1996 Canadian Open Math Challenge

Part of Canadian Open Math Challenge

Subcontests

(10)

1

Help

Determine the sum of angles

A,B,

0^\circ \leq A,B, \leq 180^\circ

\sin A + \sin B = \sqrt{\frac{3}{2}}, \cos A + \cos B = \sqrt{\frac{1}{2}}

1

Logarithrms

\log_{2n} 1994 = \log_n \left(486 \sqrt{2}\right)

n^6

1

Nice algebra problem

Determine all pairs of integers

(x,y)

which satisfy the equation

6x^2-3xy-13x+5y = -11

1

Geo

ABC

is right angled at

A

. The circle with center

A

AB

BC

AC

D

E

respectively. If

BD = 20

DC = 16

AC^2

1

Do you like baseball?

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.

1

Great problem (different wording than actual problem)

Edward starts in his house, which is at (0,0) and needs to go point (6,4), which is coordinate 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?

1

Number Theory equation

Determine the smallest positive integer,

n

, which satisfies the equation

n^3+2n^2 = b

b

is the square of an odd integer.

1

Inradius and circumradius

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

1

Three digit number and equation

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)

1

The third root

The roots of the equation

x^2+4x-5 = 0

are also the roots of the equation

2x^3+9x^2-6x-5 = 0

. What is the third root of the second equation?