🇨🇦 Canadian Mathematical Olympiad 1980–1989 — All Problems

🇨🇦 Canadian Mathematical Olympiad 1980–1989

The 1980s brought increased creativity and rising difficulty to the Canadian Mathematical Olympiad. The problems from this decade highlight strategic thinking, inventive techniques and a deeper exploration of algebra, geometry, number theory and combinatorics.

This collection brings together the problems from the ten competitions held from 1980 to 1989, without solutions.

1980 Canada National Olympiad

Problem 1. If \(a679b\) is the decimal expansion of a number in base 10, such that it is divisible by 72, determine \(a,b\).

Problem 2. 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?

Problem 3. Among all triangles having (i) a fixed angle \(A\) and (ii) an inscribed circle of fixed radius \(r\), determine which triangle has the minimum perimeter.

Problem 4. 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

\[ \frac13\left(2+\left(-\frac12\right)^n\right). \]

Problem 5. 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.

1981 Canada National Olympiad

Problem 1. For any real number \(t\), denote by \([t]\) the greatest integer which is less than or equal to \(t\). Show that the equation

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

has no real solution.

Problem 2. 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 area of triangle \(PQR\).

Problem 3. 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 in \(P\) so that every circle in \(P\) meets at least one of the lines.

Problem 4. \(P(x)\), \(Q(x)\) are two polynomials such that \(P(x)=Q(x)\) has no real solution, and

\[ P(Q(x))\equiv Q(P(x)),\qquad x\in\mathbb R. \]

Prove that \(P(P(x))=Q(Q(x))\) has no real solution.

Problem 5. 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 one performance of every other group. At least how many days did the festival last?

1982 Canada National Olympiad

Problem 1. In the diagram, \(OB_i\) is parallel and equal in length to \(A_iA_{i+1}\), for \(i=1,2,3,4\), where \(A_5=A_1\). Show that the area of \(B_1B_2B_3B_4\) is twice that of \(A_1A_2A_3A_4\).

📐 Figure required: Insert the original diagram for Problem 1 here.

Problem 2. If \(a,b,c\) are the roots of

\[ x^3-x^2-x-1=0, \]

(i) show that \(a,b,c\) 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.

Problem 3. Let \(\mathbb R^n\) be the \(n\)-dimensional Euclidean space. Determine the smallest number \(g(n)\) of 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.

Problem 4. Let \(p\) be a permutation of \(S_n=\{1,2,\ldots,n\}\). An element \(j\in S_n\) is called a fixed point if \(p(j)=j\). Let \(f_n\) be the number of permutations having no fixed points and \(g_n\) the number having exactly one fixed point. Show that

\[ |f_n-g_n|=1. \]

Problem 5. The altitudes of a tetrahedron \(ABCD\) are extended externally to points \(A',B',C',D'\), where

\[ AA'=\frac{k}{h_a},\qquad BB'=\frac{k}{h_b},\qquad CC'=\frac{k}{h_c},\qquad DD'=\frac{k}{h_d}. \]

Here \(k\) is a constant and \(h_a\) denotes the length of the altitude from vertex \(A\), etc. Prove that the centroid of tetrahedron \(A'B'C'D'\) coincides with the centroid of \(ABCD\).

1983 Canada National Olympiad

Problem 1. Find all positive integers \(w,x,y,z\) which satisfy

\[ w!=x!+y!+z!. \]

Problem 2. For each \(r\in\mathbb R\), let \(T_r\) be the transformation of the plane defined in the original CMO problem. Let \(F=\{T_r:r\in\mathbb R\}\). Find all curves \(y=f(x)\) whose graphs remain unchanged by every transformation in \(F\).

Problem 3. 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?

Problem 4. Prove that for every prime number \(p\), there are infinitely many positive integers \(n\) such that

\[ p\mid(2^n-n). \]

Problem 5. The geometric mean of \(k\) positive integers \(a_1,a_2,\ldots,a_k\) is defined to be the positive \(k\)-th root of their product. Show that the geometric mean of a set \(S\) of \(n\) positive numbers is equal to the geometric mean of the geometric means of all non-empty subsets of \(S\).

1984 Canada National Olympiad

Problem 1. Prove that the sum of the squares of 1984 consecutive positive integers cannot be the square of an integer.

Problem 2. Alice and Bob are in a hardware store. The store sells coloured sleeves that fit over keys to distinguish them. Determine the smallest number of colours needed to distinguish \(n\) keys if all the keys are to be covered, taking into account that the keys lie on a key ring and that the ring may be turned over.

Problem 3. An integer is digitally divisible if:

(a) none of its digits is zero;
(b) it is divisible by the sum of its digits.

Show that there are infinitely many digitally divisible integers.

Problem 4. 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

\[ \frac{2}{\sqrt[4]{27}}. \]

Problem 5. Given any seven real numbers, prove that there are two of them \(x,y\) such that

\[ 0\leq\frac{x-y}{1+xy}\leq\frac1{\sqrt3}. \]

1985 Canada National Olympiad

Problem 1. 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.

Problem 2. Prove or disprove that there exists an integer which is doubled when the initial digit is transferred to the end.

Problem 3. 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 every vertex of \(P_2\). Prove that

\[ x+y\geq2c. \]

Problem 4. Prove that \(2^{n-1}\) divides \(n!\) if and only if

\[ n=2^{k-1} \]

for some positive integer \(k\).

Problem 5. Let \(1<x_1<2\) and define

\[ x_{n+1}=1+x_n-\frac12x_n^2. \]

Prove that, for \(n\geq3\),

\[ |x_n-\sqrt2|<2^{-n}. \]

1986 Canada National Olympiad

Problem 1. In the diagram, line segments \(AB\) and \(CD\) have length 1, while

\[ \angle ABC=90^\circ,\qquad \angle CBD=30^\circ. \]

Find \(AC\).

📐 Figure required: Insert the original diagram for Problem 1 here.

Problem 2. A Mathlon is a competition consisting of \(M\) athletic events. Only \(A,B,C\) participated. In each event \(p_1,p_2,p_3\) points were awarded for first, second and third place, where

\[ p_1>p_2>p_3>0. \]

The final scores were 22 for \(A\), 9 for \(B\), and 9 for \(C\). \(B\) won the 100 metres. Determine \(M\) and who was second in the high jump.

Problem 3. 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.

Problem 4. For positive integers \(n,k\), define

\[ F(n,k)=\sum_{r=1}^{n}r^{\,2k-1}. \]

Prove that \(F(n,1)\) divides \(F(n,k)\).

Problem 5. Let \(u_1,u_2,\ldots\) be a sequence of integers satisfying

\[ u_{n+2}=u_{n+1}^{\,2}-u_n, \]

where \(u_1=39\) and \(u_2=45\). Prove that 1986 divides infinitely many terms of the sequence.

1987 Canada National Olympiad

Problem 1. Find all solutions of

\[ a^2+b^2=n! \]

for positive integers \(a,b,n\), with \(a\leq b\) and \(n<14\).

Problem 2. 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 \(x,y,z,b\).

Problem 3. Suppose \(ABCD\) is a parallelogram and \(E\) lies between \(B\) and \(C\). If triangles \(DEC\), \(BED\) and \(BAD\) are isosceles, what are the possible values of \(\angle DAB\)?

Problem 4. On a large flat field \(n\) people are positioned so that, for each person, the distances to all the other people are different. At a signal each person fires a water pistol at the closest person. When \(n\) is odd, show that at least one person remains dry. Is this always true when \(n\) is even?

Problem 5. For every positive integer \(n\), show that

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

where \([x]\) denotes the greatest integer less than or equal to \(x\).

1988 Canada National Olympiad

Problem 1. For what real values of \(k\) do

\[ 1988x^2+kx+8891 \]

and

\[ 8891x^2+kx+1988 \]

have a common zero?

Problem 2. A house is in the shape of a triangle, with 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?

Problem 3. Suppose \(S\) is a finite set of at least five points in the plane; some are coloured red and the others blue. No subset of three or more similarly coloured points is collinear. Show that there is a triangle whose vertices are all the same colour and at least one side contains no point of the opposite colour.

Problem 4. Let

\[ x_{n+1}=4x_n-x_{n-1},\qquad x_0=0,\quad x_1=1, \]

and

\[ y_{n+1}=4y_n-y_{n-1},\qquad y_0=1,\quad y_1=2. \]

Show that, for all \(n\geq0\),

\[ y_n^2=3x_n^2+1. \]

Problem 5. If \(S\) is a sequence of positive integers, let \(p(S)\) be the product of its members. Let \(m(S)\) be the arithmetic mean of \(p(T)\) over all non-empty subsets \(T\) of \(S\). Suppose \(S'\) is formed from \(S\) by appending one positive integer. If

\[ m(S)=13,\qquad m(S')=49, \]

find \(S'\).

1989 Canada National Olympiad

Problem 1. The integers \(1,2,\ldots,n\) are placed in order so that each value is either strictly bigger than all preceding values or strictly smaller than all preceding values. In how many ways can this be done?

Problem 2. Let \(ABC\) be a right-angled triangle of area 1. Let \(A'B'C'\) be obtained by reflecting \(A,B,C\), respectively, in their opposite sides. Find the area of \(\triangle A'B'C'\).

Problem 3. Define \(\{a_n\}\) by

\[ a_1=19891989, \]

and, for \(n>1\), let \(a_n\) be the sum of the digits of \(a_{n-1}\). Find \(a_5\).

Problem 4. There are five monkeys and five ladders, and at the top of each ladder there is a banana. Ropes connect pairs of ladders, with no two ropes attached to the same rung of the same ladder. Each monkey starts at the bottom of a different ladder. Whenever a monkey encounters a rope, it crosses to the other ladder and continues upwards. Show that, regardless of the number of ropes, each monkey gets a banana.

Problem 5. Given the numbers

\[ 1,2,2^2,\ldots,2^{n-1}, \]

for a permutation \(\sigma=x_1,x_2,\ldots,x_n\), define

\[ S_1(\sigma)=x_1,\quad S_2(\sigma)=x_1+x_2,\quad\ldots \]

and

\[ Q(\sigma)=S_1(\sigma)S_2(\sigma)\cdots S_n(\sigma). \]

Evaluate

\[ \sum_{\sigma}\frac1{Q(\sigma)}, \]

where the sum is taken over all possible permutations.

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