Question bank Math Olympiad Vault 1978

The sides $AD$ and $BC$ of a convex quadrilateral $ABCD$ are extended to meet at $E$. Let $H$ and $G$ be the midpoints of $BD$ and $AC$, respectively. Find the ratio of the area of the triangle $EHG$ to that of the quadrilateral $ABCD$.

Determine the largest real number $z$ such that
$$
\begin{gathered}
x + y + z = 5 \\
x y + y z + x z = 3
\end{gathered}
$$
and $x$, $y$ are also real.

Let $n$ be an integer. If the tens digit of $n^2$ is $7$, what is the units digit of $n^2$?

Sketch the graph of $x^{3} + x y + y^{3} = 3$.

Find all pairs $a, b$ of positive integers satisfying the equation $2 a^{2} = 3 b^{3}$.

Eve and Odette play a game on a $3 \times 3$ checkerboard, with black checkers and white checkers. The rules are as follows:
I. They play alternately.
II. A turn consists of placing one checker on an unoccupied square of the board.
III. In her turn, a player may select either a white checker or a black checker and need not always use the same colour.
IV. When the board is full, Eve obtains one point for every row, column or diagonal that has an even number of black checkers, and Odette obtains one point for every row, column or diagonal that has an odd number of black checkers.
V. The player obtaining at least five of the eight points WINS.


a. Is a $4$-$4$ tie possible? Explain.


b. Describe a winning strategy for the girl who is first to play.

An $n$-gon area $A$ is inscribed in a circle radius $R$. We take a point on each side of the polygon to form another $n$-gon. Show that it has perimeter at least $2A/R$.

If $P$, $Q$ are points in space the point $[PQ]$ is the point on the line $PQ$ on the opposite side of $Q$ to $P$ and the same distance from $Q$.

$K_0$ is a set of points in space. Given $K_n$ we derive $K_{n+1}$ by adjoining all the points $[PQ]$ with $P$ and $Q$ in $K_n$.

(1) $K_0$ contains just two points $A$ and $B$, a distance $1$ apart, what is the smallest $n$ for which $K_n$ contains a point whose distance from $A$ is at least $1000$?

(2) $K_0$ consists of three points, each pair a distance $1$ apart, find the area of the smallest convex polygon containing $K_n$.

(3) $K_0$ consists of four points, forming a regular tetrahedron with volume $1$. Let $H_n$ be the smallest convex polyhedron containing $K_n$. How many faces does $H_1$ have? What is the volume of $H_n$?

Let $a_n$ be the nearest integer to $\sqrt{n}$. Find
$$
\frac{1}{a_1} + \frac{1}{a_2} + \ldots + \frac{1}{a_{1980}}.
$$

Show that there is an infinite sequence of reals $x_1$, $x_2$, $x_3$, ... such that $|x_n|$ is bounded and for any $m \gt n$, we have $|x_m - x_n| \gt 1 / (m - n)$.

Show that for some $k$, you can find $1978$ different sizes of square with all its vertices on the graph of the function $y = k \sin x$.

$a_1, a_2, \ldots, a_n$ are real numbers. Let $b_k = (a_1 + a_2 + \ldots + a_k)/k$ for $k = 1, 2, \ldots, n$. Let
$$
C = (a_1 - b_1)^2 + (a_2 - b_2)^2 + \ldots + (a_n - b_n)^2,
$$
and
$$
D = (a_1 - b_n)^2 + (a_2 - b_n)^2 + \ldots + (a_n - b_n)^2.
$$
Show that $C \leq D \leq 2C$.

Let $x_n = (1 + \sqrt{2} + \sqrt{3})^{n}$. We may write $x_n = a_n + b_n \sqrt{2} + c_n \sqrt{3} + d_n \sqrt{6}$, where $a_n, b_n, c_n, d_n$ are integers. Find the limit as $n$ tends to infinity of $b_n / a_n$, $c_n / a_n$, $d_n / a_n$.

Given a set of $n$ non-intersecting segments in the plane. No two segments lie on the same line. Can we successively add $n-1$ additional segments so that we end up with a single non-intersecting path? Each segment we add must have as its endpoints two existing segment endpoints.

The set $\{S_0\}$ has the single member $(5, 19)$. We derive the set $\{S_{n + 1}\}$ from $\{S_n\}$ by adjoining a pair to $\{S_n\}$. If $\{S_n\}$ contains the pair $(2a, 2b)$, then we may adjoin the pair $(a, b)$. If $\{S\}$ contains the pair $(a, b)$ we may adjoin $(a+1, b+1)$. If $\{S\}$ contains $(a, b)$ and $(b, c)$, then we may adjoin $(a, c)$. Can we obtain $(1, 50)$? $(1, 100)$? If we start with $(a, b)$, with $a \lt b$, instead of $(5, 19)$, for which $n$ can we obtain $(1, n)$?

Show that there is no positive integer $n$ for which $1000^{n} - 1$ divides $1978^{n} - 1$.

Let $p(x) = x^2 + x + 1$. Show that for every positive integer $n$, the numbers $n$, $p(n)$, $p(p(n))$, $p(p(p(n)))$, ... are relatively prime.

Two players play a game by moving a piece on an $n \times n$ chessboard. The piece is initially in a corner square. Each player may move the piece to any adjacent square (which shares a side with its current square), except that the piece may never occupy the same square twice. The first player who is unable to move loses. Show that for even $n$ the first player can always win, and for odd $n$ the second player can always win. Who wins if the piece is initially on a square adjacent to the corner?

$ABCD$ is a quadrilateral. $M$ is a point inside it such that $ABMD$ is a parallelogram. $\angle CBM = \angle CDM$. Show that $\angle ACD = \angle BCM$.

Two players play a game. There is a heap of $m$ tokens and a heap of $n \lt m$ tokens. Each player in turn takes one or more tokens from the heap which is larger. The number he takes must be a multiple of the number in the smaller heap. For example, if the heaps are $15$ and $4$, the first player may take $4$, $8$ or $12$ from the larger heap. The first player to clear a heap wins. Show that if $m \gt 2n$, then the first player can always win. Find all $k$ such that if $m \gt k n$, then the first player can always win.