Part A
1 The roots of the equation x2+4x−5 = 0 are also the roots of the equation 2x3+9x2−6x−5 = 0. What is the third root of the second equation?
2 The numbers a, b, c are the digits of a three digit number which satisfy 49a + 7b + c = 286. What is the three digit number (100a + 10b + c)?
3 The vertices of a right-angled triangle are on a circle of radius R and the sides of the triangle are tangent to another circle of radius r (this is the circle that is inside triangle). If the lengths of the sides about the right angles are 16 and 30, determine the value of R + r. 4 Determine the smallest positive integer, n, which satisfies the equation n3+ 2n2 = b, where
b is the square of an odd integer.
5 Edward starts in his house, which is at (0,0) and needs to go point (6,4), which is coordi- nate for his school. However, there is a park that shaped as a square and has coordinates (2,1),(2,3),(4,1), and (4,3). There is no road for him to walk inside the park but there is a road for him to walk around the perimeter of the square. How many different shortest road routes are there from Edward’s house to his school?
6 In a 14 team baseball league, each team played each of the other teams 10 times. At the end of the season, the number of games won by each team differed from those won by the team that immediately followed it by the same amount. Determine the greatest number of games the last place team could have won, assuming that no ties were allowed.
7 Triangle ABC is right angled at A. The circle with center A and radius AB cuts BC and AC internally at D and E respectively. If BD = 20 and DC = 16, determine AC2.
8 Determine all pairs of integers (x, y) which satisfy the equation 6x2− 3xy − 13x + 5y = −11 9 If log2n1994 = logn486√2, compute n6.
10 Determine the sum of angles A, B, where 0◦≤ A, B, ≤ 180◦ and sin A + sin B = r 3 2, cos A + cos B = r 1 2 http://www.artofproblemsolving.com/
This file was downloaded from the AoPS − MathLinks Math Olympiad Resources Page Page 1 http://www.mathlinks.ro/
Canada
National Olympiad
1969
1 If a1/b1 = a2/b2 = a3/b3 and p1, p2, p3 are not all zero, show that for all n ∈ N,
a1 b1 n = p1a n 1 + p2an2 + p3an3 p1bn1 + p2bn2 + p3bn3 .
2 Determine which of the two numbers √c + 1 −√c,√c −√c − 1 is greater for any c ≥ 1. 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?
4 Let ABC be an equilateral triangle, and P be an arbitrary point within the triangle. Per- pendiculars P D, P E, P F are drawn to the three sides of the triangle. Show that, no matter where P is chosen,
P D + P E + P F AB + BC + CA =
1 2√3.
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
2ab cosC2 a + b .
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. 7 Show that there are no integers a, b, c for which a2+ b2− 8c = 6.
8 Let f be a function with the following properties:
1) f (n) is defined for every positive integer n; 2) f (n) is an integer; 3) f (2) = 2; 4) f (mn) = f (m)f (n) for all m and n; 5) f (m) > f (n) whenever m > n.
Prove that f (n) = n.
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.
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 AP Q and P BR, 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.
http://www.artofproblemsolving.com/
This file was downloaded from the AoPS − MathLinks Math Olympiad Resources Page Page 1 http://www.mathlinks.ro/
National Olympiad
1970
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.
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.
[img]http://www.artofproblemsolving.com/Forum/albumpic.php?picid = 860[/img]Asetof ballsisgiven.Eachballiscolouredredorblue, andthereisatleastoneof eachcolour.Eachballweighseither1poundor2pounds, andthereisatleastoneof eachweight.P rovethattherearetwoballshavingdif f erentweightsanddif f erentcolours.
3
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.
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 ≤ a2+ b2+ c2+ d2 ≤ 4.
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.
7 Show that from any five integers, not necessarily distinct, one can always choose three of these integers whose sum is divisible by 3.
8 Consider all line segments of length 4 with one end-point on the line y = x and the other end-point on the line y = 2x. Find the equation of the locus of the midpoints of these line segments.
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.
10 Given the polynomial
f (x) = xn+ a1xn−1+ a2xn−2+ · · · + an−1x + an
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.
http://www.artofproblemsolving.com/
This file was downloaded from the AoPS − MathLinks Math Olympiad Resources Page Page 1 http://www.mathlinks.ro/
Canada
National Olympiad
1971
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. (See diagram). Given EC = 1, find the radius of the circle.
[img]6704[/img]
2 Let x and y be positive real numbers such that x + y = 1. Show that 1 + 1 x 1 +1 y ≥ 9.
3 ABCD is a quadrilateral with AD = BC. If ∠ADC is greater than ∠BCD, prove that AC > BD.
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.
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.
6 Show that, for all integers n, n2+ 2n + 12 is not a multiple of 121.
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. 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.
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.
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.
http://www.artofproblemsolving.com/
This file was downloaded from the AoPS − MathLinks Math Olympiad Resources Page Page 1 http://www.mathlinks.ro/
National Olympiad
1972
1 Given three distinct unit circles, each of which is tangent to the other two, find the radii of the circles which are tangent to all three circles.
2 Let a1, a2, . . . , an be non-negative real numbers. Define M to be the sum of all products of
pairs aiaj (i < j), i.e.,
M = a1(a2+ a3+ · · · + an) + a2(a3+ a4+ · · · + an) + · · · + an−1an.
Prove that the square of at least one of the numbers a1, a2, . . . , andoes not exceed 2M/n(n−1).
3 a) Prove that 10201 is composite in all bases greater than 2. b) Prove that 10101 is composite in all bases.
4 Describe a construction of quadrilateral ABCD given: (i) the lengths of all four sides;
(ii) that AB and CD are parallel; (iii) that BC and DA do not intersect.
5 Prove that the equation x3+ 113 = y3 has no solution in positive integers x and y.
6 Let a and b be distinct real numbers. Prove that there exist integers m and n such that am + bn < 0, bm + an > 0.
7 a) Prove that the values of x for which x = (x2+1)/198 lie between 1/198 and 197.99494949 · · · . b) Use the result of problem a) to prove that
√
2 < 1.41421356. c) Is it true that
√
2 < 1.41421356?
8 During a certain election campaign, p different kinds of promises are made by the different political parties (p > 0). While several political parties may make the same promise, any two parties have at least one promise in common; no two parties have exactly the same set of promises. Prove that there are no more than 2p−1.
9 Four distinct lines L1, L2, L3, L4are given in the plane: L1 and L2 are respectively parallel to
L3 and L4. Find the locus of a point moving so that the sum of its perpendicular distances
from the four lines is constant.
10 What is the maximum number of terms in a geometric progression with common ratio greater than 1 whose entries all come from the set of integers between 100 and 1000 inclusive?
http://www.artofproblemsolving.com/
This file was downloaded from the AoPS − MathLinks Math Olympiad Resources Page Page 1 http://www.mathlinks.ro/