Question bank Math Olympiad Vault 2009

Evaluate the sum:
$$
11^{2}-1^{2}+12^{2}-2^{2}+13^{2}-3^{2}+\ldots+20^{2}-10^{2}
$$

Given that $a + b + c = 5$ and that $1 \leq a, b, c \leq 2$, what is the minimum possible value of $\frac{1}{a+b} + \frac{1}{b+c}$?

What is the period of the function $f(x) = \cos(\cos(x))$?

How many subsets $A$ of $\{1,2,3,4,5,6,7,8,9,10\}$ have the property that no two elements of $A$ sum to $11$?

A polyhedron has faces that are all either triangles or squares. No two square faces share an edge, and no two triangular faces share an edge. What is the ratio of the number of triangular faces to the number of square faces?

Find the maximum value of $x+y$, given that $x^{2}+y^{2}-3y-1=0$.

There are 15 stones placed in a line. In how many ways can you mark 5 of these stones so that there are an odd number of stones between any two of the stones you marked?

Let $\triangle ABC$ be an equilateral triangle with height $13$, and let $O$ be its center. Point $X$ is chosen at random from all points inside $\triangle ABC$. Given that the circle of radius $1$ centered at $X$ lies entirely inside $\triangle ABC$, what is the probability that this circle contains $O$?

A set of points is convex if the points are the vertices of a convex polygon (that is, a non-selfintersecting polygon with all angles less than or equal to $180^{\circ}$). Let $S$ be the set of points $(x, y)$ such that $x$ and $y$ are integers and $1 \leq x, y \leq 26$. Find the number of ways to choose a convex subset of $S$ that contains exactly 98 points.

Compute
$$
\prod_{n=0}^{\infty}\left(1-\left(\frac{1}{2}\right)^{3^{n}}+\left(\frac{1}{4}\right)^{3^{n}}\right)
$$

Compute
$$
1 \cdot 2^{2} + 2 \cdot 3^{2} + 3 \cdot 4^{2} + \cdots + 19 \cdot 20^{2}
$$

If $f(x) = \frac{x}{x+1}$, what is $f(f(f(f(2009))))$?

Given that $\sin A + \sin B = 1$ and $\cos A + \cos B = 3/2$, what is the value of $\cos (A - B)$?

A knight begins on the lower-left square of a standard chessboard. How many squares could the knight end up at after exactly 2009 legal knight's moves? (A knight's move is 2 squares either horizontally or vertically, followed by 1 square in a direction perpendicular to the first.)

Find all pairs of integer solutions $(n, m)$ to
$$
2^{3^{n}} = 3^{2^{m}} - 1
$$

Consider a square, inside which is inscribed a circle, inside which is inscribed a square, inside which is inscribed a circle, and so on, with the outermost square having side length $1$. Find the difference between the sum of the areas of the squares and the sum of the areas of the circles.

Simplify: $i^{0} + i^{1} + \cdots + i^{2009}$.

A cube has side length $1$. Find the product of the lengths of the diagonals of this cube (a diagonal is a line between two vertices that is not an edge).

Tanks has a pile of 5 blue cards and 5 red cards. Every morning, he takes a card and throws it down a well. What is the probability that the first card he throws down and the last card he throws down are the same color?

In how many distinct ways can you color each of the vertices of a tetrahedron either red, blue, or green such that no face has all three vertices the same color? (Two colorings are considered the same if one coloring can be rotated in three dimensions to obtain the other.)