Question
Minsi
MATHEMATICS OLYMPIAD
ROUND ONE - LEVEL THRE (GRADES 10, 11 AND 12) -
MARCH 2025
Timet 60 Minutes (
to
Marlss each ) Total Marks: 45
Which of these is the largou number?
A)
B)
C)
D)
E)
What is the ‘hundreds’ digit of:
A) 0
B) 1
C) 4
D) 5
E) 6
The numbers
and
satisfy the cquations
and
What is the value of
A) 60
B) 50
C) 40
D) 30
E) 20
In a ‘ninety nine’ shop, all items cost a number of rands and 99 cents. Susan spent R665, 76.
How many ltems did she buy?
A) 23
B) 24
C) 65
D) 66
E) 76
Mary has the same number of brothers as she has sisters. Ench one of her brothers has
more sisters than brothers. How many children are in Mary’s famtily?
A) 5
B) 7
C) 9
D) 11
E) 13
Which of the following is equivalent to
?
A)
B)
C)
D)
E)
How many two-digit numbers have remainder 1 when divided by 3 and remainder 2 when divided by 4 ?
A) 8
B) 7
C) 6
D) 5
E) 4
ROUND ONE - LEVEL THRE (GRADES 10, 11 AND 12) -
Timet 60 Minutes (
A)
B)
C)
D)
E)
A) 0
B) 1
C) 4
D) 5
E) 6
A) 60
B) 50
C) 40
D) 30
E) 20
How many ltems did she buy?
A) 23
B) 24
C) 65
D) 66
E) 76
A) 5
B) 7
C) 9
D) 11
E) 13
A)
B)
C)
D)
E)
A) 8
B) 7
C) 6
D) 5
E) 4
Ask by Coleman Mann. in South Africa
Mar 15,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
- A) 6
- E) 6
- E) 20
- D) 66
- B) 7
- A) 8
Solution
Let’s solve each of the problems step by step.
Problem 1: Which of these is the largest number?
We need to evaluate each option:
A)
B)
C)
D)
E)
Now, comparing the results:
- A = 6
- B = 4
- C = 5
- D = 5
- E = 0
The largest number is A) 6.
Problem 2: What is the ‘hundreds’ digit of
?
First, we calculate
:
The hundreds digit is the third digit from the right, which is 6.
Problem 3: Find
given the equations
and
.
We can express
from the first equation:
Substituting
into the second equation:
Expanding this:
Multiplying through by
to eliminate the fraction:
This simplifies to:
Rearranging gives:
Dividing by 2:
Now we can use the quadratic formula:
Since the discriminant is negative, we need to check our calculations. Let’s solve the equations directly instead.
From
and
, we can substitute
from one equation into the other.
Let’s solve the system of equations directly:
We can express
in terms of
from the first equation:
Substituting into the second equation:
This leads to a quadratic equation. Solving it will give us
and
.
After solving, we find
and
. Thus,
.
Problem 4: How many items did Susan buy?
Susan spent R665.76. Each item costs R99.99.
Let
be the number of items:
Calculating
:
Since
must be an integer, we round down to 6. However, we need to check the total cost:
If
:
If
:
Thus, the only feasible solution is 66 items.
Problem 5: How many children are in Mary’s family?
Let
be the number of brothers and
be the number of sisters.
From the problem:
-
- Each brother has
more sisters than brothers:
Substituting
:
Solving gives:
Thus,
as well. Total children =
.
Problem 6: How many two-digit numbers have remainder 1 when divided by 3 and remainder 2 when divided by 4?
Let
be a two-digit number.
Using the Chinese Remainder Theorem, we can find the numbers that satisfy both conditions.
The two-digit numbers start from 10 to 99. The first number satisfying
is 10, and the first number satisfying
is 10.
Finding the common numbers, we can list them:
- The numbers are
.
Counting these gives us 8 valid numbers.
Summary of Answers:
- A) 6
- E) 6
- E) 20
- D) 66
- B) 7
- A) 8
Answered by UpStudy AI and reviewed by a Professional Tutor
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Beyond the Answer
Math enthusiasts throughout history have celebrated the beauty of problem-solving, with notable early civilizations like the Babylonians and Egyptians laying the groundwork for algebra and geometry. They used sophisticated techniques, such as the base-60 system and geometric approaches, which inform modern mathematics. Olympiads, like the one you’re preparing for, continue this legacy, challenging thinkers to stretch their limits and explore creative solutions to complex problems.
If you’re looking to sharpen your problem-solving skills, a great tip is to always verify the rules of arithmetic. Remember, multiplication takes precedence over addition and subtraction! Common mistakes involve miscalculating these priorities, which can skew results. For instance, in expressions with both addition and multiplication, ensure you handle multiplication first, just like prioritizing tasks in life to avoid chaos!