Class 6 Olympiad math worksheet with questions on geometry, algebra, and number patterns.

Let's solve each question one by one from the given Olympiad worksheet for Class 6.

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(1) Meenakshi introduces Pradip as the son of the brother of her mother. How is Pradip related to Meenakshi?

Step-by-step reasoning:
- Meenakshi’s mother has a brother → That’s Meenakshi’s maternal uncle.
- The son of that brother (the maternal uncle) is Meenakshi’s cousin.
- So, Pradip is Meenakshi’s cousin.

✔ Answer: Pradip is Meenakshi’s cousin.

---

(2) How many more squares need to be shaded to cover 50% of the total area?

We are given a grid. Let's count:

- The grid is 6 rows × 6 columns = 36 squares in total.
- Currently shaded squares: Let's count them.

Looking at the image:
- Row 1: 0
- Row 2: 1 (center)
- Row 3: 1 (center)
- Row 4: 1 (center)
- Row 5: 1 (left), 1 (right) → 2
- Row 6: 1 (left), 1 (right) → 2

Wait — actually, looking carefully:

From the image:
- Row 1: no shading
- Row 2: 1 shaded square (middle)
- Row 3: 1 shaded square (middle)
- Row 4: 1 shaded square (middle)
- Row 5: 2 shaded (left and right)
- Row 6: 2 shaded (left and right)

So total shaded = 1 + 1 + 1 + 2 + 2 = 7 squares

Total squares = 6 × 6 = 36

50% of 36 = 18 squares

So we need 18 - 7 = 11 more squares to be shaded.

✔ Answer: 11

---

(3) If P = 3x² + 3x + 6, Q = 9x² - 7x - 5, R = -3x² + 7x + 8, find (P - Q + R)

We compute:

$$
P - Q + R = (3x^2 + 3x + 6) - (9x^2 - 7x - 5) + (-3x^2 + 7x + 8)
$$

Step-by-step:

First, expand the subtraction:

$$
= 3x^2 + 3x + 6 - 9x^2 + 7x + 5 - 3x^2 + 7x + 8
$$

Now combine like terms:

- $ x^2 $: $ 3 - 9 - 3 = -9x^2 $
- $ x $: $ 3 + 7 + 7 = 17x $
- Constants: $ 6 + 5 + 8 = 19 $

So,

$$
P - Q + R = -9x^2 + 17x + 19
$$

✔ Answer: $ \boxed{-9x^2 + 17x + 19} $

---

(4) If the number in the center is the sum of all other numbers, find value of 'a'

Given a cross-shaped arrangement:

```
a 3
13
5 1
```

The center is 13, and it says: center = sum of all other numbers

So:

$$
13 = a + 3 + 5 + 1
$$

$$
13 = a + 9
\Rightarrow a = 13 - 9 = 4
$$

✔ Answer: $ \boxed{4} $

---

(5) Find the missing number

We have this pattern:

```
27 24 28
15 ? 24 4 25 33 24 8 38
35 28 36
```

It seems like three separate groups of 3×3 grids or sequences?

But let's look at the structure:

There are three vertical columns of numbers, each with a top, middle, and bottom number.

Let’s label them:

#### Column 1:
- Top: 27
- Middle: 15, ?, 24 → Wait, maybe not.

Actually, it looks like:

Each "block" has:
- Top number
- Middle row: two numbers on left and right, and a center?
Wait, better to see the pattern.

Looking closely:

It appears there are three separate 3×3 grids, but only some values are shown.

Alternatively, perhaps each group is a 3-number sequence.

Wait — let's re-analyze:

It seems like three separate triangular patterns?

No, better idea: Look at the sums or relationships.

Let’s try this:

There are three sets of numbers:

#### Set 1:
- Top: 27
- Middle: 15, ?, 24
- Bottom: 35

#### Set 2:
- Top: 24
- Middle: 25, 4, 33
- Bottom: 28

#### Set 3:
- Top: 28
- Middle: 24, 8, 38
- Bottom: 36

Wait — perhaps the middle row is three numbers: left, center, right.

But in the diagram:

```
27 24 28
15 ? 25 4 24 8 38
35 28 36
```

So it's three separate columns:

- Column A: 27 (top), 15 and ? (middle), 35 (bottom)
- Column B: 24 (top), 25, 4, 33 (middle), 28 (bottom)
- Column C: 28 (top), 24, 8, 38 (middle), 36 (bottom)

Wait — maybe each column is a 3-row block?

But in the middle row, there are three numbers per block.

Ah! It might be that each block is a 3×3 grid, but only some entries are visible.

But the layout suggests:

- Each block has:
- Top: one number
- Middle: three numbers (left, center, right)
- Bottom: one number

And possibly the sum of top + bottom = sum of middle row?

Let’s test.

#### Block 1:
- Top: 27
- Bottom: 35
- Sum: 27 + 35 = 62
- Middle: 15, ?, 24 → sum = 15 + ? + 24 = 39 + ?
Set equal: 39 + ? = 62 → ? = 23 → Not among options.

But options are 6, 7, 9, 8 → so probably not.

Alternative idea: Maybe the center number is derived from others.

Wait — look at Block 2:

Middle: 25, 4, 33 → sum = 62
Top: 24, Bottom: 28 → sum = 52 → no

Try another idea: Perhaps the product or difference?

Wait — look at the numbers around the missing one.

In first block:

- Top: 27
- Left: 15
- Right: 24
- Bottom: 35

Missing is in the middle.

Is there a pattern?

Maybe the middle number is such that:

Sum of top and bottom = sum of left and right + middle?

Try:

For Block 1: 27 + 35 = 62
Left + right = 15 + 24 = 39
Then middle = 62 - 39 = 23 → again not matching options.

Wait — maybe the middle number is the average?

Or perhaps the missing number is part of a pattern across blocks.

Let’s look at the middle row across all blocks:

Block 1: 15, ?, 24
Block 2: 25, 4, 33
Block 3: 24, 8, 38

Look at the center numbers: ?, 4, 8 → increasing by 4? Then ? = 0? No.

Wait — look at top, bottom, and center.

Another idea: In each block, the sum of the outer numbers equals the center?

Wait — in Block 2: middle row is 25, 4, 33 → maybe the center is 4, and 25 + 33 = 58 → not related.

Wait — look at Block 2:

- Top: 24
- Middle: 25, 4, 33
- Bottom: 28

Check: 25 + 33 = 58, 24 + 28 = 52 → not helpful.

Wait — what if the middle number is the difference?

Try: 25 - 4 = 21, 33 - 4 = 29 → no.

Another idea: Look at the first block:

- Top: 27
- Left: 15
- Right: 24
- Bottom: 35

What if: 27 + 35 = 62, 15 + 24 = 39 → 62 - 39 = 23 → still not helpful.

Wait — maybe the missing number is between 15 and 24?

But 15 and 24 are on the same row.

Wait — perhaps the middle number is the average?

(15 + 24)/2 = 19.5 → not integer.

Wait — maybe it's a pattern across columns.

Look at the first numbers of each block:

- Block 1: 15, 27, 35
- Block 2: 25, 24, 28
- Block 3: 24, 28, 36

Not clear.

Wait — look at the last block:

- Top: 28
- Middle: 24, 8, 38
- Bottom: 36

Notice: 24 + 38 = 62, 28 + 36 = 64 → close.

Wait — try: 28 + 36 = 64, 24 + 38 = 62 → no.

Wait — maybe the center number is related.

In Block 2: center is 4
In Block 3: center is 8

Now, look at the top and bottom:

Block 2: top=24, bottom=28 → sum=52
Block 3: top=28, bottom=36 → sum=64

Center: 4 and 8 → doubling? 4 → 8

But Block 1: top=27, bottom=35 → sum=62 → if pattern, center should be 12? But options are small.

Wait — maybe the missing number is 7?

Let’s think differently.

Perhaps the sum of the four corners of each block equals something?

Wait — maybe it's a magic square style.

Wait — another idea: Look at Block 2:

Numbers: 24 (top), 25, 4, 33 (middle), 28 (bottom)

Can we see: 25 + 33 = 58, 24 + 28 = 52 → no.

Wait — 25 - 24 = 1, 33 - 28 = 5 → not helpful.

Wait — look at Block 3:

- Top: 28
- Middle: 24, 8, 38
- Bottom: 36

Try: 24 + 38 = 62, 28 + 36 = 64 → difference of 2.

Block 2: 25 + 33 = 58, 24 + 28 = 52 → diff 6.

No.

Wait — maybe the middle number is the average of top and bottom?

Block 2: (24+28)/2 = 26 → but center is 4 → no.

Wait — maybe the number in the middle is not the center of the row, but rather a separate cell?

Wait — the layout is:

```
27 24 28
15 ? 25 4 24 8 38
35 28 36
```

So it's three separate columns, each with:

- Top: one number
- Middle: three numbers (left, center, right)
- Bottom: one number

So each column has 5 numbers.

Let’s consider each column separately.

#### Column 1:
- Top: 27
- Middle: 15, ?, 24
- Bottom: 35

Is there a relationship?

Try: 27 + 35 = 62
15 + 24 = 39
62 - 39 = 23 → so missing number = 23? But not in options.

Wait — maybe the sum of top and bottom equals sum of left and right?

27 + 35 = 62
15 + 24 = 39 → not equal.

Wait — maybe the middle number is the difference?

27 - 15 = 12, 24 - 15 = 9 → no.

Another idea: Maybe the number in the middle is the average of top and bottom?

(27 + 35)/2 = 31 → not matching.

Wait — look at Column 2:

- Top: 24
- Middle: 25, 4, 33
- Bottom: 28

Now check: 25 + 33 = 58
24 + 28 = 52 → not equal.

But 25 - 24 = 1, 33 - 28 = 5 → no.

Wait — notice: 25 + 33 = 58, 24 + 28 = 52 → difference 6.

Column 3: 24 + 38 = 62, 28 + 36 = 64 → difference 2.

No.

Wait — maybe the center number is key.

In Column 2: center is 4
In Column 3: center is 8

Now, look at the left and right numbers:

Col 2: left=25, right=33 → sum=58
Col 3: left=24, right=38 → sum=62

But centers: 4 and 8 → ratio 2:1

Now Col 1: left=15, right=24 → sum=39 → if center is x, then maybe x = ?

If pattern: center = (right - left)/something?

Col 2: 33 - 25 = 8, center = 4 → half
Col 3: 38 - 24 = 14, center = 8 → not half

No.

Wait — Col 2: 33 - 25 = 8, center = 4 → half
Col 3: 38 - 24 = 14, center = 8 → not half

No.

Wait — maybe the center number is the average of the top and bottom?

Col 2: (24 + 28)/2 = 26 → but center is 4 → no.

Wait — perhaps the number in the center is the difference between top and bottom?

28 - 24 = 4 → yes! Center is 4.

Col 3: top=28, bottom=36 → 36 - 28 = 8 → center is 8 → matches!

Oh! So center = bottom - top

For Col 2: 28 - 24 = 4 → correct
For Col 3: 36 - 28 = 8 → correct

So for Col 1: center = bottom - top = 35 - 27 = 8

So missing number is 8

✔ Answer: 8 → Option d. 8

---

(6) What will be the shape at 13th position?

Sequence of shapes:

Given:

```
● □ ◆ ◇ ● □ ◆ ◇ ● □ ◆ ◇ ...
```

Wait — let's list them:

Position: 1 2 3 4 5 6 7 8 9 10 11 12 13 ...

Shapes:
1. ● (circle)
2. □ (square)
3. ◆ (diamond)
4. ◇ (rhombus?) — wait, looks like a diamond too.

Wait — in the image:

It shows:
- Red circle
- Blue square
- Purple diamond
- Gray hexagon

Wait — let's look at the actual sequence:

From the image:

Sequence:
1. ● (red circle)
2. □ (blue square)
3. ◆ (purple diamond)
4. ◇ (gray hexagon)
5. ● (red circle)
6. □ (blue square)
7. ◆ (purple diamond)
8. ◇ (gray hexagon)
9. ● (red circle)
10. □ (blue square)
11. ◆ (purple diamond)
12. ◇ (gray hexagon)
13. ?

So it's repeating every 4 shapes:

Pattern: ●, □, ◆, ◇

Cycle length = 4

To find 13th position:

Divide 13 by 4: 13 ÷ 4 = 3 remainder 1

Remainder 1 → corresponds to the 1st shape in the cycle → ● (circle)

But wait — option a is gray hexagon (◇), b is red circle (●), c is blue square (□), d is purple diamond (◆)

So 13th is ● → red circle → option b

✔ Answer: b. red circle

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✔ Final Answers:

1. Cousin
2. 11
3. $ \boxed{-9x^2 + 17x + 19} $
4. 4
5. d. 8
6. b. red circle

Let me know if you want explanations in simpler language!