Dataset Viewer
id
string | source
string | problem
string | solutions
list |
|---|---|---|---|
CMO-1969-1
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1969.pdf
|
Show that if \( a_1/b_1 = a_2/b_2 = a_3/b_3 \) and \( p_1, p_2, p_3 \) are not all zero, then
\[
\left( \frac{a_1}{b_1} \right)^n = \frac{p_1 a_1^n + p_2 a_2^n + p_3 a_3^n}{p_1 b_1^n + p_2 b_2^n + p_3 b_3^n}
\]
for every positive integer \( n \).
|
[] |
CMO-1969-2
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1969.pdf
|
Determine which of the two numbers \( \sqrt{c+1} - \sqrt{c} \), \( \sqrt{c} - \sqrt{c-1} \) is greater for any \( c \geq 1 \).
|
[] |
CMO-1969-3
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1969.pdf
|
Let \( c \) be the length of the hypotenuse of a right angle triangle whose other two sides have lengths \( a \) and \( b \). Prove that \( a + b \leq \sqrt{2}c \). When does the equality hold?
|
[] |
CMO-1969-4
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1969.pdf
|
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,
\[
\frac{PD + PE + PF}{AB + BC + CA} = \frac{1}{2\sqrt{3}}.
\]
|
[] |
CMO-1969-5
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1969.pdf
|
Let \( ABC \) be a triangle with sides of lengths \( a, b \) and \( c \). Let the bisector of the angle \( C \) cut \( AB \) in \( D \). Prove that the length of \( CD \) is
\[
\frac{2ab \cos \frac{C}{2}}{a + b}.
\]
|
[] |
CMO-1969-6
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1969.pdf
|
Find the sum of \( 1 \cdot 1! + 2 \cdot 2! + 3 \cdot 3! + \cdots + (n - 1)(n - 1)! + n \cdot n! \), where \( n! = n(n - 1)(n - 2) \cdots 2 \cdot 1 \).
|
[] |
CMO-1969-7
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1969.pdf
|
Show that there are no integers \( a, b, c \) for which \( a^2 + b^2 - 8c = 6 \).
|
[] |
CMO-1969-8
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1969.pdf
|
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 \( f(n) = n \).
|
[] |
CMO-1969-9
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1969.pdf
|
Show that for any quadrilateral inscribed in a circle of radius 1, the length of the shortest side is less than or equal to \( \sqrt{2} \).
|
[] |
CMO-1969-10
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1969.pdf
|
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 \( 2/9 \).
|
[] |
CMO-1970-1
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1970.pdf
|
Find all number triples \( (x, y, z) \) such that when any one of these numbers is added to the product of the other two, the result is 2.
|
[] |
CMO-1970-2
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1970.pdf
|
Given a triangle \( ABC \) with angle \( A \) obtuse and with altitudes of length \( h \) and \( k \) as shown in the diagram, prove that \( a + h \geq b + k \). Find under what conditions \( a + h = b + k \).
|
[] |
CMO-1970-3
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1970.pdf
|
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 2 balls having different weights and different colours.
|
[] |
CMO-1970-4
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1970.pdf
|
a) Find all positive integers with initial digit 6 such that the integer formed by deleting this 6 is \( 1/25 \) of the original integer.
b) Show that there is no integer such that deletion of the first digit produces a result which is \( 1/35 \) of the original integer.
|
[] |
CMO-1970-5
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1970.pdf
|
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
\[
2 \leq a^2 + b^2 + c^2 + d^2 \leq 4.
\]
|
[] |
CMO-1970-6
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1970.pdf
|
Given three non-collinear points \( A, B, C \), construct a circle with centre \( C \) such that the tangents from \( A \) and \( B \) to the circle are parallel.
|
[] |
CMO-1970-7
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1970.pdf
|
Show that from any five integers, not necessarily distinct, one can always choose three of these integers whose sum is divisible by 3.
|
[] |
CMO-1970-8
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1970.pdf
|
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.
|
[] |
CMO-1970-9
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1970.pdf
|
Let \( f(n) \) be the sum of the first \( n \) terms of the sequence
\[
0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, \ldots
\]
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 \( s > t \).
|
[] |
CMO-1970-10
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1970.pdf
|
Given the polynomial
\[
f(x) = x^n + a_1 x^{n-1} + a_2 x^{n-2} + \cdots + a_{n-1} x + a_n
\]
with integral coefficients \( a_1, a_2, \ldots, a_n \), 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 \( f(k) = 8 \).
|
[] |
CMO-1971-1
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1971.pdf
|
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.
|
[] |
CMO-1971-2
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1971.pdf
|
Let \( x \) and \( y \) be positive real numbers such that \( x + y = 1 \). Show that
\[
\left(1 + \frac{1}{x}\right)\left(1 + \frac{1}{y}\right) \geq 9.
\]
|
[] |
CMO-1971-3
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1971.pdf
|
ABCD is a quadrilateral with \( AD = BC \). If \( \angle ADC \) is greater than \( \angle BCD \), prove that \( AC > BD \).
|
[] |
CMO-1971-4
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1971.pdf
|
Determine all real numbers \( a \) such that the two polynomials \( x^2 + ax + 1 \) and \( x^2 + x + a \) have at least one root in common.
|
[] |
CMO-1971-5
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1971.pdf
|
Let
\[
p(x) = a_0 x^n + a_1 x^{n-1} + \cdots + a_{n-1} x + a_n,
\]
where the coefficients \( a_i \) are integers. If \( p(0) \) and \( p(1) \) are both odd, show that \( p(x) \) has no integral roots.
|
[] |
CMO-1971-6
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1971.pdf
|
Show that, for all integers \( n \), \( n^2 + 2n + 12 \) is not a multiple of 121.
|
[] |
CMO-1971-7
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1971.pdf
|
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 deleting its middle digit. Determine all \( n \) such that \( n/m \) is an integer.
|
[] |
CMO-1971-8
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1971.pdf
|
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 \( r \).
|
[] |
CMO-1971-9
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1971.pdf
|
Two flag poles of heights \( h \) 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.
|
[] |
CMO-1971-10
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1971.pdf
|
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.
|
[] |
CMO-1972-1
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1972.pdf
|
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.
|
[] |
CMO-1972-2
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1972.pdf
|
Let \( a_1, a_2, \ldots, a_n \) be non-negative real numbers. Define \( M \) to be the sum of all products of pairs \( a_i a_j (i < j) \), i.e.,
\[
M = a_1(a_2 + a_3 + \cdots + a_n) + a_2(a_3 + a_4 + \cdots + a_n) + \cdots + a_{n-1}a_n.
\]
Prove that the square of at least one of the numbers \( a_1, a_2, \ldots, a_n \) does not exceed \( \frac{2M}{n(n - 1)} \).
|
[] |
CMO-1972-3
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1972.pdf
|
a) Prove that 10201 is composite in any base greater than 2.
b) Prove that 10101 is composite in any base.
|
[] |
CMO-1972-4
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1972.pdf
|
Describe a construction of a 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.
|
[] |
CMO-1972-5
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1972.pdf
|
Prove that the equation \( x^3 + 11^3 = y^3 \) has no solution in positive integers \( x \) and \( y \).
|
[] |
CMO-1972-6
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1972.pdf
|
Let \( a \) and \( b \) be distinct real numbers. Prove that there exist integers \( m \) and \( n \) such that \( am + bn < 0 \), \( bm + an > 0 \).
|
[] |
CMO-1972-7
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1972.pdf
|
a) Prove that the values of \( x \) for which \( x = (x^2 + 1)/198 \) lie between 1/198 and 197.99494949\ldots
b) Use the result of a) to prove that \( \sqrt{2} < 1.41421356 \).
c) Is it true that \( \sqrt{2} < 1.41421356 \)?
|
[] |
CMO-1972-8
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1972.pdf
|
During a certain election campaign, \( p \) different kinds of promises are made by the various political parties (\( p > 0 \)). While several 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 \( 2^{p-1} \) parties.
|
[] |
CMO-1972-9
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1972.pdf
|
Four distinct lines \( L_1, L_2, L_3, L_4 \) are given in the plane: \( L_1 \) and \( L_2 \) are respectively parallel to \( L_3 \) and \( L_4 \). Find the locus of a point moving so that the sum of its perpendicular distances from the four lines is constant.
|
[] |
CMO-1972-10
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1972.pdf
|
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?
|
[] |
CMO-1973-1
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1973.pdf
|
(i) Solve the simultaneous inequalities, \( x < \frac{1}{x} \) and \( x < 0 \); i.e., find a single inequality equivalent to the two given simultaneous inequalities.
(ii) What is the greatest integer which satisfies both inequalities \( 4x + 13 < 0 \) and \( x^2 + 3x > 16 \)?
(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
\[
\frac{1}{\log_2 36} + \frac{1}{\log_{36} 36}.
\]
|
[] |
CMO-1973-2
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1973.pdf
|
Find all the real numbers which satisfy the equation \( |x + 3| - |x - 1| = x + 1 \). (Note: \( |a| = a \) if \( a \geq 0 \); \( |a| = -a \) if \( a < 0 \).)
|
[] |
CMO-1973-3
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1973.pdf
|
Prove that if \( p \) and \( p + 2 \) are both prime integers greater than 3, then 6 is a factor of \( p + 1 \).
|
[] |
CMO-1973-4
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1973.pdf
|
The figure shows a (convex) polygon with nine vertices. The six diagonals which have been drawn dissect the polygon into the seven triangles:
\( P_0P_1P_3, P_3P_6P_0, P_0P_6P_7, P_0P_7P_8, P_1P_2P_3, P_3P_4P_5, P_4P_5P_6 \).
In how many ways can these triangles be labelled with the names \( \Delta_1, \Delta_2, \Delta_3, \Delta_4, \Delta_5, \Delta_6, \Delta_7 \) so that \( P_i \) is a vertex of triangle \( \Delta_i \) for \( i = 1, 2, 3, 4, 5, 6, 7 \)? Justify your answer.
|
[] |
CMO-1973-5
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1973.pdf
|
For every positive integer \( n \), let
\[
h(n) = 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n}.
\]
For example, \( h(1) = 1, h(2) = 1 + \frac{1}{2}, h(3) = 1 + \frac{1}{2} + \frac{1}{3} \). Prove that for \( n = 2, 3, 4, \ldots \)
\[
n + h(1) + h(2) + h(3) + \cdots + h(n - 1) = nh(n).
\]
|
[] |
CMO-1973-6
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1973.pdf
|
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 \( Y \)).
|
[] |
CMO-1973-7
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1973.pdf
|
Observe that
\[
\frac{1}{1} = \frac{1}{2} + \frac{1}{2};\quad \frac{1}{2} = \frac{1}{3} + \frac{1}{6};\quad \frac{1}{3} = \frac{1}{4} + \frac{1}{12};\quad \frac{1}{4} = \frac{1}{5} + \frac{1}{20}.
\]
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
\[
\frac{1}{n} = \frac{1}{i(i+1)} + \frac{1}{(i+1)(i+2)} + \frac{1}{(i+2)(i+3)} + \cdots + \frac{1}{j(j+1)}.
\]
|
[] |
CMO-1974-1
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1974.pdf
|
i) If \( x = \left(1 + \frac{1}{n}\right)^n \) and \( y = \left(1 + \frac{1}{n}\right)^{n+1} \), show that \( y^n = x^{n+1} \).
ii) Show that, for all positive integers \( n \),
\[
1^2 - 2^2 + 3^2 - 4^2 + \cdots + (-1)^n(n - 1)^2 + (-1)^{n+1}n^2 = (-1)^{n+1}(1 + 2 + \cdots + n).
\]
|
[] |
CMO-1974-2
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1974.pdf
|
Let \( ABCD \) be a rectangle with \( BC = 3AB \). Show that if \( P, Q \) are the points on side \( BC \) with \( BP = PQ = QC \), then
\[
\angle DBC + \angle DPC = \angle DQC.
\]
|
[] |
CMO-1974-3
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1974.pdf
|
Let
\[
f(x) = a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n
\]
be a polynomial with coefficients satisfying the conditions:
\[
0 \leq a_i \leq a_0, \quad i = 1, 2, \ldots, n.
\]
Let \( b_0, b_1, \ldots, b_{2n} \) be the coefficients of the polynomial
\[
(f(x))^2 = (a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n)^2
= b_0 + b_1 x + b_2 x^2 + \cdots + b_{n+1} x^{n+1} + \cdots + b_{2n} x^{2n}.
\]
Prove that
\[
b_{n+1} \leq \frac{1}{2} (f(1))^2.
\]
|
[] |
CMO-1974-4
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1974.pdf
|
Let \( n \) be a fixed positive integer. To any choice of \( n \) real numbers satisfying
\[
0 \leq x_i \leq 1, \quad i = 1, 2, \ldots, n,
\]
there corresponds the sum
\[(*) \quad \sum_{1 \leq i < j \leq n} |x_i - x_j|\]
\[
= |x_1 - x_2| + |x_1 - x_3| + |x_1 - x_4| + \cdots + |x_1 - x_{n-1}| + |x_1 - x_n|
+ |x_2 - x_3| + |x_2 - x_4| + \cdots + |x_2 - x_{n-1}| + |x_2 - x_n|
+ |x_3 - x_4| + \cdots + |x_3 - x_{n-1}| + |x_3 - x_n|
+ \cdots + |x_{n-2} - x_{n-1}| + |x_{n-2} - x_n|
+ |x_{n-1} - x_n|.
\]
Let \( S(n) \) denote the largest possible value of the sum (*). Find \( S(n) \).
|
[] |
CMO-1974-5
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1974.pdf
|
Given a circle with diameter \( AB \) and a point \( X \) on the circle different from \( A \) and \( B \), let \( t_a, t_b \) and \( t_x \) be the tangents to the circle at \( A, B \) and \( X \) respectively. Let \( Z \) be the point where line \( AX \) meets \( t_b \) and \( Y \) the point where line \( BX \) meets \( t_a \). Show that the three lines \( YZ, t_x \) and \( AB \) are either concurrent (i.e., all pass through the same point) or parallel.
|
[] |
CMO-1974-6
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1974.pdf
|
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.)
|
[] |
CMO-1974-7
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1974.pdf
|
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). 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 (\( 0 \leq x < 12 \)) 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 \( 0 \leq x < 12 \).
|
[] |
CMO-1975-1
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1975.pdf
|
Simplify
\[
\left( \frac{1 \cdot 2 \cdot 4 + 2 \cdot 4 \cdot 8 + \cdots + n \cdot 2n \cdot 4n}{1 \cdot 3 \cdot 9 + 2 \cdot 6 \cdot 18 + \cdots + n \cdot 3n \cdot 9n} \right)^{1/3}.
\]
|
[] |
CMO-1975-2
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1975.pdf
|
A sequence of numbers \( a_1, a_2, a_3, \ldots \) satisfies
(i) \( a_1 = \frac{1}{2} \),
(ii) \( a_1 + a_2 + \cdots + a_n = n^2 a_n \quad (n \geq 1) \).
Determine the value of \( a_n \quad (n \geq 1) \).
|
[] |
CMO-1975-3
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1975.pdf
|
For each real number \( r \), \( [r] \) denotes the largest integer less than or equal to \( r \), e.g., \( [6] = 6 \), \( [\pi] = 3 \), \( [-1.5] = -2 \). Indicate on the \( (x, y) \)-plane the set of all points \( (x, y) \) for which \( [x]^2 + [y]^2 = 4 \).
|
[] |
CMO-1975-4
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1975.pdf
|
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.
|
[] |
CMO-1975-5
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1975.pdf
|
Let \( A, B, C, D \) be 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 \( QS \).
|
[] |
CMO-1975-6
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1975.pdf
|
(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.
|
[] |
CMO-1975-7
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1975.pdf
|
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 \( 2\pi \). Is the function \( \sin(x^2) \) periodic? Prove your assertion.
|
[] |
CMO-1975-8
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1975.pdf
|
Let \( k \) be a positive integer. Find all polynomials
\[
P(x) = a_0 + a_1 x + \cdots + a_n x^n,
\]
where the \( a_i \) are real, which satisfy the equation
\[
P(P(x)) = \{P(x)\}^k.
\]
|
[] |
CMO-1976-1
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1976.pdf
|
Given four weights in geometric progression and an equal arm balance, show how to find the heaviest weight using the balance only twice.
|
[] |
CMO-1976-2
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1976.pdf
|
Suppose
\[n(n+1)a_{n+1} = n(n-1)a_n - (n-2)a_{n-1}\]
for every positive integer \( n \geq 1 \).
Given that \( a_0 = 1, a_1 = 2 \), find
\[\frac{a_0}{a_1} + \frac{a_1}{a_2} + \frac{a_2}{a_3} + \cdots + \frac{a_{50}}{a_{51}}.\]
|
[] |
CMO-1976-3
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1976.pdf
|
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 from grade eight participated in the chess tournament? Is the solution unique?
|
[] |
CMO-1976-4
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1976.pdf
|
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 \[\frac{AC}{CB} = \frac{QC}{CP}.\] Describe, with proof, the locus of the point \( P \).
|
[] |
CMO-1976-5
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1976.pdf
|
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.
|
[] |
CMO-1976-6
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1976.pdf
|
If \( A, B, C, D \) are four points in space, such that
\[\angle ABC = \angle BCD = \angle CDA = \angle DAB = \pi/2,\]
prove that \( A, B, C, D \) lie in a plane.
|
[] |
CMO-1976-7
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1976.pdf
|
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 \( x^2 - 2xy + y^2 \) satisfies this condition). Given that \( (x - y) \) is a factor of \( P(x, y) \), show that \( (x - y)^2 \) is a factor of \( P(x, y) \).
|
[] |
CMO-1976-8
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1976.pdf
|
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.
|
[] |
CMO-1977-1
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1977.pdf
|
If \( f(x) = x^2 + x \), prove that the equation \( 4f(a) = f(b) \) has no solutions in positive integers \( a \) and \( b \).
|
[] |
CMO-1977-2
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1977.pdf
|
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 \( \angle OPA \) is a maximum.
|
[] |
CMO-1977-3
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1977.pdf
|
\( N \) is an integer whose representation in base \( b \) is 777. Find the smallest positive integer \( b \) for which \( N \) is the fourth power of an integer.
|
[] |
CMO-1977-4
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1977.pdf
|
Let
\[
p(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0
\]
and
\[
q(x) = b_m x^m + b_{m-1} x^{m-1} + \cdots + b_1 x + b_0
\]
be two polynomials with integer coefficients. Suppose that all the coefficients of the product \( p(x) \cdot q(x) \) 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.
|
[] |
CMO-1977-5
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1977.pdf
|
A right circular cone of 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 (see diagram). What is the minimum distance from the vertex \( V \) to this path?
|
[] |
CMO-1977-6
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1977.pdf
|
Let \( 0 < u < 1 \) and define
\[
u_1 = 1 + u, \quad u_2 = \frac{1}{u_1} + u, \quad \ldots, \quad u_{n+1} = \frac{1}{u_n} + u, \quad n \geq 1.
\]
Show that \( u_n > 1 \) for all values of \( n = 1, 2, 3, \ldots \).
|
[] |
CMO-1977-7
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1977.pdf
|
A rectangular city is exactly \( m \) blocks long and \( n \) blocks wide (see diagram). 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 \( f(m,n) \leq 2^{mn} \).
|
[] |
CMO-1978-1
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1978.pdf
|
Let \( n \) be an integer. If the tens digit of \( n^2 \) is 7, what is the units digit of \( n^2 \)?
|
[] |
CMO-1978-2
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1978.pdf
|
Find all pairs \( a, b \) of positive integers satisfying the equation \( 2a^2 = 3b^3 \).
|
[] |
CMO-1978-3
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1978.pdf
|
Determine the largest real number \( z \) such that
\[
x + y + z = 5
\]
and
\[
xy + yz + zx = 3
\]
and \( x, y \) are also real.
|
[] |
CMO-1978-4
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1978.pdf
|
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 \( ABCD \).
|
[] |
CMO-1978-5
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1978.pdf
|
Eve and Odette play a game on a \( 3 \times 3 \) 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 every 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.
|
[] |
CMO-1978-6
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1978.pdf
|
Sketch the graph of \( x^3 + xy + y^3 = 3 \).
|
[] |
CMO-1979-1
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1979.pdf
|
Given: (i) \( a, b > 0 \); (ii) \( a, A_1, A_2, b \) is an arithmetic progression; (iii) \( a, G_1, G_2, b \) is a geometric progression. Show that
\[
A_1 A_2 \geq G_1 G_2
\]
|
[] |
CMO-1979-2
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1979.pdf
|
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.
Note: (i) A tetrahedron is a triangular pyramid, and (ii) a dihedral angle is the interior angle between a pair of faces.
|
[] |
CMO-1979-3
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1979.pdf
|
Let \( a, b, c, d, e \) be integers such that \( 1 \leq a < b < c < d < e \). 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. \([4,6] = 12\)).
|
[] |
CMO-1979-4
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1979.pdf
|
A dog standing at the centre of a circular arena sees a rabbit at the wall. The rabbit runs around 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.
|
[] |
CMO-1979-5
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1979.pdf
|
A walk consists of a sequence of steps of length 1 taken in directions north, south, east or west. A walk is self-avoiding if it never passes through the same point twice. Let \( f(n) \) denote 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
\[
2^n < f(n) \leq 4 \cdot 3^{n-1}.
\]
|
[] |
CMO-1980-1
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1980.pdf
|
If \( a679b \) is a five digit number (in base 10) which is divisible by 72, determine \( a \) and \( b \).
|
[] |
CMO-1980-2
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1980.pdf
|
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?
|
[] |
CMO-1980-3
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1980.pdf
|
Among all triangles having (i) a fixed angle \( A \) and (ii) an inscribed circle of fixed radius \( r \), determine which triangle has the least perimeter.
|
[] |
CMO-1980-4
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1980.pdf
|
A gambling student tosses a fair coin and scores one point for each head that turns up and two points for each tail. Prove that the probability of the student scoring exactly \( n \) points is \[ \frac{1}{3} [2 + (-\tfrac{1}{2})^n]. \]
|
[] |
CMO-1980-5
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1980.pdf
|
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.
|
[] |
CMO-1981-1
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1981.pdf
|
For any real number \( t \), denote by \( [t] \) the greatest integer which is less than or equal to \( t \). For example: \( [8] = 8 \), \( [\pi] = 3 \) and \( [-5/2] = -3 \). Show that the equation
\[
[x] + [2x] + [4x] + [8x] + [16x] + [32x] = 12345
\]
has no real solution.
|
[] |
CMO-1981-2
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1981.pdf
|
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 of the area of triangle \( PQR \).
|
[] |
CMO-1981-3
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1981.pdf
|
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 an infinite sequence of lines (first line, second line, third line, etc.) in \( P \) so that every circle in \( P \) meets at least one of the lines. (A point is not considered to be a circle.)
|
[] |
CMO-1981-4
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1981.pdf
|
\( P(x) \) and \( Q(x) \) are two polynomials that satisfy the identity \( P(Q(x)) \equiv Q(P(x)) \) for real numbers \( x \). If the equation \( P(x) = Q(x) \) has no real solution, show that the equation \( P(P(x)) = Q(Q(x)) \) also has no real solution.
|
[] |
CMO-1981-5
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1981.pdf
|
Eleven 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?
|
[] |
CMO-1982-1
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1982.pdf
|
In the diagram, \( OB_i \) is parallel and equal in length to \( A_iA_{i+1} \) for \( i = 1, 2, 3 \) and \( 4 (A_5 = A_1) \). Show that the area of \( B_1B_2B_3B_4 \) is twice that of \( A_1A_2A_3A_4 \).
|
[] |
CMO-1982-2
|
https://cms.math.ca/wp-content/uploads/2019/07/exam1982.pdf
|
If \( a, b \) and \( c \) are the roots of the equation \( x^3 - x^2 - x - 1 = 0 \),
(i) show that \( a, b \) and \( c \) 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.
|
[] |
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