1997 Canadian Open Math Challenge

Part of Canadian Open Math Challenge

Subcontests

(12)

1

COMC 1997 #12

Find all real values of x, y, and z such that

x - \sqrt{yz} = 42

y - \sqrt{xz}=6

z-\sqrt{xy}=30

1

COMC 1997 #11

In an isosceles right-angled triangle AOB, points P; Q and S are chosen on sides OB, OA, and AB respectively such that a square PQRS is formed as shown. If the lengths of OP and OQ are a and b respectively, and the area of PQRS is 2 5 that of triangle AOB, determine a : b. [asy]
pair A = (0,3); pair B = (0,0); pair C = (3,0); pair D = (0,1.5); pair E = (0.35,0); pair F = (1.2,1.8); pair J = (0.17,0); pair Y = (0.17,0.75);
pair Z = (1.6,0.2); draw(A--B); draw(B--C); draw(C--A); draw(D--F--Z--E--D); draw("

O

", B, dir(180)); draw("

B

", A, dir(45)); draw("

A

", C, dir(45)); draw("

Q

", E, dir(45)); draw("

P

", D, dir(45)); draw("

R

", Z, dir(45)); draw("

S

", F, dir(45)); draw("

a

", Y, dir(210)); draw("

b

", J, dir(100)); [/asy]

1

COMC 1997 #10

Consider the ten numbers

ar, ar^2, ar^3, ... , ar^{10}

. If their sum is 18 and the sum of their reciprocals is 6, determine their product.

1

COMC 1997 #9

The straight line

l_{1}

x-2y+10 = 0

meets the circle with equation

x^2 + y^2 = 100

at B in the first quadrant. A line through B, perpendicular to

l_{1}

cuts the y-axis at P (0, t). Determine the value of

t

1

COMC 1997 #8

An hourglass is formed from two identical cones. Initially, the upper cone is filled with sand and the lower one is empty. The sand flows at a constant rate from the upper to the lower cone. It takes exactly one hour to empty the upper cone. How long does it take for the depth of sand in the lower cone to be half the depth of sand in the upper cone? (Assume that the sand stays level in both cones at all times.)

1

COMC 1997 #7

Determine the minimum value of

f (x)

where f (x) = (3 sin x - 4 cos x - 10)(3 sin x + 4 cos x - 10).

1

COMC 1997 #6

The triangle ABC has sides AB = 137, AC = 241, and BC =200. There is a point D, on BC, such that both incircles of triangles ABD and ACD touch AD at the same point E. Determine the length of CD. [asy]
pair A = (2,6); pair B = (0,0); pair C = (10,0); pair D = (3.5,0) ; pair E = (3.1,2); draw(A--B); draw(B--C); draw(C--A); draw (A--D); dot ((3.1,1.7)); label ("E", E, dir(45)); label ("A", A, dir(45)); label ("B", B, dir(45)); label ("C", C, dir(45)); label ("D", D, dir(45)); draw(circle((1.8,1.3),1.3)); draw(circle((4.9,1.7),1.75)); [/asy]

1

COMC 1997 #5

Two cubes have their faces painted either red or blue. The 1st cube has five red faces and one blue face. When the two cubes are rolled simultaneously, the probability that the two top faces show the same color is

\frac{1}{2}

. How many red faces are there on the second cube?

1

COMC 1997 #4

Given three distinct digits a, b and c, it is possible, by choosing two digits at a time, to form six two-digit numbers. Determine all possible sets {a, b, c} for which the sum of the six two-digits numbers is 484.

1

COMC 1997 #3

Determine all points on the straight line which joins

(4, 11)

(16, 1)

and whose coordinates are positive integers.

1

COMC 1997 #2

Solve the following system of equations:

xy^2 = 108

\frac{x^3}{y}= 10^{10}

1

COMC 1997 #1

In triangle ABC,

\angle

A equals 120 degrees. A point D is inside the triangle such that

\angle

\times \angle

\angle

\times \angle

ACD. Determine the measure, in degrees, of

\angle

BDC. [asy]
pair A = (5,4); pair B = (0,0); pair C = (10,0); pair D = (5,2.5) ; draw(A--B); draw(B--C); draw(C--A); draw (B--D--C); label ("A", A, dir(45)); label ("B", B, dir(45)); label ("C", C, dir(45)); label ("D", D, dir(45));
[/asy]