10/2003

## Theory

x a x b = x (a + b)
x a y a = (xy) a
(x a) b = x (ab)
x (a/b) = bth root of (x a) = ( bth (x) ) a
x (-a) = 1 / x a
x (a - b) = x a / x b

Solutions from the Previous Issue:

1. A train, traveling at constant speed, takes 20 seconds from the time it first enters a tunnel that is 300 meters long until it completely emerges from the tunnel. One of the stationary ceiling lights in the tunnel is directly above the train for 10 seconds. Find the length of the train.

Solution:

300 meters.

Let x be the length of the train. Then the train travels 300 + x meters in 20 seconds. Since the light is above the train for 10 seconds, the train travels x meters in 10 seconds, so it would travel 2x meters in 20 seconds. Thus, 300 + x = 2x, so x = 300 meters.

1. Three cubes, whose edges are 2, 6, and 8 centimeters long, are glued together at their faces. Compute the minimum surface area possible for the resulting figure.

Solution:

536 square centimeters.

Before gluing, the total surface area is 6(4 + 36 + 64) = 6(104), or 624, square centimeters. The minimum surface area is achieved if each cube is glued to both others, as shown. Twice the surface area of one face of the smaller cube is removed in each gluing. Therefore, we must subtract 2(4 + 4 + 36), or 88, square centimeters from 624 square centimeters to get 536 square centimeters.

1. Determine the values of the three different digits d, a, and n given the following:

Solution:

d = 4, a = 9, and n = 5.

Looking at the tens-digit sum, we see that a is either 0 or 9, depending on whether a + n is smaller than 10 in the ones place. However, if a = 0, then n = d according to the ones-place sum.

Since we know that a, d, and n are unequal, we see that a = 9 and a + n is at least 10, as is n + a + 1, "carrying" from the ones place to the tens place. We therefore conclude that d + d + 1 = a = 9, so d = 4. Finally, we see that n = 5, giving us

1. The point (a, b) is reflected over the y-axis to the point (c, d), which is reflected over the x-axis to the point (e, f). Compute ab - ef.

Solution:

0.

When the point (a, b) is reflected about the y-axis, it "lands" at the point (-a, b); and when that point is reflected about the x-axis, it lands on (-a, -b). Therefore, e = -a and f = -b, so ab - ef = ab - ab = 0.

Proposed Exercises:

Simplify the following , where x ≥ 0 and y ≥ 0

1.

2.

3.