prize?

A cat is going up a stairwell with ten stairs. However, instead of walking up the stairs one at a time, the cat jumps, going either two or three stairs up at each step (though if necessary, it will just walk the last step). How many different ways can the cat go from the bottom to the top?

If $p$ and $q$ are positive integers and $\frac{2008}{2009} \lt \frac{p}{q} \lt \frac{2009}{2010}$, what is the minimum value of $p$?

Point $A$ lies at $(0,4)$ and point $B$ lies at $(3,8)$. Find the $x$-coordinate of the point $X$ on the $x$-axis maximizing $\angle A X B$.

Determine the last two digits of $17^{17}$, written in base 10.

Solve the equation
$$
\sqrt{x+\sqrt{4 x+\sqrt{16 x+\sqrt{\ldots+\sqrt{4^{2008} x+3}}}}}-\sqrt{x}=1
$$
Express your answer as a reduced fraction with the numerator and denominator written in their prime factorization.

Let $ABC$ be a right triangle with $\angle A = 90^{\circ}$. Let $D$ be the midpoint of $AB$ and let $E$ be a point on segment $AC$ such that $AD = AE$. Let $BE$ meet $CD$ at $F$. If $\angle BFC = 135^{\circ}$, determine $BC / AB$.