In a non-leap year, what is the date of the middle day of the year?

**Solution**: There are 365 days in a non-leap year, so the middle day is day 183. There are 31+28+31+30+31+30=181 days in the first six months; so the middle day is **2**^{nd} July.

Convert 10,000 minutes into days, hours and minutes.

**Solution: **There are 60 minutes in 1 hour, and 24×60=1440 minutes in a day. So there are 6×1440=8640 minutes in 6 days, leaving 10,000-8640=1360 minutes. This is 80 minutes less than 1 day. So 10,000 minutes is equivalent to **6 days, 22 hours and 40 minutes**.

In how many ways can 100 be written as a sum of consecutive positive even numbers?

**Solution**: There are **two **ways to write 100 as a sum of consecutive positive even numbers: 22+24+26+28 and 16+18+20+22+24.

Two numbers have a sum of 2 and a product of -960. What is their difference?

**Solution**: The two numbers are 32 and -30, so their difference is **62**

Two numbers have a sum of 1 and a product of -5/16. What is their difference?

**Solution: **The two numbers are 5/4 and -1/4, so their difference is **3/2 **(or **5**)

How many of the first 100 square numbers are divisible by 4?

**Solution**: If you square an even number (a multiple of 2), the result is a multiple of 2×2=4. If you square an odd number, the result is still odd, so not divisible by 4. Hence, exactly half of the first 100 square numbers are divisible by 4, that is **50 **in total.

A fair six-sided die, with faces numbered 1, 2, 3, 4, 5, 6, is rolled twice. What is the probability that the product of the two numbers rolled is greater than the sum?

**Solution: **There are 6×6=36 possible pairs of numbers that could be rolled. Of these, most have a product greater than their sum. The exceptions are when one or both numbers are equal to 1 (this occurs in 11 pairs), and when both numbers are 2 (this occurs in 1 pair). So there are 36-11-1=24 pairs for which the product is greater than the sum, and the required probability is 24/36=**2/3**

Wanda walks one mile north, then half a mile south, then half of half a mile north, then half of half of half a mile south. How far north or south is she from where she started?

**Solution**: Wanda is 1-1/2+1/4-1/8=8/8-4/8+2/8-1/8=**5/8 of a mile north **of where she started.

Which integer is closest to the cube root of 15?

**Solution: **The cube root of 8 is 2 and the cube root of 27 is 3. So the cube root of 15 is between 2 and 3. To decide which is closer, we can cube 2.5. We have 25x25x25=625×25=15625, so 2.5×2.5×2.5=15.625. Hence the cube root of 15.625 is 2.5. This means that the cube root of 15 is between 2 and 2.5. Therefore the closest integer to the cube root of 15 is **2**.

In a hexagon, what is the greatest possible number of interior right-angles?

**Solution**: Six is impossible – the interior angle sum would then be 540 degrees instead of the required 720 degrees. The answer is **five **(make the hexagon look like a letter L).

Martha thinks of a whole number. She squares it, then adds her original number, ending up with a prime. What number or numbers could she have been thinking of?

**Solution: **Whether Martha begins with an odd number or an even number, the result will be even. But the only even prime is 2. It quickly becomes clear that large numbers (positive and negative) produce results that are also large; an analysis of numbers close to zero yields **1 **and **-2**.

Bananas come in bunches of 3 (small bunch) or 5 (large bunch). Tim wants to purchase exactly 32 bananas; how many different combinations of bunches could he purchase?

*Solution*: The possible bunch combinations are 4 small and 4 large, or 9 small and 1 large, so there are just **two **combinations with a total of 32 bananas

How many three-digit numbers contain the digit 3?

*Solution*: There are 9x10x10=900 three-digit numbers, of which 8x9x9=648 do not contain the digit 3, so the required total is 900-648=**252**

Five angles in a hexagon are 110, 120, 130, 140 and 150 degrees. What is the sixth angle?

*Solution*: The angles in a hexagon sum to 720, so the sixth angle is 720-(110+120+130+140+150)=720-650=**70 degrees**

The reciprocal of a number is 1 divided by that number. For example, the reciprocal of 4 is 1/4, and the reciprocal of 2/5 is 5/2. Find all numbers that are quadruple their reciprocal.

*Solution*: Let x be such a number. Then x = 4/x, so x^2 = 4, so x=2 or -2. So there are two such numbers: **2 **and **-2**.

What is the sum of all the odd numbers between -1000 and 990?

*Solution*: Most of the numbers in the sum cancel each other out; this happens with the numbers between -990 and 990. So the required sum is (-999)+(-997)+(-995)+(-993)+(-991) = **-4975**

Find three positive integers less than 1000, each of which is both a square and a cube.

*Solution*: There are only three such integers; all of them are sixth powers. 1x1x1x1x1x1=1, 2x2x2x2x2x2=64 and 3x3x3x3x3x3=729. **1, 64, 729**

A rectangle has a length 4 more than its width, and a perimeter of 28. What is its area?

**Solution:** The dimensions are 9 and 5, so the area is **45**

A rectangle has a perimeter of 21 and an area of 5. What is its length?

**Solution:** The dimensions are ½ and 10, so its length is **10**

If x is 2/3 of y, and z is 4/5 of x, what fraction of z is y?

**Solution**: z=(4/5)x=(4/5)(2/3)y=(8/15)y. So y=(15/8)z. That is, y is **15/8** of z.

What is the largest three-digit multiple of 6 that is still a multiple of 6 when its digits are reversed?

**Solution**: A number is divisible by 6 precisely when it is even and its digits add to a multiple of 3. So we require the first and last digits to be even. The largest number satisfying the given conditions is **894**.

Find three positive integers, each of which has an equal number of even factors and odd factors.

**Solution**: Any number divisible by 2 but not by 4 will do. The first few examples are 2 (factors are 1,2), 6 (factors are 1, 2, 3, 6) and 10 (factors are 1, 2, 5, 10)

Find three positive integers, each of which has three times as many even factors as odd factors.

**Solution**: Any number divisible by 8 but not by 16 will do. The first few examples are 8 (factors are 1, 2, 4, 8), 24 (factors are 1, 2, 3, 4, 6, 8, 12, 24) and 40 (factors are 1, 2, 4, 5, 8, 10, 20, 40)

The reciprocal of a number is the result of dividing 1 by the number. For example, the reciprocal of 4 is 1/4. What is the reciprocal of 1/2 + 1/3?

**Solution**: 1/2 + 1/3 = 3/6 + 2/6 = 5/6, and the reciprocal of 5/6 is **6/5**

A positive integer (whole number) n is divisible by 14 and 15. How many other positive integers are guaranteed to be factors of n?

**Solution**: Since n is divisible by 14 and 15, it must be divisible by the lowest common multiple of 14 and 15, which is 210. It must also be divisible by all the factors of 210: 1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210. So there are 16 factors of 210, of which we have already counted the factors 14 and 15; hence there are **14 **other positive integers guaranteed to be factors of n

Which negative number is four times as far away from 41 as from 2??

**Solution:-11 **(it differs from 2 by 13 and from 41 by 52)

What is the smallest cube divisible by 6 and 10?

**Solution**: The cube must be divisible by 2, 3 and 5. Further, in a cube, the exact power to which any prime appears in its prime factorisation must be a multiple of 3. So the smallest cube divisible by 6 and 10 is 2 cubed x 3 cubed x 5 cubed = **27000**

Four football teams play each other exactly once, with 3 points for a win, 1 for a draw and 0 for a loss. The teams are then ranked according to the number of points. Teams with the same number of points are awarded the same rank.

1) What is the smallest possible total number of points for the highest ranked team(s)?

2) What is the greatest possible total number of points for the lowest ranked team(s)?

*There are 6 games in total, and in each game, either 2 or 3 points are awarded overall. So the total number of points awarded is between 12 and 18, and the average number of points per team is between 3 and 4.5*

1) **Solution**: The highest ranked team cannot have fewer points than the average number of points per team, so they must score at least 3 points. This can happen is every match is a draw. So the smallest possible number of points for the highest ranked team is **3**

2) **Solution**: The lowest ranked team cannot have more points than the average number of points per team, so they cannot score more than 4 points. This can happen is every team wins one, draws one and loses one of their matches. So the greatest possible number of points for the lowest ranked team is **4**