Question bank Math Olympiad Vault 2000

Compute the sum $S=\sum_{i=0}^{101} \frac{x_{i}^{3}}{1-3 x_{i}+3 x_{i}^{2}}$ for $x_{i}=\frac{i}{101}$.
Answer: $S=51$.

Let $A B C$ be a triangle. Let $M$ and $N$ be the points in which the median and angle bisector, respectively, at $A$ meet the side $B C$. Let $Q$ and $P$ be the points in which the perpendicular at $N$ to $N A$ meets $M A$ and $B A$, respectively, and $O$ be the point in which the perpendicular at $P$ to $B A$ meets $A N$ produced. Prove that $Q O$ is perpendicular to $B C$.

Let $n$, $k$ be given positive integers with $n \gt k$. Prove that
$$
\frac{1}{n+1} \cdot \frac{n^{n}}{k^{k}(n-k)^{n-k}} \lt \frac{n!}{k!(n-k)!} \lt \frac{n^{n}}{k^{k}(n-k)^{n-k}}.
$$

Prove that any integer greater than or equal to $7$ can be written as a sum of two relatively prime integers, both greater than $1$. (Two integers are relatively prime if they share no common positive divisor other than $1$. For example, $22$ and $15$ are relatively prime, and thus $37=22+15$ represents the number $37$ in the desired way.)

Let $x_{1}, x_{2}, \ldots, x_{n}$ be positive numbers, with $n \geq 2$. Prove that
$$
\left(x_{1}+\frac{1}{x_{1}}\right)\left(x_{2}+\frac{1}{x_{2}}\right) \cdots\left(x_{n}+\frac{1}{x_{n}}\right) \geq\left(x_{1}+\frac{1}{x_{2}}\right)\left(x_{2}+\frac{1}{x_{3}}\right) \cdots\left(x_{n-1}+\frac{1}{x_{n}}\right)\left(x_{n}+\frac{1}{x_{1}}\right) .
$$

Prove that there exists a set $S$ of $3^{1000}$ points in the plane such that for each point $P$ in $S$, there are at least 2000 points in $S$ whose distance to $P$ is exactly 1 inch.

Alice plays the following game of solitaire on a $20 \times 20$ chessboard. She begins by placing 100 pennies, 100 nickels, 100 dimes, and 100 quarters on the board so that each of the 400 squares contains exactly one coin. She then chooses 59 of these coins and removes them from the board. After that, she removes coins, one at a time, subject to the following rules:
- A penny may be removed only if there are four squares of the board adjacent to its square (up, down, left, and right) that are vacant (do not contain coins). Squares "off the board" do not count towards this four: for example, a non-corner square bordering the edge of the board has three adjacent squares, so a penny in such a square cannot be removed under this rule, even if all three adjacent squares are vacant.
- A nickel may be removed only if there are at least three vacant squares adjacent to its square. (And again, "off the board" squares do not count.)
- A dime may be removed only if there are at least two vacant squares adjacent to its square ("off the board" squares do not count).
- A quarter may be removed only if there is at least one vacant square adjacent to its square ("off the board" squares do not count).
Alice wins if she eventually succeeds in removing all the coins. Prove that it is impossible for her to win.

I have three opaque jars filled with coins. One contains only nickels; one contains only dimes; one contains some nickels and some dimes. Also, the three jars are labeled "Nickels," "Dimes," and "Nickels and Dimes," but it is known that all of the jars are labeled incorrectly. You are allowed to choose one jar and look at one coin randomly drawn from it. How can you determine which jar is which?

2000 distinct positive integers are written down, and it so happens that the product of any 3 different numbers from this list is a square. Prove that each one of them is a square.

Given $n+1$ distinct integers from the set $\{1,2, \ldots, 2n\}$ ($n \geq 1$), prove that some two of them are relatively prime.

Two sets of points in the coordinate plane are given: $\{(-1,1),(-1,2), \ldots,(-1,2000)\}$ and $\{(1,1),(1,2), \ldots,(1,2000)\}$. $2000$ line segments are drawn connecting these points so that each point in the first set is connected to exactly one point in the second set, and vice versa. Find, with proof, the sum of the $y$-intercepts of the segments.

Equilateral triangle $A B C$ is inscribed in a circle. Let $D$ be the midpoint of $A B$, and let $E$ be the midpoint of $A C$. The ray $\overrightarrow{D E}$ meets the circle at $P$. Prove that $D E^{2} = D P \cdot P E$.

Do there exist positive integers $x, y, z$ such that $28x + 30y + 31z = 365$? What is the largest positive integer $n$ such that the equation $28x + 30y + 31z = n$ cannot be solved in positive integers $x, y, z$?

An $n \times n$ matrix of integers is called "golden" if, for every row and every column, their union contains all of the numbers $1,2,3, \ldots, 2 n-1$. Find all golden matrices (of all sizes).

Martha is a chicken who lives in a coop whose finitely many residents have a definite "pecking order": for every pair of distinct chickens, exactly one pecks the other. (However, it is not necessarily true that if $X$ pecks $Y$ and $Y$ pecks $Z$, then $X$ pecks $Z$.) A chicken $X$ is called a "leader" of the coop if every other chicken is pecked by $X$ or pecked by a chicken who is pecked by $X$. Prove that if no one in the coop pecks more chickens than Martha, then Martha is a leader.

Find all functions $f$ from the set $\mathbf{R}$ of real numbers to itself such that $f(x y+1)=x f(y)+2$ for all $x, y \in \mathbf{R}$.

Three grasshoppers are on a straight line. Every second one grasshopper jumps. It jumps across one (but not across two) of the other grasshoppers. Prove that after 1999 seconds the grasshoppers cannot be in the initial position.

Let $x_{1} = x_{2} = 1$ and $x_{n+2} = 2 x_{n+1} + 8 x_{n} - 1$ for $n \geq 1$. Prove that $x_{n}$ is a square for all $n$.

Let $ABC$ be a triangle and $D, E, F, G, H, I, J$ points on sides $BC, CA, AB, BC, CA, AB, BC$ respectively, such that $EF$ and $HI$ are parallel to $BC$; $IJ$ and $FG$ are parallel to $CA$; and $DE, GH$ are parallel to $AB$. Prove that $D = J$.

If $a, b, c, d, e, f$ are real numbers such that $a+b+c+d+e+f=0$ and $a^{3}+b^{3}+c^{3}+d^{3}+e^{3}+f^{3}=0$, prove that $(a+c)(a+d)(a+e)(a+f)=(b+c)(b+d)(b+e)(b+f)$.