1974
1 i) If x = 1 + 1 n n and y = 1 +1 n n+1 , show that yx= xy. ii) Show that, for all positive integers n,12− 22+ 32− 42+ · · · + (−1)n(n − 1)2+ (−1)n+1n2= (−1)n+1(1 + 2 + · · · + n). 2 Let ABCD be a rectangle with BC = 3AB. Show that if P, Q are the points on side BC
with BP = P Q = QC, then
∠DBC + ∠DP C = ∠DQC. 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 = a0+ a1x + a2x2+ · · · anxn = b0+ b1x + b2x2+ · · · + b2nx2n. Prove that bn+1≤ 1 2(f (1)) 2 .
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
X
1≤i<j≤n
|xi− xj|.
Let S(n) denote the largest possible value of this sum. Find S(n).
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National Olympiad
1974
5 Given a circle with diameter AB and a point X on the circle different from A and B, let ta,
tb and tx be the tangents to the circle at A, B and X respectively. Let Z be the point where
line AX meets tb and Y the point where line BX meets ta. Show that the three lines Y Z, tx
and AB are either concurrent (i.e., all pass through the same point) or parallel. [img]6762[/img]
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.)
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 (see diagram). [img]6763[/img]
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.
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Canada
National Olympiad
1975
1 Simplify 1 · 2 · 4 + 2 · 4 · 8 + · · · + n · 2n · 4n 1 · 3 · 9 + 2 · 6 · 18 + · · · + n · 3n · 9n 13 2 A sequence of numbers a1, a2, a3, ... satisfies(i) a1 =
1
2 (ii) a1+ a2+ · · · + an= n
2a
n (n ≥ 1)
Determine the value of an (n ≥ 1).
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.
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.
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 P R is perpendicular to QS.
6 (i) 15 chairs are equally placed around a circular table on which are name cards for 15 guests. The guest 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.
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.
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
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National Olympiad
1976
1 Given four weights in geometric proression and an equal arm balance, show how to find the heaviest weight using the balance only twice.
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
a0 a1 +a1 a2 + a2 a3 + · +a50 a51 .
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?
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 .
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.
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.
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).
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.
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Canada
National Olympiad
1989
1 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?
2 Let ABC be a right angled triangle of area 1. Let A0B0C0be the points obtained by reflecting A, B, C respectively, in their opposite sides. Find the area of 4A0B0C0.
3 Define {an}n=1 as follows: a1 = 19891989; an, n > 1, is the sum of the digits of an−1. What
is the value of a5?
4 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.
5 Given the numbers 1, 2, 22, . . . , 2n−1, for a specific permutation σ = x1, x2, . . . , xn of these
numbers we define S1(σ) = x1, S2(σ) = x1+x2, . . . and Q(σ) = S1(σ)S2(σ)···Sn(σ). Evaluate
X
1/Q(σ), where the sum is taken over all possible permutations.
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National Olympiad
1992
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.
2 For x, y, z ≥ 0, establish the inequality
x(x − z)2+ y(y − z)2 ≥ (x − z)(y − z)(x + y − z) and determine when equality holds.
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 P U QV .
[img]http://i250.photobucket.com/albums/gg265/geometry101/CMO1992Number3.jpg[/img] 4 Solve the equation
x2+ x
2
(x + 1)2 = 3
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?
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Canada
National Olympiad
1993
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.
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, . . . .
3 In triangle ABC, the medians to the sides AB and AC are perpendicular. Prove that cot B + cot C ≥ 2
3.
4 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?
5 Let y1, y2, y3, . . . be a sequence such that y1= 1 and, for k > 0, is defined by the relationship:
y2k = ( 2yk if k is even 2yk+ 1 if k is odd y2k+1= ( 2yk if k is odd 2yk+ 1 if k is even
Show that the sequence takes on every positive integer value exactly once.
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National Olympiad
1996
1 If α, β, and γ are the roots of x3− x − 1 = 0, compute 1 + α 1 − α+
1 + β 1 − β +
1 + γ 1 − γ.
2 Find all real solutions to the following system of equations. Carefully justify your answer. 4x2 1 + 4x2 = y 4y2 1 + 4y2 = z 4z2 1 + 4z2 = x
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.
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. 5 Let r1, r2, . . ., rm be a given set of m positive rational numbers such that
m X k=1 rk = 1. Define the function f by f (n) = n − m X k=1
[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.
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Canada
National Olympiad
1998
1 Determine the number of real solutions a to the equation: 1 2 a + 1 3 a + 1 5 a = a .
Here, if x is a real number, then [ x ] denotes the greatest integer that is less than or equal to x. Find all real numbers x such that:
x = r x − 1 x + r 1 −1 x Let n be a natural number such that n ≥ 2. Show that
1 n + 1 1 +1 3 + · · · + 1 2n − 1 > 1 n 1 2+ 1 4 + · · · + 1 2n .
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. 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
a2+ b2 ab + 1 = m
2
if and only if (a, b) is of the form (an, an+1) for some n ≥ 0.
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National Olympiad
1999
1 Find all real solutions to the equation 4x2− 40bxc + 51 = 0.
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.
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.
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.
5 Let x, y, and z be non-negative real numbers satisfying x + y + z = 1. Show that x2y + y2z + z2x ≤ 4
27 and find when equality occurs.
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Canada
National Olympiad
2000
1 At 12:00 noon, Anne, Beth and Carmen begin running laps around a circular track of length 300 meters, all starting from the same point on the track. Each jogger maintains a constant speed in one of the two possible directions for an indefinite period of time. Show that if Anne’s speed is different from the other two speeds, then at some later time Anne will be at least 100 meters from each of the other runners. (Here, distance is measured along the shorter of the two arcs separating two runners.)
2 A permutation of the integers 1901, 1902, · · · , 2000 is a sequence a1, a2, · · · , a100in which each
of those integers appears exactly once. Given such a permutation, we form the sequence of partial sums
s1 = a1, s2 = a1+ a2, s3 = a1+ a2+ a3, . . . , s100 = a1+ a2+ · · · + a100.
How many of these permutations will have no terms of the sequence s1, . . . , s100 divisible by three?
Let A = (a1, a2, · · · , a2000) be a sequence of integers each lying in the interval [−1000, 1000]. Sup-
pose that the entries in A sum to 1. Show that some nonempty subsequence of A sums to zero. Let ABCD be a convex quadrilateral with ∠CBD = 2∠ADB, ∠ABD = 2∠CDB and AB = CB.
Prove that AD = CD.
Suppose that the real numbers a1, a2, . . . , a100 satisfy
0 ≤ a100 ≤ a99≤ · · · ≤ a2 ≤ a1,
a1+ a2 ≤ 100
a3+ a4+ · · · + a100 ≤ 100.
Determine the maximum possible value of a21+ a22+ · · · + a2100, and find all possible sequences a1, a2, . . . , a100 which achieve this maximum.
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National Olympiad
2001
1 Randy: ”Hi Rachel, that’s an interesting quadratic equation you have written down. What are its roots?”
Rachel: ”The roots are two positive integers. One of the roots is my age, and the other root is the age of my younger brother, Jimmy.”
Randy: ”That is very neat! Let me see if I can figure out how old you and Jimmy are. That shouldn’t be too difficult since all of your coefficients are integers. By the way, I notice that the sum of the three coefficients is a prime number.”
Rachel: ”Interesting. Now figure out how old I am.”
Randy: ”Instead, I will guess your age and substitute it for x in your quadratic equation . . . darn, that gives me −55, and not 0.”
Rachel: ”Oh, leave me alone!”
(1) Prove that Jimmy is two years old. (2) Determine Rachel’s age.
3 Let ABC be a triangle with AC > AB. Let P be the intersection point of the perpendicular bisector of BC and the internal angle bisector of ∠A. Construct points X on AB (extended) and Y on AC such that P X is perpendicular to AB and P Y is perpendicular to AC. Let Z be the intersection point of XY and BC. Determine the value of BZ
ZC.
4 Let n be a positive integer. Nancy is given a rectangular table in which each entry is a positive integer. She is permitted to make either of the following two moves:
(1) select a row and multiply each entry in this row by n;
(2) select a column and subtract n from each entry in this column. Find all possible values of n for which the following statement is true:
Given any rectangular table, it is possible for Nancy to perform a finite sequence of moves to create a table in which each entry is 0.
5 Let P0, P1, P2 be three points on the circumference of a circle with radius 1, where P1P2 =
t < 2. For each i ≥ 3, define Pi to be the centre of the circumcircle of 4Pi−1Pi−2Pi−3.
(1) Prove that the points P1, P5, P9, P13, · · · are collinear.
(2) Let x be the distance from P1 to P1001, and let y be the distance from P1001 to P2001.
Determine all values of t for which 500r x
y is an integer.
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Canada
National Olympiad
2002
1 Let S be a subset of {1, 2, . . . , 9}, such that the sums formed by adding each unordered pair of distinct numbers from S are all different. For example, the subset {1, 2, 3, 5} has this property, but {1, 2, 3, 4, 5} does not, since the pairs {1, 4} and {2, 3} have the same sum, namely 5. What is the maximum number of elements that S can contain?
2 Call a positive integer n practical if every positive integer less than or equal to n can be written as the sum of distinct divisors of n.
For example, the divisors of 6 are 1, 2, 3, and 6. Since
1 = 1, 2 = 2, 3 = 3, 4 = 1 + 3, 5 = 2 + 3, 6 = 6, we see that 6 is practical.
Prove that the product of two practical numbers is also practical. 3 Prove that for all positive real numbers a, b, and c,
a3 bc + b3 ca + c3 ab ≥ a + b + c and determine when equality occurs.
4 Let Γ be a circle with radius r. Let A and B be distinct points on Γ such that AB < √
3r. Let the circle with centre B and radius AB meet Γ again at C. Let P be the point inside Γ such that triangle ABP is equilateral. Finally, let the line CP meet Γ again at Q.
Prove that P Q = r.
5 Let N = {0, 1, 2, . . .}. Determine all functions f : N → N such that xf (y) + yf (x) = (x + y)f (x2+ y2) for all x and y in N.
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National Olympiad
2003
1 Consider a standard twelve-hour clock whose hour and minute hands move continuously. Let m be an integer, with 1 ≤ m ≤ 720. At precisely m minutes after 12:00, the angle made by the hour hand and minute hand is exactly 1◦. Determine all possible values of m.
2 Find the last three digits of the number 200320022001. 3 Find all real positive solutions (if any) to
x3+ y3+ z3= x + y + z, and x2+ y2+ z2= xyz.
4 Prove that when three circles share the same chord AB, every line through A different from AB determines the same ratio XY
Y Z, where X is an arbitrary point different from B on the first circle while Y and Z are the points where AX intersects the other two circles (labelled so that Y is between X and Z).
5 Let S be a set of n points in the plane such that any two points of S are at least 1 unit apart. Prove there is a subset T of S with at least n
7 points such that any two points of T are at least√3 units apart.
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Canada
National Olympiad
2006
1 Let f (n, k) be the number of ways of distributing k candies to n children so that each child receives at most 2 candies. For example f (3, 7) = 0, f (3, 6) = 1, f (3, 4) = 6. Determine the value of f (2006, 1) + f (2006, 4) + . . . + f (2006, 1000) + f (2006, 1003) + . . . + f (2006, 4012). 2 Let ABC be acute triangle. Inscribe a rectangle DEF G in this triangle such that D ∈
AB, E ∈ AC, F ∈ BC, G ∈ BC. Describe the locus of (i.e., the curve occupied by) the intersections of the diagonals of all possible rectangles DEF G.
3 In a rectangular array of nonnegative reals with m rows and n columns, each row and each column contains at least one positive element. Moreover, if a row and a column intersect in a positive element, then the sums of their elements are the same. Prove that m = n.
4 Consider a round-robin tournament with 2n + 1 teams, where each team plays each other team exactly one. We say that three teams X, Y and Z, form a cycle triplet if X beats Y , Y beats Z and Z beats X. There are no ties. a)Determine the minimum number of cycle triplets possible. b)Determine the maximum number of cycle triplets possible.
5 The vertices of a right triangle ABC inscribed in a circle divide the circumference into three arcs. The right angle is at A, so that the opposite arc BC is a semicircle while arc BC and