HKIMO 2022 Heat round grade 7

There are some chickens and rabbits in a cage. The number of chickens is 4 times as the number of rabbits. The total number of legs of chicken’s is 280 fewer than that of rabbit’s. How many rabbit(s) is / are there?

Mathematical Equivalence

ON

Answer explanation

Let's assume that there are x rabbits in the cage. Since the number of chickens is 4 times as the number of rabbits, then the number of chickens in the cage is 4x.

Each chicken has two legs, and each rabbit has four legs. Therefore, the total number of legs of the chickens is 8x and the total number of legs of the rabbits is 4x.

The problem states that the total number of legs of chicken's is 280 fewer than that of rabbit's. We can express this as an equation:

4x * 2 = 4x + 280 - 8x

Simplifying this equation, we get:

8x = 560

x = 70

Therefore, there are 70 rabbits in the cage.

How many time(s) will a straight line be made by hour hand and minute hand between 6pm and 7pm?

Mathematical Equivalence

ON

Answer explanation

Between 6pm and 7pm, the hour hand will move from the 6 to the 7, while the minute hand will complete one full rotation around the clock face. The straight line will be made once every hour when the minute hand passes over the hour hand. Therefore, the straight line will be made once between 6pm and 7pm.

There are 5 pairs of white chopsticks, 6 pairs of yellow chopsticks and 7 pairs of brown chopsticks mixed together. Close your eyes. If you want to get 2 pairs of chopsticks with different colour, at least how many piece(s) of chopstick(s) is / are needed to be taken?

Mathematical Equivalence

ON

Answer explanation

To get 2 pairs of chopsticks with different colors, you would need to take at least 5 pieces of chopsticks.

Here's why:

To get 2 pairs of chopsticks, we need 4 chopsticks in total.

If we take 4 chopsticks of the same color, we won't have 2 pairs with different colors.

Therefore, we need to take at least 1 chopstick of a different color.

The worst-case scenario is that we pick chopsticks of the same color 4 times in a row, and then on the 5th pick, we get a chopstick of a different color.

So, we need to take at least 5 chopsticks to guarantee that we get 2 pairs of chopsticks with different colors.

Andy goes west for 20km, then goes north for 21km. How far is he now from the original position?

Mathematical Equivalence

ON

Answer explanation

Andy travels a distance of 20 km towards west and 21 km towards north. Using the Pythagorean theorem, we can calculate the total distance he is now from his original position.

The distance he covered on the west side is the base of the right-angled triangle, so it can be represented by "a" and the distance he covered on the north side is the height of the right-angled triangle, so it can be represented by "b".

Using the formula, c = sqrt(a^2 + b^2), we can find the hypotenuse, which represents the total distance from the original position.

Therefore, c = sqrt(20^2 + 21^2) = sqrt(841) = 29 km

So, Andy is now 29 km away from his original position.

Find the average of the following sequence.

2002、2004、2006、...、2998、3000、3002

Mathematical Equivalence

ON

Answer explanation

To find the average of this sequence, we need to add up all the numbers and divide by the total count. The first term is 2002, the last term is 3002, and the common difference is 2.

So we can use the arithmetic series formula to find the sum of this sequence:

S = (n/2) * (a + l),

where S is the sum of the sequence, n is the number of terms, a is the first term, and l is the last term.

We can find the number of terms by subtracting the first term from the last term, dividing by the common difference, and adding 1:

n = (l - a)/d + 1 = (3002 - 2002)/2 + 1 = 501.

Plugging in the values, we get:

S = (501/2) * (2002 + 3002) = 1,501,501.

Finally, we can find the average by dividing the sum by the number of terms:

average = S/n = 1,501,501/501 = 3,000.

Therefore, the average of the sequence is 3,000.

Find the value of x if

|2y-x+13|+|y+7|=0.

Mathematical Equivalence

ON

Answer explanation

To solve the given equation |2y-x+13|+|y+7|=0, we can use the fact that the sum of two absolute values is always non-negative. Therefore, the only way for the left-hand side to equal zero is if both absolute values are equal to zero. This means that 2y-x+13=0 and y+7=0. Solving these two equations simultaneously, we get x=2y+13 and y=-7. Substituting the value of y in the equation for x, we get x=2*(-7)+13=-1. Therefore, the value of x is -1.

Given a and b are positive real numbers and satisfy equations

a + b = 20 and ab = 4, find the value of a^2 + b^2

Mathematical Equivalence

ON

Answer explanation

We can use the identity (a + b)^2 = a^2 + 2ab + b^2 to express a^2 + b^2 in terms of a and b:

(a + b)^2 = a^2 + 2ab + b^2

Substituting a + b = 20 and ab = 4, we get:

(20)^2 = a^2 + 2(4) + b^2

Simplifying, we get:

400 = a^2 + b^2 + 8

a^2 + b^2 = 392

Therefore, the value of a^2 + b^2 is 392.

Factorise x^2-12x-45

Answer explanation

To factorize x^2-12x-45, we need to find two numbers that multiply to give -45 and add to give -12. These numbers are -15 and 3. Therefore, we can write x^2-12x-45 as (x-15)(x+3).

Mathematical Equivalence

ON

Answer explanation

According to the given inequality, we have: -5 ≤ ((x-7)x²-9)/3 < 3. Simplifying this inequality, we get: -30 ≤ (x-7)x²-9 < 9. Further simplifying, we get: (x-7)x² ≤ 21. Now, we can use trial and error method to find out the integral solution(s) for x that satisfies this inequality. The possible integral values of x can be -4, -3, -2, -1, 0, 1, 2, 3. By substituting these values, we can see that only x = -3 satisfies the inequality. Therefore, there is only one integral solution for x which is x = -3.

Find the value of y if (41-2y)-3(y+7) =0

Mathematical Equivalence

ON

Answer explanation

Expanding the brackets, we get:

41 - 2y - 3y - 21 = 0

Combining like terms, we get:

20 - 5y = 0

Adding 5y to both sides of the equation, we get:

20 = 5y

Dividing both sides of the equation by 5, we get:

y = 4

Therefore, the value of y is 4.