2010 MATHEMATICS TEAM CHALLENGE
SENIOR SECONDARY
RELAY CONTEST
Time: 1 hour
Calculators may be used
Each question is worth 5 points
Total of 100 points
R1 (5 points)
(95 points remaining)
2
3
4
5
Find an integer solution to the equation 1  x  x  x  x  x .
R2 (5 points)
(90 points remaining)
A person cashes a cheque at the bank. By mistake, the teller pays the number of cents as
dollars and the number of dollars as cents. The person spends $3.50 before noticing the
mistake. Then, on counting the money, the person finds that the remaining money is exactly
the doubled amount of the cheque. For what amount was the cheque issued?
R3 (5 points)
(85 points remaining)
Find the sum of digits for the integer formed by the product 16 8  530 .
R4 (5 points)
(80 points remaining)
2
We say that N is an automorphic number if the value of N ends with the string of digits
forming N . For example, 6 is automorphic since 6 2 ends in 6. Find one 2-digit automorphic
number.
R5 (5 points)
(75 points remaining)
Find log 5 12 in terms of a and b , where a  log 10 2 and b  log 10 3 .
R6 (5 points)
(70 points remaining)
What is the least positive multiple of 15 that is made up of only the digits 0, 4 and 7, each
appearing the same number of times?
R7 (5 points)
(65 points remaining)
Find the exact area of the regular octagon formed by cutting equal isosceles right triangles
from the corners of a square with sides of length one unit.
R8 (5 points)
(60 points remaining)
2
2
If x  y  4 and xy  12 , what is the value of x  5 xy  y ?
R9 (5 points)
(55 points remaining)
Find two 2-digit numbers that increase by 75% when their digits are reversed.
R10 (5 points)
(50 points remaining)
The triangle ABC has ACB  120 , AC  6 and BC  2 . The internal bisector of ACB
meets the side AB at the point D .Determine the length of CD .
R11 (5 points)
(45 points remaining)
How many positive integers less than 10000 contain the digit 7 at least once?
R12 (5 points)
(40 points remaining)
A box contains 11 balls numbered 1, 2, 3, …, 11. If 6 balls are drawn simultaneously at
random, what is the probability that the sum of the numbers drawn is odd?
R13 (5 points)
(35 points remaining)
Find the number of different 7-digit numbers that can be made by rearranging the digits in the
number 3853345.
R14 (5 points)
(30 points remaining)
Two circles of radius r are externally tangent. They are also internally tangent to the two
sides of a right triangle of sides 3, 4 and 5 with the hypotenuse of the triangle being tangent to
both circles. Find r .
R15 (5 points)


Determine two integers x such that x  3x  1
2
x 1
(25 points remaining)
 1.
R16 (5 points)
(20 points remaining)
A boat travels at 30 km per hour going downstream and only 22 km per hour going upstream.
To go between the cities A and B on the river takes 4 hours less one way than another. What
is the distance between the cities?
R17 (5 points)
(15 points remaining)
A gardener owns a riding mower and a push mower. It takes him 3 hours to cut the entire
lawn with the push mower, but only 75 minutes with the riding mower. One day he cuts a
portion of the lawn with the push mower and the rest with the riding mower. If the total time
to mow the lawn was 96 minutes, what fraction of the lawn was cut with the riding mower?
R18 (5 points)
(10 points remaining)
Find the largest positive integer less than 122003 that is both a perfect square and a perfect
cube.
R19 (5 points)
(5 points remaining)
Mr and Mrs Johnson are at a party with three other married couples. Since some of the guests
are not familiar with each other, various handshakes take place. No one shakes hands with his
or her spouse, and no one shakes hands with himself or herself. After all of the handshakes
have been made, Mrs Johnson asks the other seven people how many hands they shook.
Surprisingly, they all give different answers. How many hands did Mr Johnson shake?
R20 (5 points)
Find the sum of the digits of 10 2010  2010 .
(0 points remaining)
SENIOR SECONDARY MATHS TEAM CHALLENGE (2010)
Relay Answer Sheet
Problem
R 1 (5 points)
R 2 (5 points)
R 3 (5 points)
R 4 (5 points)
Change
R 5 (5 points)
R 6 (5 points)
R 7 (5 points)
R 8 (5 points)
Change
R 9 (5 points)
R 10 (5 points)
R 11 (5 points)
R 12 (5 points)
Change
R 13 (5 points)
R 14 (5 points)
R 15 (5 points)
R 16 (5 points)
Change
R 17 (5 points)
R 18 (5 points)
R 19 (5 points)
R 20 (5 points)
Answer
Attempts √ or ×
5 4 3 2 1
Score Progressive
Score
-1
$14.32
4
25 or 76
Change
Change
Change
Change
Change
Change
Change
Change
2𝑎 + 𝑏
1−𝑎
400447770
2√2 - 2 or
0.8284
-20
Any two of the
following:
12, 24, 36,48
3
2
3439
118
231
420
5
7
Any two of :
-1,0,1, 3
330 km
4
5
6
7 or 117649
3
18079
Total
School:___________________________________
Team 1:
Team 2: