OSSM Middle School Mathematics Contest:

An Awesome Contest

Solutions

1. In how many years after 2003 will the number representing the year be a perfect square?

The square root of 2003 is 44.7548… The least integer above this is 45, and 45 squared is 2025. So, in 2025 – 2003 = 22 years, the year will be a perfect square.

2. A sector (pie-shaped piece) of a circle graph representing 40% of the data should have a central angle of what degree measure?

The total central angle in a circle is 360 degrees. 40% of this is equivalent to 40/100 = .4 of 360. The degree measure of the sector is therefore .4 x 360 = 144 degrees.

3. How many cubic feet are there in ten boxes if each box is a cube measuring six feet on a side?

The volume of one of these cubes is length3 = 63 = 216 cubic feet. There are ten identical boxes, so the total volume is 216 x 10 = 2160 cubic feet.

4. Mary has an average of 84% on five math tests. What score must she make on the next test to raise her average to 86%?

Let’s call her first five test scores a, b, c, d, and e. The average of these is given by:

 

and the sum of the scores is:

 

Now consider her sixth test grade f. Now her average test grade is:

 

 

We can group the first five variables in the numerator, since we already found their sum, and then solve for f:

 

 

Mary must get a 96 percent on her last test to have an average test grade of 86.

5. What is the least positive integer, n, for which ( 10π – 1 )( 10 π – 2 )( 10 π – 3 ) …            ( 10 π – n ) < 0?

The first few terms in this series are decreasing positive numbers (for example, 10 π – 1 = 30.4159, 10 π – 2 = 29.4159, 10 π – 3 = 28.4159,). For the product to be negative, an odd number of the factors must be negative. We can find the first n to make a term negative with (10 π – n ) = 0, so n = 31.4159. Any n greater than this will work, and since it must be an integer, n is 32.  

6. What is the largest possible area of a rectangle with integer sides and a perimeter of 26?

Let’s call the dimensions of the rectangle x and y. Then the perimeter = 26 = 2x + 2y. Since x + y = 13 and both are integers, the choices are 0 and 13, 1 and 12, 2 and 11 … 6 and 7. Testing the first few choices shows that area increases as x and y become closer together, so one side length is 6 and the other is 7. The area is then 6 x 7 = 42.

7. A cone has radius 1 inch and height 4 inches. What is the number of inches in the radius of a sphere of ice cream which has the same volume as the cone?  Vcone = 1/3 π r2 h,  V­sphere = 4/3 π r3.

First let’s calculate the volume of the cone. Substituting 1 for r and 4 for h in the given equation, we have V = 1/3 π x 12 x 4 = 4/3 Œ in3. To find the radius of a sphere with the same volume, we can substitute this value in the second equation. Then we get:

The radius of the sphere must be 1 inch.

  8. After being painted, a solid wooden cube whose edge is 4 cm is cut into 64 small 1-cm cubes. How many of these small cubes have exactly two painted faces?

In the four by four large cube, the corner pieces have three sides painted, the inner edge pieces have two, and the center pieces have one, as shown in this illustration:

  We can see that in the top and bottom levels there are eight pieces with two painted faces, and four in the middle two layers. Altogether, there are 24 cubes with exactly two painted faces.

9. If one pair of opposite sides of a rectangle is increased by 20% and the other pair of opposites sides is decreased by 20%, by what percent does the area of the rectangle increase or decrease (indicate which)?

Lets call the width of the rectangle x and the length y. The original area of the rectangle is then A = xy. Suppose x is increased by 20% and y is decreased. The new width is 1.20x and the new length is 0.80y. The new area is then A = (1.20x)(0.80y) = 0.96xy. 0.96xy – xy = -0.04xy.  -0.04xy / xy = -0.04, so the area changed by a 4% decrease.

10. What is the 50th digit in the decimal expansion of 1/7?

First evaluate the first few digits of 1/7: 0.142857142857… We can quickly see a repeating pattern of the six digits 142857. Numbering the digits, every multiple of six falls on a seven. 50 is 6 x 8 + 2, and two digits past any 7 in the sequence is 4.

11. What is the measure in degrees of the angle formed by the minute and hour hands of a clock at 2:20?  

First let’s determine the position of the minute hand. 20 is one third past the hour, so it is exactly 360/3 = 120 degrees past vertical. The hour hand should also be one third past the current hour. In this case it is one third of the way between the 2 and the 3. 21/3 / 12 = 7/36, so the hour hand should be 7/36 x 360 = 70 degrees past vertical. The difference between these two locations is 120 – 70 = 50 degrees.

12. How many three-digit numbers are divisible by both 2 and 3?

For a number to be divisible by both 2 and 3 it must be divisible by 2 x 3 = 6. Between 100 and 999 there are 900 numbers, and 1 in 6 includes 150 of them. So, there are 150 three-digit numbers divisible by 2 and 3.  

13. How many three-digits numbers are divisible by either 2 or 3?

Consider the first six natural numbers. We can see that only two out of six consecutive integers, 1 and 5 in this case, are not divisible by 2 or 3. There are 900 numbers between 100 and 999, and 4 out of 6 includes 600 of them. So, There are 600 three-digit numbers divisible by either 2 or 3.  

14. Which is larger, 2150 or 3100?

This question can be approached more easily if rephrased: Which is larger, an or bn, which amounts to which is larger, a or b.  2150 = (23)50,  3100 = (32)50.  850 < 950 so 2150 < 3100.

15. A factorial is denoted by n! and means (n)(n-1)(n-2)(n-3)…(3)(2)(1). How many zeroes are there at the end of 100!?

First we must find what combinations of numbers produce zeroes at the end of the number when multiplied. By factoring the first numbers like this, 10 = 2 x 5, 100 = 4 x 25 and 1000 = 8 x 125, we can see that zeroes result from powers of two multiplied by the same power of five (the number of zeroes produced is the power that 2 and 5 is raised to). There are far more multiples of two than five to make combinations of this type, so simply counting the multiples of powers of five from 1 to 100 gives the total number of zeroes on the product. There are 100/5 = 20 multiples of five, and 100 / 25 = 4 of those are also multiples of 52, so twenty-four zeroes appear on the end of 100!.

16.  There are 101 red marbles and 101 black marbles in a box.  Let Ps be the probability that two marbles drawn at random (without replacement) are the same color, and let Pd be the probability that they are different colors.  Find Ps/Pd as a fraction in lowest terms.

The probability of drawing a red marble first is 101/202.  The probability of then drawing a second red marble is 100/201.  The same

probabilities hold for drawing two black marbles.  So . 

The probability of drawing a red first (or a black first) is 101/202 and the probability of then drawing one of the other color is 101/201. 

So .   Therefore   .