1990 Canada National Olympiad
Problem 1. A competition involving n ≥ 2 players was held over k days. In each day, the players received scores of 1, 2, 3, ..., 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.
Problem 2. 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.
Problem 3. 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.
Problem 4. A particle can travel at speeds up to 2 m/s along the x-axis, and up to 1 m/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.
Problem 5. The function f : ℕ → 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 ≤ f(n + 1) − f(n) ≤ 1. Find all n for which f(n) = 1025.
1991 Canada National Olympiad
Problem 1. Show that the equation x² + y⁵ = z³ has infinitely many solutions in integers x, y, z for which xyz ≠ 0.
Problem 2. 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.)
Problem 3. 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.
Problem 4. Can ten distinct numbers a₁, a₂, b₁, b₂, b₃, c₁, c₂, d₁, d₂, d₃ be chosen from {0, 1, 2, ..., 14}, so that the 14 differences |a₁ − b₁|, |a₁ − b₂|, |a₁ − b₃|, |a₂ − b₁|, |a₂ − b₂|, |a₂ − b₃|, |c₁ − d₁|, |c₁ − d₂|, |c₁ − d₃|, |c₂ − d₁|, |c₂ − d₂|, |c₂ − d₃|, |a₁ − c₁|, and |a₂ − c₂| are all distinct?
Problem 5. The sides of an equilateral triangle ABC are divided into n equal parts (n ≥ 2). For each point on a side, we draw the lines parallel to other sides of the triangle ABC. For each n ≥ 2, find the number of existing parallelograms.
1992 Canada National Olympiad
Problem 1. 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.
Problem 2. For x, y, z ≥ 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.
Problem 3. 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.
Problem 4. Solve the equation $$x^2 + \frac{x^2}{(x+1)^2} = 3$$
Problem 5. 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 ≤ k ≤ 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?
1993 Canada National Olympiad
Problem 1. 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.
Problem 2. 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, ...
Problem 3. In triangle ABC, the medians to the sides AB and AC are perpendicular. Prove that $$\cot B + \cot C \geq \frac{2}{3}$$
Problem 4. A number 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?
Problem 5. Let y₁, y₂, y₃, ... be a sequence such that y₁ = 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.
1994 Canada National Olympiad
Problem 1. Evaluate $$\sum_{n=1}^{1994} \left((-1)^n \cdot \frac{n^2+n+1}{n!}\right)$$
Problem 2. Prove that (√2 − 1)ⁿ ∀n ∈ ℤ⁺ can be represented as √m − √(m − 1) for some m ∈ ℤ⁺.
Problem 3. 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-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)-th vote as on the n-th vote; but if his response is different from that of both his neighbours on the n-th vote, then his response on the (n + 1)-th vote will be different from his response on the n-th vote. Prove that, however everybody responded on the first vote, there will be a time after which nobody's response will ever change.
Problem 4. Let AB be a diameter of a circle Ω and P be any point not on the line through AB. Suppose that the line through PA cuts Ω again at U, and the line through PB cuts Ω at V. Note that in case of tangency, U may coincide with A or V might coincide with B. Also, if P is on Ω then P = U = V. Suppose that |PU| = s|PA| and |PV| = t|PB| for some 0 ≤ s, t ∈ ℝ. Determine cos ∠APB in terms of s, t.
Problem 5. 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 ∠EDH = ∠FDH.
1995 Canada National Olympiad
Problem 1. Let f(x) = 9ˣ/(9ˣ+3). Evaluate $$\sum_{i=1}^{1995} f\left(\frac{i}{1996}\right)$$
Problem 2. Let {a, b, c} ∈ ℝ⁺. Prove that aᵃbᵇcᶜ ≥ (abc)^((a+b+c)/3).
Problem 3. Define a boomerang as a quadrilateral whose opposite sides do not intersect and one of whose internal angles is greater than 180°. Let C be a convex polygon with s sides. The interior region of C is the union of q quadrilaterals, none of whose interiors overlap each other. b of these quadrilaterals are boomerangs. Show that q ≥ b + (s−2)/2.
Problem 4. Let n be a constant positive integer. Show that for only non-negative integers k, the Diophantine equation Σᵢ₌₁ⁿ xᵢ³ = y³ᵏ⁺² has infinitely many solutions in the positive integers xᵢ, y.
Problem 5. u is a real parameter such that 0 < u < 1. For 0 ≤ x ≤ u, f(x) = 0. For u ≤ x ≤ 1, f(x) = 1 − √(ux) + √((1 − u)(1 − x))/2. The sequence {uₙ} is define recursively as follows: u₁ = f(1) and uₙ = f(uₙ₋₁) ∀n ∈ ℕ, n ≠ 1. Show that there exists a positive integer k for which uₖ = 0.
1996 Canada National Olympiad
Problem 1. If α, β, and γ are the roots of x³ − x − 1 = 0, compute $$\frac{1+\alpha}{1-\alpha} + \frac{1+\beta}{1-\beta} + \frac{1+\gamma}{1-\gamma}$$
Problem 2. Find all real solutions to the following system of equations. Carefully justify your answer. $$\begin{cases} 4x^2/(1 + 4x^2) = y \ 4y^2/(1 + 4y^2) = z \ 4z^2/(1 + 4z^2) = x \end{cases}$$
Problem 3. We denote an arbitrary permutation of the integers 1, 2, ..., n by a₁, a₂, ..., aₙ. Let f(n) denote the number of these permutations such that: (1) a₁ = 1; (2): |aᵢ − aᵢ₊₁| ≤ 2, i = 1, ..., n − 1. Determine whether f(1996) is divisible by 3.
Problem 4. Let triangle ABC be an isosceles triangle with AB = AC. Suppose that the angle bisector of its angle ∠B meets the side AC at a point D and that BC = BD + AD. Determine ∠A.
Problem 5. Let r₁, r₂, ..., rₘ be a given set of m positive rational numbers such that Σᵏ₌₁ᵐ rₖ = 1. Define the function f by f(n) = n − Σᵏ₌₁ᵐ [rₖn] for each positive integer n. Determine the minimum and maximum values of f(n). Here [x] denotes the greatest integer less than or equal to x.
1997 Canada National Olympiad
Problem 1. Determine the number of pairs of positive integers x, y such that x ≤ y, gcd(x, y) = 5! and lcm(x, y) = 50!.
Problem 2. The closed interval A = [0, 50] is the union of a finite number of closed intervals, each of length 1. Prove that some of the intervals can be removed so that those remaining are mutually disjoint and have total length greater than 25.
Note: For reals a ≤ b, the closed interval [a, b] := {x ∈ ℝ : a ≤ x ≤ b} has length b − a; disjoint intervals have empty intersection.
Problem 3. Prove that $$\frac{1}{1999} < \prod_{i=1}^{999} \frac{2i-1}{2i} < \frac{1}{44}$$
Problem 4. The point O is situated inside the parallelogram ABCD such that ∠AOB + ∠COD = 180°. Prove that ∠OBC = ∠ODC.
Problem 5. Write the sum $$\sum_{i=0}^{n} \frac{(-1)^i \binom{n}{i}}{i^3+9i^2+26i+24}$$ as the ratio of two explicitly defined polynomials with integer coefficients.
1998 Canada National Olympiad
Problem 1. Determine the number of real solutions a to the equation: $$\left[\frac{1}{2}a\right] + \left[\frac{1}{3}a\right] + \left[\frac{1}{5}a\right] = a$$ Here, if x is a real number, then [x] denotes the greatest integer that is less than or equal to x.
Problem 2. Find all real numbers x such that: $$x = \sqrt{x - \frac{1}{x}} + \sqrt{1 - \frac{1}{x}}$$
Problem 3. Let n be a natural number such that n ≥ 2. Show that $$\frac{1}{n+1}\left(1 + \frac{1}{3} + \cdots + \frac{1}{2n-1}\right) > \frac{1}{n}\left(\frac{1}{2} + \frac{1}{4} + \cdots + \frac{1}{2n}\right)$$
Problem 4. Let ABC be a triangle with ∠BAC = 40° and ∠ABC = 60°. Let D and E be the points lying on the sides AC and AB, respectively, such that ∠CBD = 40° and ∠BCE = 70°. Let F be the point of intersection of the lines BD and CE. Show that the line AF is perpendicular to the line BC.
Problem 5. Let m be a positive integer. Define the sequence a₀, a₁, a₂, ... by a₀ = 0, a₁ = m, and aₙ₊₁ = m²aₙ − aₙ₋₁ for n = 1, 2, 3, ... Prove that an ordered pair (a, b) of non-negative integers, with a ≤ b, gives a solution to the equation $$(a^2 + b^2)/(ab + 1) = m^2$$ if and only if (a, b) is of the form (aₙ, aₙ₊₁) for some n ≥ 0.
1999 Canada National Olympiad
Problem 1. Find all real solutions to the equation 4x² − 40⌊x⌋ + 51 = 0.
Problem 2. Let ABC be an equilateral triangle of altitude 1. A circle with radius 1 and center on the same side of AB as C rolls along the segment AB. Prove that the arc of the circle that is inside the triangle always has the same length.
Problem 3. Determine all positive integers n with the property that n = (d(n))². Here d(n) denotes the number of positive divisors of n.
Problem 4. Suppose a₁, a₂, ..., a₈ are eight distinct integers from {1, 2, ..., 16, 17}. Show that there is an integer k > 0 such that the equation aᵢ − aⱼ = k has at least three different solutions. Also, find a specific set of 7 distinct integers from {1, 2, ..., 16, 17} such that the equation aᵢ − aⱼ = k does not have three distinct solutions for any k > 0.
Problem 5. Let x, y, and z be non-negative real numbers satisfying x + y + z = 1. Show that $$x^2y + y^2z + z^2x \leq \frac{4}{27}$$ and find when equality occurs.
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