Navvye/Olympiad-Problems-CSV huggingface dataset:
1994 Canada National Olympiad
Evaluate $\sum_{n=1}^{1994}{\left((-1)^{n}\cdot\left(\frac{n^2 + n + 1}{n!}\right)\right)}$ .
1994 Canada National Olympiad
Prove that $(\sqrt{2}-1)^n$ $\forall n\in \mathbb{Z}^{+}$ can be represented as $\sqrt{m}-\sqrt{m-1}$ for some $m\in \mathbb{Z}^{+}$.
1994 Canada National Olympiad
$25$ men sit around a circular table. Every hour there is a vote, and each must respond yes or no. Each man behaves as follows: on the $n^{\text{th}}$, vote if his response is the same as the response of at least one of the two people he sits between, then he will respond the same way on the $(n+1)^{\text{th}}$ vote as on the $n^{\text{th}}$ vote; but if his response is different from that of both his neighbours on the $n^{\text{th}}$ vote, then his response on the $(n+1)^{\text{th}}$ vote will be different from his response on the $n^{\text{th}}$ vote. Prove that, however everybody responded on the first vote, there will be a time after which nobody's response will ever change.
1994 Canada National Olympiad
Let $AB$ be a diameter of a circle $\Omega$ and $P$ be any point not on the line through $AB$. Suppose that the line through $PA$ cuts $\Omega$ again at $U$, and the line through $PB$ cuts $\Omega$ at $V$. Note that in case of tangency, $U$ may coincide with $A$ or $V$ might coincide with $B$. Also, if $P$ is on $\Omega$ then $P=U=V$. Suppose that $|PU|=s|PA|$ and $|PV|=t|PB|$ for some $0\le s,t\in \mathbb{R}$. Determine $\cos \angle APB$ in terms of $s,t$.
1994 Canada National Olympiad
Let $ABC$ be an acute triangle. Let $AD$ be the altitude on $BC$, and let $H$ be any interior point on $AD$. Lines $BH,CH$, when extended, intersect $AC,AB$ at $E,F$ respectively. Prove that $\angle EDH=\angle FDH$.
1993 Canada National Olympiad
Determine a triangle for which the three sides and an altitude are four consecutive integers and for which this altitude partitions the triangle into two right triangles with integer sides. Show that there is only one such triangle.
1993 Canada National Olympiad
Show that the number $x$ is rational if and only if three distinct terms that form a geometric progression can be chosen from the sequence

$$x, ~ x+1, ~ x+2,~ x+3,\ldots . $$
1993 Canada National Olympiad
In triangle $ABC,$ the medians to the sides $\overline{AB}$ and $\overline{AC}$ are perpendicular. Prove that $\cot B+\cot C\ge \frac23.$
1993 Canada National Olympiad
Anumber of schools took part in a tennis tournament. No two players from the same school played against each other. Every two players from different schools played exactly one match against each other. A match between two boys or between two girls was called a single and that between a boy and a girl was called a mixed single. The total number of boys differed from the total number of girls by at most 1. The total number of singles differed from the total number of mixed singles by at most 1. At most how many schools were represented by an odd number of players?
1993 Canada National Olympiad
Let $y_{1}, y_{2}, y_{3},\ldots$ be a sequence such that $y_{1}=1$ and, for $k>0,$ is defined by the relationship:

$$y_{2k}=\begin{cases}2y_{k}& \text{if}~k~ \text{is even}\\ 2y_{k}+1 & \text{if}~k~ \text{is odd}\end{cases}$$$$y_{2k+1}=\begin{cases}2y_{k}& \text{if}~k~ \text{is odd}\\ 2y_{k}+1 & \text{if}~k~ \text{is even}\end{cases}$$Show that the sequence takes on every positive integer value exactly once.
1992 Canada National Olympiad
Prove that the product of the first $ n$ natural numbers is divisible by the sum of the first $ n$ natural numbers if and only if $ n+1$ is not an odd prime.
1992 Canada National Olympiad
For $ x,y,z \geq 0,$ establish the inequality



$$ x(x-z)^2 + y(y-z)^2 \geq (x-z)(y-z)(x+y-z)$$



and determine when equality holds.
1992 Canada National Olympiad
In the diagram, $ ABCD$ is a square, with $ U$ and $ V$ interior points of the sides $ AB$ and $ CD$ respectively. Determine all the possible ways of selecting $ U$ and $ V$ so as to maximize the area of the quadrilateral $ PUQV$.



//cdn.artofproblemsolving.com/images/adf30de296d208bb2f36792cf94608619204b78f.jpg
1992 Canada National Olympiad
Solve the equation



$$ x^2 + \frac{x^2}{(x+1)^2} = 3$$
1992 Canada National Olympiad
A deck of $ 2n+1$ cards consists of a joker and, for each number between 1 and $ n$ inclusive, two cards marked with that number. The $ 2n+1$ cards are placed in a row, with the joker in the middle. For each $ k$ with $ 1 \leq k \leq n,$ the two cards numbered $ k$ have exactly $ k-1$ cards between them. Determine all the values of $ n$ not exceeding 10 for which this arrangement is possible. For which values of $ n$ is it impossible?
1991 Canada National Olympiad
Show that the equation $x^2+y^5=z^3$ has infinitely many solutions in integers $x, y,z$ for which $xyz \neq 0$.
1991 Canada National Olympiad
Let $n$ be a fixed positive integer. Find the sum of all positive integers with the property that in base $2$ each has exactly $2n$ digits, consisting of $n$ 1's and $n$ 0's. (The first digit cannot be $0$.)
1991 Canada National Olympiad
Let $C$ be a circle and $P$ a given point in the plane. Each line through $P$ which intersects $C$ determines a chord of $C$. Show that the midpoints of these chords lie on a circle.
1991 Canada National Olympiad
Can ten distinct numbers $a_1, a_2, b_1, b_2, b_3, c_1, c_2, d_1, d_2, d_3$ be chosen from $\{0, 1, 2, \ldots, 14\}$, so that the $14$ differences $|a_1 - b_1|$, $|a_1 - b_2|$, $|a_1 - b_3|$, $|a_2 - b_1|$, $|a_2 - b_2|$, $|a_2 - b_3|$, $|c_1 - d_1|$, $|c_1 - d_2|$, $|c_1 - d_3|$, $|c_2 - d_1|$, $|c_2 - d_2|$, $|c_2 - d_3|$, $|a_1 - c_1|$, and $|a_2 - c_2|$ are all distinct?
1991 Canada National Olympiad
The sides of an equilateral triangle $ABC$ are divided into $n$ equal parts $(n \geq 2) .$ For each point on a side, we draw the lines parallel to other sides of the triangle $ABC,$ e.g. for $n=3$ we have the following diagram:

[asy]

unitsize(150);

defaultpen(linewidth(0.7));

int n = 3; /* # of vertical lines, including AB */

pair A = (0,0), B = dir(-30), C = dir(30);

draw(A--B--C--cycle,linewidth(2)); dot(A,UnFill(0)); dot(B,UnFill(0)); dot(C,UnFill(0));

label("$A$",A,W); label("$C$",C,NE); label("$B$",B,SE);

for(int i = 1; i < n; ++i) {

draw((i*A+(n-i)*B)/n--(i*A+(n-i)*C)/n);

draw((i*B+(n-i)*A)/n--(i*B+(n-i)*C)/n);

draw((i*C+(n-i)*A)/n--(i*C+(n-i)*B)/n);

}

[/asy]

For each $n \geq 2,$ find the number of existing parallelograms.
1990 Canada National Olympiad
A competition involving $n\ge 2$ players was held over $k$ days. In each day, the players received scores of $1,2,3,\ldots , n$ points with no players receiving the same score. At the end of the $k$ days, it was found that each player had exactly $26$ points in total. Determine all pairs $(n,k)$ for which this is possible.
1990 Canada National Olympiad
$\frac{n(n + 1)}{2}$ distinct numbers are arranged at random into $n$ rows. The first row has $1$ number, the second has $2$ numbers, the third has $3$ numbers and so on. Find the probability that the largest number in each row is smaller than the largest number in each row with more numbers.
1990 Canada National Olympiad
The feet of the perpendiculars from the intersection point of the diagonals of a convex cyclic quadrilateral to the sides form a quadrilateral $q$. Show that the sum of the lengths of each pair of opposite sides of $q$ is equal.
1990 Canada National Olympiad
A particle can travel at speeds up to $ \frac{2m}{s}$ along the $ x$-axis, and up to $ \frac{1m}{s}$ elsewhere in the plane. Provide a labelled sketch of the region which can be reached within one second by the particle starting at the origin.
1990 Canada National Olympiad
The function $f : \mathbb N \to \mathbb R$ satisfies $f(1) = 1, f(2) = 2$ and $$f (n+2) = f(n+2 - f(n+1) ) + f(n+1 - f(n) ).$$ Show that $0 \leq f(n+1) - f(n) \leq 1$. Find all $n$ for which $f(n) = 1025$.
1989 Canada National Olympiad
The integers $ 1,2,...,n$ are placed in order so that each value is either strictly bigger than all the preceding values or is strictly smaller than all preceding values. In how many ways can this be done?
1989 Canada National Olympiad
Let $ ABC$ be a right angled triangle of area 1. Let $ A'B'C'$ be the points obtained by reflecting $ A,B,C$ respectively, in their opposite sides. Find the area of $ \triangle A'B'C'.$
1989 Canada National Olympiad
Define $ \{ a_n \}_{n=1}$ as follows: $ a_1 = 1989^{1989}; \ a_n, n > 1,$ is the sum of the digits of $ a_{n-1}$. What is the value of $ a_5$?
1989 Canada National Olympiad
There are 5 monkeys and 5 ladders and at the top of each ladder there is a banana. A number of ropes connect the ladders, each rope connects two ladders. No two ropes are attached to the same rung of the same ladder. Each monkey starts at the bottom of a different ladder. The monkeys climb up the ladders but each time they encounter a rope they climb along it to the other ladder at the end of the rope and then continue climbing upwards. Show that, no matter how many ropes there are, each monkey gets a banana.
1989 Canada National Olympiad
Given the numbers $ 1,2,2^2, \ldots ,2^{n-1}$, for a specific permutation $ \sigma = x_1,x_2, \ldots, x_n$ of these numbers we define $ S_1(\sigma) = x_1$, $ S_2(\sigma)=x_1+x_2$, $ \ldots$ and $ Q(\sigma)=S_1(\sigma)S_2(\sigma)\cdot \cdot \cdot S_n(\sigma)$. Evaluate $ \sum 1/Q(\sigma)$, where the sum is taken over all possible permutations.
1988 Canada National Olympiad
For what real values of $k$ do $1988x^2 + kx + 8891$ and $8891x^2 + kx + 1988$ have a common zero?
1988 Canada National Olympiad
A house is in the shape of a triangle, perimeter $P$ metres and area $A$ square metres.  The garden consists of all the land within 5 metres of the house.  How much land do the garden and house together occupy?
1988 Canada National Olympiad
Suppose that $S$ is a finite set of at least five points in the plane; some are coloured red, the others are coloured blue.  No subset of three or more similarly coloured points is collinear.  Show that there is a triangle

(i) whose vertices are all the same colour, and

(ii) at least one side of the triangle does not contain a point of the opposite colour.
1988 Canada National Olympiad
Let $x_{n + 1} = 4x_n - x_{n - 1}$, $x_0 = 0$, $x_1 = 1$, and $y_{n + 1} = 4y_n - y_{n - 1}$, $y_0 = 1$, $y_1 = 2$.  Show that for all $n \ge 0$ that $y_n^2 = 3x_n^2 + 1$.
1988 Canada National Olympiad
If $S$ is a sequence of positive integers let $p(S)$ be the product of the members of $S$. Let $m(S)$ be the arithmetic mean of $p(T)$ for all non-empty subsets $T$ of $S$. Suppose that $S'$ is formed from $S$ by appending an additional positive integer. If $m(S) = 13$ and $m(S') = 49$, find $S'$.
1987 Canada National Olympiad
Find all solutions of $a^2 + b^2 = n!$ for positive integers $a$, $b$, $n$ with $a \le b$ and  $n < 14$.
1987 Canada National Olympiad
The number 1987 can be written as a three digit number $xyz$ in some base $b$.  If $x + y + z = 1 + 9 + 8 + 7$, determine all possible values of $x$, $y$, $z$, $b$.
1987 Canada National Olympiad
Suppose $ABCD$ is a parallelogram and $E$ is a point between $B$ and $C$ on the line $BC$.  If the triangles $DEC$, $BED$ and $BAD$ are isosceles what are the possible values for the angle $DAB$?
1987 Canada National Olympiad
On a large, flat field $n$ people are positioned so that for each person the distances to all the other people are different.  Each person holds a water pistol and at a given signal fires and hits the person who is closest.  When $n$ is odd show that there is at least one person left dry.  Is this always true when $n$ is even?
1987 Canada National Olympiad
For every positive integer $n$ show that

$$[\sqrt{4n + 1}] = [\sqrt{4n + 2}] = [\sqrt{4n + 3}] = [\sqrt{n} + \sqrt{n + 1}]$$

where $[x]$ is the greatest integer less than or equal to $x$ (for example $[2.3] = 2$, $[\pi] = 3$, $[5] = 5$).
1986 Canada National Olympiad
In the diagram line segments $AB$ and $CD$ are of length 1 while angles $ABC$ and $CBD$ are $90^\circ$ and $30^\circ$ respectively.  Find $AC$.



[asy]

import geometry;

import graph;



unitsize(1.5 cm);



pair A, B, C, D;



B = (0,0);

D = (3,0);

A = 2*dir(120);

C = extension(B,dir(30),A,D);



draw(A--B--D--cycle);

draw(B--C);

draw(arc(B,0.5,0,30));



label("$A$", A, NW);

label("$B$", B, SW);

label("$C$", C, NE);

label("$D$", D, SE);

label("$30^\circ$", (0.8,0.2));

label("$90^\circ$", (0.1,0.5));



perpendicular(B,NE,C-B);

[/asy]
1986 Canada National Olympiad
A Mathlon is a competition in which there are $M$ athletic events.  Such a competition was held in which only $A$, $B$, and $C$ participated.  In each event $p_1$ points were awarded for first place, $p_2$ for second and $p_3$ for third, where $p_1 > p_2 > p_3 > 0$ and $p_1$, $p_2$, $p_3$ are integers.  The final scores for $A$ was 22, for $B$ was 9 and for $C$ was also 9.  $B$ won the 100 metres.  What is the value of $M$ and who was second in the high jump?
1986 Canada National Olympiad
A chord $ST$ of constant length slides around a semicircle with diameter $AB$. $M$ is the midpoint of $ST$ and $P$ is the foot of the perpendicular from $S$ to $AB$. Prove that $\angle SPM$ is constant for all positions of $ST$.
1986 Canada National Olympiad
For all positive integers $n$ and $k$, define $F(n,k) = \sum_{r = 1}^n r^{2k - 1}$.  Prove that $F(n,1)$ divides $F(n,k)$.
1986 Canada National Olympiad
Let $u_1$, $u_2$, $u_3$, $\dots$ be a sequence of integers satisfying the recurrence relation $u_{n + 2} = u_{n + 1}^2 - u_n$.  Suppose $u_1 = 39$ and $u_2 = 45$.  Prove that 1986 divides infinitely many terms of the sequence.
1985 Canada National Olympiad
The lengths of the sides of a triangle are 6, 8 and 10 units.  Prove that there is exactly one straight line which simultaneously bisects the area and perimeter of the triangle.
1985 Canada National Olympiad
Prove or disprove that there exists an integer which is doubled when the initial digit is transferred to the end.
1985 Canada National Olympiad
Let $P_1$ and $P_2$ be regular polygons of 1985 sides and perimeters $x$ and $y$ respectively.  Each side of $P_1$ is tangent to a given circle of circumference $c$ and this circle passes through each vertex of $P_2$.  Prove $x + y \ge 2c$.  (You may assume that $\tan \theta \ge \theta$ for $0 \le \theta < \frac{\pi}{2}$.)
1985 Canada National Olympiad
Prove that $2^{n - 1}$ divides $n!$ if and only if $n = 2^{k - 1}$ for some positive integer $k$.
1985 Canada National Olympiad
Let $1 < x_1 < 2$ and, for $n = 1$, 2, $\dots$, define $x_{n + 1} = 1 + x_n - \frac{1}{2} x_n^2$.  Prove that, for $n \ge 3$, $|x_n - \sqrt{2}| < 2^{-n}$.
1984 Canada National Olympiad
Prove that the sum of the squares of $1984$ consecutive positive integers cannot be the square of an integer.
1984 Canada National Olympiad
Alice and Bob are in a hardware store. The store sells coloured sleeves that fit over keys to distinguish them. The following conversation takes place:

Alice: Are you going to cover your keys?

Bob: I would like to, but there are only $7$ colours and I have $8$ keys.

Alice: Yes, but you could always distinguish a key by noticing that the red key next to the green key was different from the red key next to the blue key.

Bob: You must be careful what you mean by "next to" or "three keys over from" since you can turn the key ring over and the keys are arranged in a circle.

Alice: Even so, you don't need $8$ colours.

Problem: What is the smallest number of colours needed to distinguish $n$ keys if all the keys are to be covered.
1984 Canada National Olympiad
An integer is digitally divisible if both of the following conditions are fulfilled:

$(a)$ None of its digits is zero;

$(b)$ It is divisible by the sum of its digits

e.g. $322$ is digitally divisible. Show that there are infinitely many digitally divisible integers.
1984 Canada National Olympiad
An acute triangle has unit area. Show that there is a point inside the triangle whose distance from each of the vertices is at least $\frac{2}{\sqrt[4]{27}}$.
1984 Canada National Olympiad
Given any $7$ real numbers, prove that there are two of them $x,y$ such that $0\le\frac{x-y}{1+xy}\le\frac{1}{\sqrt{3}}$.
1983 Canada National Olympiad
Find all positive integers $w$, $x$, $y$ and $z$ which satisfy $w! = x! + y! + z!$.
1983 Canada National Olympiad
For each $r\in\mathbb{R}$ let $T_r$ be the transformation of the plane that takes the point $(x, y)$ into the point $(2^r x; r2^r x+2^r y)$. Let $F$ be the family of all such transformations (i.e. $F = \{T_r : r\in\mathbb{R}\}$). Find all curves $y = f(x)$ whose graphs remain unchanged by every transformation in $F$.
1983 Canada National Olympiad
The area of a triangle is determined by the lengths of its sides. Is the volume of a tetrahedron determined by the areas of its faces?
1983 Canada National Olympiad
Prove that for every prime number $p$, there are infinitely many positive integers $n$ such that $p$ divides $2^n - n$.
1983 Canada National Olympiad
The geometric mean (G.M.) of $k$ positive integers $a_1$, $a_2$, $\dots$, $a_k$ is defined to be the (positive) $k$-th root of their product.  For example, the G.M. of 3, 4, 18 is 6.  Show that the G.M. of a set $S$ of $n$ positive numbers is equal to the G.M. of the G.M.'s of all non-empty subsets of $S$.
1982 Canada National Olympiad
In the diagram, $OB_i$ is parallel and equal in length to $A_i A_{i + 1}$ for $i = 1$, 2, 3, and 4 ($A_5 = A_1$).  Show that the area of $B_1 B_2 B_3 B_4$ is twice that of $A_1 A_2 A_3 A_4$.



[asy]

unitsize(1 cm);



pair O;

pair[] A, B;



O = (0,0);

A[1] = (0.5,-3);

A[2] = (2,0);

A[3] = (-0.2,0.5);

A[4] = (-1,0);

B[1] = A[2] - A[1];

B[2] = A[3] - A[2];

B[3] = A[4] - A[3];

B[4] = A[1] - A[4];



draw(A[1]--A[2]--A[3]--A[4]--cycle);

draw(B[1]--B[2]--B[3]--B[4]--cycle);

draw(O--B[1]);

draw(O--B[2]);

draw(O--B[3]);

draw(O--B[4]);



label("$A_1$", A[1], S);

label("$A_2$", A[2], E);

label("$A_3$", A[3], N);

label("$A_4$", A[4], W);

label("$B_1$", B[1], NE);

label("$B_2$", B[2], W);

label("$B_3$", B[3], SW);

label("$B_4$", B[4], S);

label("$O$", O, E);

[/asy]
1982 Canada National Olympiad
If $a$, $b$ and $c$ are the roots of the equation $x^3 - x^2 - x - 1 = 0$,

(i) show that $a$, $b$ and $c$ are distinct:

(ii) show that

$$\frac{a^{1982} - b^{1982}}{a - b} + \frac{b^{1982} - c^{1982}}{b - c} + \frac{c^{1982} - a^{1982}}{c - a}$$

is an integer.
1982 Canada National Olympiad
Let $\mathbb{R}^n$ be the $n$-dimensional Euclidean space.  Determine the smallest number $g(n)$ of a points of a set in $\mathbb{R}^n$ such that every point in $\mathbb{R}^n$ is an irrational distance from at least one point in that set.
1982 Canada National Olympiad
Let $p$ be a permutation of the set $S_n = \{1, 2, \dots, n\}$.  An element $j \in S_n$ is called a fixed point of $p$ if $p(j) = j$.  Let $f_n$ be the number of permutations having no fixed points, and $g_n$ be the number with exactly one fixed point.  Show that $|f_n - g_n| = 1$.
1982 Canada National Olympiad
The altitudes of a tetrahedron $ABCD$ are extended externally to points $A'$, $B'$, $C'$, and $D'$, where $AA' = k/h_a$, $BB' = k/h_b$, $CC' = k/h_c$, and $DD' = k/h_d$.  Here, $k$ is a constant and $h_a$ denotes the length of the altitude of $ABCD$ from vertex $A$, etc.  Prove that the centroid of tetrahedron $A'B'C'D'$ coincides with the centroid of $ABCD$.
1981 Canada National Olympiad
For any real number $t$, denote by $[t]$ the greatest integer which is less than or equal to $t$.  For example: $[8] = 8$, $[\pi] = 3$, and $[-5/2] = -3$.  Show that the equation

$$[x] + [2x] + [4x] + [8x] + [16x] + [32x] = 12345$$

has no real solution.
1981 Canada National Olympiad
Given a circle of radius $r$ and a tangent line $\ell$ to the circle through a given point $P$ on the circle.  From a variable point $R$ on the circle, a perpendicular $RQ$ is drawn to $\ell$ with $Q$ on $\ell$.  Determine the maximum of the area of triangle $PQR$.
1981 Canada National Olympiad
Given a finite collection of lines in a plane $P$, show that it is possible to draw an arbitrarily large circle in $P$ which does not meet any of them.  On the other hand, show that it is possible to arrange a countable infinite sequence of lines (first line, second line, third line, etc.) in $P$ so that every circle in $P$ meets at least one of the lines.  (A point is not considered to be a circle.)
1981 Canada National Olympiad
$P(x),Q(x)$ are two polynomials such that $P(x)=Q(x)$ has no real solution, and $P(Q(x))\equiv Q(P(x))\forall x\in\mathbb{R}$. Prove that $P(P(x))=Q(Q(x))$ has no real solution.
1981 Canada National Olympiad
$11$ theatrical groups participated in a festival. Each day, some of the groups were scheduled to perform while the remaining groups joined the general audience. At the conclusion of the festival, each group had seen, during its days off, at least $1$ performance of every other group. At least how many days did the festival last?
1980 Canada National Olympiad
If $a679b$ is the decimal expansion of a number in base $10$, such that it is divisible by  $72$, determine $a,b$.
1980 Canada National Olympiad
The numbers from $1$ to $50$ are printed on cards. The cards are shuffled and then laid out face up in $5$ rows of $10$ cards each. The cards in each row are rearranged to make them increase from left to right. The cards in each column are then rearranged to make them increase from top to bottom. In the final arrangement, do the cards in the rows still increase from left to right?
1980 Canada National Olympiad
Among all triangles having (i) a fixed angle $A$ and (ii) an inscribed circle of fixed radius $r$, determine which triangle has the least minimum perimeter.
1980 Canada National Olympiad
A gambling student tosses a fair coin. She gains $1$ point for each head that turns up, and gains $2$ points for each tail that turns up. Prove that the probability of the student scoring exactly $n$ points is $\frac{1}{3}\cdot\left(2+\left(-\frac{1}{2}\right)^{n}\right)$.
1980 Canada National Olympiad
A parallelepiped has the property that all cross sections, which are parallel to any fixed face $F$, have the same perimeter as $F$. Determine whether or not any other polyhedron has this property.



Typesetter's Note: I believe that proof of existence or non-existence suffices.
1979 Canada National Olympiad
Given: (i) $a$, $b > 0$; (ii) $a$, $A_1$, $A_2$, $b$ is an arithmetic progression; (iii) $a$, $G_1$, $G_2$, $b$ is a geometric progression.  Show that

$$A_1 A_2 \ge G_1 G_2.$$
1979 Canada National Olympiad
It is known in Euclidean geometry that the sum of the angles of a triangle is constant.  Prove, however, that the sum of the dihedral angles of a tetrahedron is not constant.
1979 Canada National Olympiad
Let $a$, $b$, $c$, $d$, $e$ be integers such that $1 \le a < b < c < d < e$.  Prove that

$$\frac{1}{[a,b]} + \frac{1}{[b,c]} + \frac{1}{[c,d]} + \frac{1}{[d,e]} \le \frac{15}{16},$$

where $[m,n]$ denotes the least common multiple of $m$ and $n$ (e.g. $[4,6] = 12$).
1979 Canada National Olympiad
A dog standing at the centre of a circular arena sees a rabbit at the wall.  The rabbit runs round the wall and the dog pursues it along a unique path which is determined by running at the same speed and staying on the radial line joining the centre of the arena to the rabbit.  Show that the dog overtakes the rabbit just as it reaches a point one-quarter of the way around the arena.
1979 Canada National Olympiad
A walk consists of a sequence of steps of length 1 taken in the directions north, south, east, or west.  A walk is self-avoiding if it never passes through the same point twice.  Let $f(n)$ be the number of $n$-step self-avoiding walks which begin at the origin.  Compute $f(1)$, $f(2)$, $f(3)$, $f(4)$, and show that

$$2^n < f(n) \le 4 \cdot 3^{n - 1}.$$
1978 Canada National Olympiad
Let $n$ be an integer.  If the tens digit of $n^2$ is 7, what is the units digit of $n^2$?
1978 Canada National Olympiad
Find all pairs of $a$, $b$ of positive integers satisfying the equation $2a^2 = 3b^3$.
1978 Canada National Olympiad
Determine the largest real number $z$ such that

\begin{align*}

x + y + z = 5 \\
xy + yz + xz = 3

\end{align*}

and $x$, $y$ are also real.