1969 Canada National Olympiad
Problem 1. If a₁/b₁ = a₂/b₂ = a₃/b₃ and p₁, p₂, p₃ are not all zero, show that for all n ∈ ℕ, $$\frac{a_1}{b_1}^n = \frac{p_1a_1^n + p_2a_2^n + p_3a_3^n}{p_1b_1^n + p_2b_2^n + p_3b_3^n}$$
Problem 2. Determine which of the two numbers √(c + 1) − √c, √c − √(c − 1) is greater for any c ≥ 1.
Problem 3. Let c be the length of the hypotenuse of a right angle triangle whose two other sides have lengths a and b. Prove that a + b ≤ c√2. When does the equality hold?
Problem 4. Let ABC be an equilateral triangle, and P be an arbitrary point within the triangle. Perpendiculars PD, PE, PF are drawn to the three sides of the triangle. Show that, no matter where P is chosen, $$\frac{PD + PE + PF}{AB + BC + CA} = \frac{1}{2\sqrt{3}}$$
Problem 5. Let ABC be a triangle with sides of length a, b and c. Let the bisector of the angle C cut AB in D. Prove that the length of CD is $$\frac{2ab \cos \frac{C}{2}}{a + b}$$
Problem 6. Find the sum of 1·1! + 2·2! + 3·3! + ··· + (n−1)(n−1)! + n·n!, where n! = n(n−1)(n−2)···2·1.
Problem 7. Show that there are no integers a, b, c for which a² + b² − 8c = 6.
Problem 8. Let f be a function with the following properties:
- f(n) is defined for every positive integer n;
- f(n) is an integer;
- f(2) = 2;
- f(mn) = f(m)f(n) for all m and n;
- f(m) > f(n) whenever m > n.
Prove that f(n) = n.
Problem 9. Show that for any quadrilateral inscribed in a circle of radius 1, the length of the shortest side is less than or equal to √2.
Problem 10. Let ABC be the right-angled isosceles triangle whose equal sides have length 1. P is a point on the hypotenuse, and the feet of the perpendiculars from P to the other sides are Q and R. Consider the areas of the triangles APQ and PBR, and the area of the rectangle QCRP. Prove that regardless of how P is chosen, the largest of these three areas is at least 2/9.
1970 Canada National Olympiad
Problem 1. Find all number triples (x, y, z) such that when any of these numbers is added to the product of the other two, the result is 2.
Problem 2. Given a triangle ABC with angle A obtuse and with altitudes of length h and k as shown in the diagram, prove that a + h ≥ b + k. Find under what conditions a + h = b + k.
Problem 3. A set of balls is given. Each ball is coloured red or blue, and there is at least one of each colour. Each ball weighs either 1 pound or 2 pounds, and there is at least one of each weight. Prove that there are two balls having different weights and different colours.
Problem 4. a) Find all positive integers with initial digit 6 such that the integer formed by deleting 6 is 1/25 of the original integer. b) Show that there is no integer such that the deletion of the first digit produces a result that is 1/35 of the original integer.
Problem 5. A quadrilateral has one vertex on each side of a square of side-length 1. Show that the lengths a, b, c and d of the sides of the quadrilateral satisfy the inequalities $$2 \leq a^2 + b^2 + c^2 + d^2 \leq 4$$
Problem 6. Given three non-collinear points A, B, C, construct a circle with centre C such that the tangents from A and B are parallel.
Problem 7. Show that from any five integers, not necessarily distinct, one can always choose three of these integers whose sum is divisible by 3.
Problem 8. Consider all line segments of length 4 with one end-point on the line y = x and the other endpoint on the line y = 2x. Find the equation of the locus of the midpoints of these line segments.
Problem 9. Let f(n) be the sum of the first n terms of the sequence 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, ... a) Give a formula for f(n). b) Prove that f(s + t) − f(s − t) = st where s and t are positive integers and s > t.
Problem 10. Given the polynomial $$f(x) = x^n + a_1x^{n-1} + a_2x^{n-2} + \cdots + a_{n-1}x + a_n$$ with integer coefficients a₁, a₂, ..., aₙ, and given also that there exist four distinct integers a, b, c and d such that f(a) = f(b) = f(c) = f(d) = 5, show that there is no integer k such that f(k) = 8.
1971 Canada National Olympiad
Problem 1. DEB is a chord of a circle such that DE = 3 and EB = 5. Let O be the centre of the circle. Join OE and extend OE to cut the circle at C. Given EC = 1, find the radius of the circle.
Problem 2. Let x and y be positive real numbers such that x + y = 1. Show that $$\left(1 + \frac{1}{x}\right)\left(1 + \frac{1}{y}\right) \geq 9$$
Problem 3. ABCD is a quadrilateral with AD = BC. If ∠ADC is greater than ∠BCD, prove that AC > BD.
Problem 4. Determine all real numbers a such that the two polynomials x² + ax + 1 and x² + x + a have at least one root in common.
Problem 5. Let p(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ··· + a₁x + a₀, where the coefficients aᵢ are integers. If p(0) and p(1) are both odd, show that p(x) has no integral roots.
Problem 6. Show that, for all integers n, n² + 2n + 12 is not a multiple of 121.
Problem 7. Let n be a five digit number (whose first digit is non-zero) and let m be the four digit number formed from n by removing its middle digit. Determine all n such that n/m is an integer.
Problem 8. A regular pentagon is inscribed in a circle of radius r. P is any point inside the pentagon. Perpendiculars are dropped from P to the sides, or the sides produced, of the pentagon. a) Prove that the sum of the lengths of these perpendiculars is constant. b) Express this constant in terms of the radius r.
Problem 9. Two flag poles of height h and k are situated 2a units apart on a level surface. Find the set of all points on the surface which are so situated that the angles of elevation of the tops of the poles are equal.
Problem 10. Suppose that n people each know exactly one piece of information, and all n pieces are different. Every time person A phones person B, A tells B everything that A knows, while B tells A nothing. What is the minimum number of phone calls between pairs of people needed for everyone to know everything? Prove your answer is a minimum.
1973 Canada National Olympiad
Problem 1. (i) Solve the simultaneous inequalities, x < 1/(4x) and x < 0; i.e. find a single inequality equivalent to the two simultaneous inequalities. (ii) What is the greatest integer that satisfies both inequalities 4x + 13 < 0 and x² + 3x > 16. (iii) Give a rational number between 11/24 and 6/13. (iv) Express 100000 as a product of two integers neither of which is an integral multiple of 10. (v) Without the use of logarithm tables evaluate 1/log₂36 + 1/log₃36.
Problem 2. Find all real numbers that satisfy the equation |x + 3| − |x − 1| = x + 1. (Note: |a| = a if a ≥ 0; |a| = −a if a < 0.)
Problem 3. Prove that if p and p + 2 are prime integers greater than 3, then 6 is a factor of p + 1.
Problem 4. The figure shows a (convex) polygon with nine vertices. The six diagonals which have been drawn dissect the polygon into the seven triangles: P₀P₁P₃, P₀P₃P₆, P₀P₆P₇, P₀P₇P₈, P₁P₂P₃, P₃P₄P₆, P₄P₅P₆. In how many ways can these triangles be labeled with the names Δ₁, Δ₂, Δ₃, Δ₄, Δ₅, Δ₆, Δ₇ so that Pᵢ is a vertex of triangle Δᵢ for i = 1, 2, 3, 4, 5, 6, 7? Justify your answer.
Problem 5. For every positive integer n, let h(n) = 1 + 1/2 + 1/3 + ··· + 1/n. For example, h(1) = 1, h(2) = 1 + 1/2, h(3) = 1 + 1/2 + 1/3. Prove that for n = 2, 3, 4, ..., $$n + h(1) + h(2) + h(3) + \cdots + h(n-1) = nh(n)$$
Problem 6. If A and B are fixed points on a given circle not collinear with centre O of the circle, and if XY is a variable diameter, find the locus of P (the intersection of the line through A and X and the line through B and Y).
Problem 7. Observe that 1/1 = 1/2 + 1/2; 1/2 = 1/3 + 1/6; 1/3 = 1/4 + 1/12; 1/4 = 1/5 + 1/20. State a general law suggested by these examples, and prove it. Prove that for any integer n greater than 1 there exist positive integers i and j such that $$\frac{1}{n} = \frac{1}{i(i+1)} + \frac{1}{(i+1)(i+2)} + \frac{1}{(i+2)(i+3)} + \cdots + \frac{1}{j(j+1)}$$
1974 Canada National Olympiad
Problem 1. i) If x = (1 + 1/n)ⁿ and y = (1 + 1/n)ⁿ⁺¹, show that y/x = x/y. ii) Show that, for all positive integers n, $$1^2 - 2^2 + 3^2 - 4^2 + \cdots + (-1)^n(n-1)^2 + (-1)^{n+1}n^2 = (-1)^{n+1}(1 + 2 + \cdots + n)$$
Problem 2. Let ABCD be a rectangle with BC = 3AB. Show that if P, Q are the points on side BC with BP = PQ = QC, then ∠DBC + ∠DPC = ∠DQC.
Problem 3. Let f(x) = a₀ + a₁x + a₂x² + ··· + aₙxⁿ be a polynomial with coefficients satisfying the conditions: 0 ≤ aᵢ ≤ a₀, i = 1, 2, ..., n. Let b₀, b₁, ..., b₂ₙ be the coefficients of the polynomial $$(f(x))^2 = b_0 + b_1x + b_2x^2 + \cdots + b_{2n}x^{2n}$$ Prove that bₙ₊₁ ≤ (1/2)(f(1))².
Problem 4. Let n be a fixed positive integer. To any choice of real numbers satisfying 0 ≤ xᵢ ≤ 1, i = 1, 2, ..., n, there corresponds the sum $$\sum_{1 \leq i < j \leq n} |x_i - x_j|$$ Let S(n) denote the largest possible value of this sum. Find S(n).
Problem 5. Given a circle with diameter AB and a point X on the circle different from A and B, let tₐ, tᵦ and tₓ be the tangents to the circle at A, B and X respectively. Let Z be the point where line AX meets tᵦ and Y the point where line BX meets tₐ. Show that the three lines YZ, tₓ and AB are either concurrent (i.e., all pass through the same point) or parallel.
Problem 6. An unlimited supply of 8-cent and 15-cent stamps is available. Some amounts of postage cannot be made up exactly, e.g., 7 cents, 29 cents. What is the largest unattainable amount, i.e., the amount, say n, of postage which is unattainable while all amounts larger than n are attainable? (Justify your answer.)
Problem 7. A bus route consists of a circular road of circumference 10 miles and a straight road of length 1 mile which runs from a terminus to the point Q on the circular road. It is served by two buses, each of which requires 20 minutes for the round trip. Bus No. 1, upon leaving the terminus, travels along the straight road, once around the circle clockwise and returns along the straight road to the terminus. Bus No. 2, reaching the terminus 10 minutes after Bus No. 1, has a similar route except that it proceeds counterclockwise around the circle. Both buses run continuously and do not wait at any point on the route except for a negligible amount of time to pick up and discharge passengers. A man plans to wait at a point P which is x miles (0 ≤ x < 12) from the terminus along the route of Bus No. 1 and travel to the terminus on one of the buses.
Assuming that he chooses to board that bus which will bring him to his destination at the earliest moment, there is a maximum time w(x) that his journey (waiting plus travel time) could take.
Find w(2); find w(4). For what value of x will the time w(x) be the longest? Sketch a graph of y = w(x) for 0 ≤ x < 12.
1975 Canada National Olympiad
Problem 1. Simplify $$\left(\frac{1 \cdot 2 \cdot 4 + 2 \cdot 4 \cdot 8 + \cdots + n \cdot 2n \cdot 4n}{1 \cdot 3 \cdot 9 + 2 \cdot 6 \cdot 18 + \cdots + n \cdot 3n \cdot 9n}\right)^{1/3}$$
Problem 2. A sequence of numbers a₁, a₂, a₃, ... satisfies (i) a₁ = 1/2 (ii) a₁ + a₂ + ··· + aₙ = n²aₙ (n ≥ 1)
Determine the value of aₙ (n ≥ 1).
Problem 3. For each real number r, [r] denotes the largest integer less than or equal to r, e.g. [6] = 6, [π] = 3, [−1.5] = −2. Indicate on the (x, y)-plane the set of all points (x, y) for which [x]² + [y]² = 4.
Problem 4. For a positive number such as 3.27, 3 is referred to as the integral part of the number and .27 as the decimal part. Find a positive number such that its decimal part, its integral part, and the number itself form a geometric progression.
Problem 5. A, B, C, D are four "consecutive" points on the circumference of a circle and P, Q, R, S are points on the circumference which are respectively the midpoints of the arcs AB, BC, CD, DA. Prove that PR is perpendicular to QS.
Problem 6. (i) 15 chairs are equally placed around a circular table on which are name cards for 15 guests. The guests fail to notice these cards until after they have sat down, and it turns out that no one is sitting in the correct seat. Prove that the table can be rotated so that at least two of the guests are simultaneously correctly seated. (ii) Give an example of an arrangement in which just one of the 15 guests is correctly seated and for which no rotation correctly places more than one person.
Problem 7. A function f(x) is periodic if there is a positive number p such that f(x+p) = f(x) for all x. For example, sin x is periodic with period 2π. Is the function sin(x²) periodic? Prove your assertion.
Problem 8. Let k be a positive integer. Find all polynomials P(x) = a₀ + a₁x + ··· + aₙxⁿ, where the aᵢ are real, which satisfy the equation P(P(x)) = {P(x)}ᵏ.
1976 Canada National Olympiad
Problem 1. Given four weights in geometric progression and an equal arm balance, show how to find the heaviest weight using the balance only twice.
Problem 2. Suppose n(n + 1)aₙ₊₁ = n(n − 1)aₙ − (n − 2)aₙ₋₁ for every positive integer n ≥ 1. Given that a₀ = 1, a₁ = 2, find $$\frac{a_0}{a_1} + \frac{a_1}{a_2} + \frac{a_2}{a_3} + \cdots + \frac{a_{50}}{a_{51}}$$
Problem 3. Two grade seven students were allowed to enter a chess tournament otherwise composed of grade eight students. Each contestant played once with each other contestant and received one point for a win, one half point for a tie and zero for a loss. The two grade seven students together gained a total of eight points and each grade eight student scored the same number of points as his classmates. How many students for grade eight participated in the chess tournament? Is the solution unique?
Problem 4. Let AB be a diameter of a circle, C be any fixed point between A and B on this diameter, and Q be a variable point on the circumference of the circle. Let P be the point on the line determined by Q and C for which AC/CB = QC/CP. Describe, with proof, the locus of the point P.
Problem 5. Prove that a positive integer is a sum of at least two consecutive positive integers if and only if it is not a power of two.
Problem 6. If A, B, C, D are four points in space, such that ∠ABC = ∠BCD = ∠CDA = ∠DAB = π/2, prove that A, B, C, D lie in a plane.
Problem 7. Let P(x, y) be a polynomial in two variables x, y such that P(x, y) = P(y, x) for every x, y (for example, the polynomial x² − 2xy + y² satisfies this condition). Given that (x − y) is a factor of P(x, y), show that (x − y)² is a factor of P(x, y).
Problem 8. Each of the 36 line segments joining 9 distinct points on a circle is coloured either red or blue. Suppose that each triangle determined by 3 of the 9 points contains at least one red side. Prove that there are four points such that the 6 segments connecting them are all red.
1977 Canada National Olympiad
Problem 1. If f(x) = x² + x, prove that the equation 4f(a) = f(b) has no solutions in positive integers a and b.
Problem 2. Let O be the centre of a circle and A a fixed interior point of the circle different from O. Determine all points P on the circumference of the circle such that the angle ∠OPA is a maximum.
Problem 3. N is an integer whose representation in base b is 777. Find the smallest integer b for which N is the fourth power of an integer.
Problem 4. Let p(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ··· + a₁x + a₀ and q(x) = bₘxᵐ + aₘ₋₁xᵐ⁻¹ + ··· + b₁x + b₀ be two polynomials with integer coefficients. Suppose that all the coefficients of the product p(x)·q(x) are even but not all of them are divisible by 4. Show that one of p(x) and q(x) has all even coefficients and the other has at least one odd coefficient.
Problem 5. A right circular cone has base radius 1 cm and slant height 3 cm is given. P is a point on the circumference of the base and the shortest path from P around the cone and back to P is drawn. What is the minimum distance from the vertex V to this path?
Problem 6. Let 0 < u < 1 and define u₁ = 1 + u, u₂ = 1/u₁ + u, ..., uₙ₊₁ = 1/uₙ + u, n ≥ 1. Show that uₙ > 1 for all values of n = 1, 2, 3, ...
Problem 7. A rectangular city is exactly m blocks long and n blocks wide. A woman lives in the southwest corner of the city and works in the northeast corner. She walks to work each day but, on any given trip, she makes sure that her path does not include any intersection twice. Show that the number f(m, n) of different paths she can take to work satisfies f(m, n) ≤ 2^(mn).
1978 Canada National Olympiad
Problem 1. Let n be an integer. If the tens digit of n² is 7, what is the units digit of n²?
Problem 2. Find all pairs of a, b of positive integers satisfying the equation 2a² = 3b³.
Problem 3. Determine the largest real number z such that x + y + z = 5, xy + yz + xz = 3, and x, y are also real.
Problem 4. 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.
Problem 5. Eve and Odette play a game on a 3 × 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 very 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.
Problem 6. Sketch the graph of x³ + xy + y³ = 3.
1979 Canada National Olympiad
Problem 1. Given: (i) a, b > 0; (ii) a, A₁, A₂, b is an arithmetic progression; (iii) a, G₁, G₂, b is a geometric progression. Show that A₁A₂ ≥ G₁G₂.
Problem 2. 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.
Problem 3. Let a, b, c, d, e be integers such that 1 ≤ a < b < c < d < e. Prove that $$\frac{1}{[a,b]} + \frac{1}{[b,c]} + \frac{1}{[c,d]} + \frac{1}{[d,e]} \leq \frac{15}{16}$$ where [m, n] denotes the least common multiple of m and n (e.g. [4, 6] = 12).
Problem 4. 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.
Problem 5. 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 2ⁿ < f(n) ≤ 4·3ⁿ⁻¹.
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