It so happens that the triangle whose sides are 4, 5, and 6 has a special property: Its largest angle is twice its smallest angle. Make calculations that support this assertion.

## Pre-Calculus

## Rhombi, Locus of Vertex

and are noncollinear rays (endpoint A). Rhombi are drawn, having one vertex at A, one vertex on , and one vertex on . What is the locus of the fourth vertex?

## Sin x = x/100

How many solutions are there for the equation sin *x* = *x*/100?

## Mongoes Eat Booms

Suppose that each of the following statements is true:

Either flems are tall or mongoes are not dangerous.

Flems are not tall.

Which of the following conclusions must also be true?

- Mongoes eat booms.
- Mongoes do not eat booms.
- Flems are not dangerous.
- Mongoes are dangerous.
- None of these statements is a possible conclusion.

## Subsistence Allowance, Passport

A notice goes up on the company bulletin board:

“All those who have not already received their subsistence allowance this week should report to the Deputy Assistant Bursar (D.A.B.) at 9 am next Monday, unless they also failed to get it last week, in which case they should not report to anybody anywhere, unless they are over 21 and/or have a maternal grandmother who is still alive, in which case they should either report to the Substantive Acting Registrar at 10 pm on Tuesday or to the D.A.B. at 12 noon on Wednesday according to whether their surname begins with a letter in the first half of the alphabet or not, unless their paternal grandmother does not have or did not have a passport, in which case they should report to the Warden as soon as possible, unless they have already passed their driving test and/or do not have a bicycle, in which case they should take a couple of aspirins and go quietly home to bed.”

Ms. Zola Hooberry, age 20, has not received her subsistence allowance this week or last. She has a bicycle but has not passed her driving test. Both her grandmothers are still alive and have passports.

What should she do?

## Slope of Angle Bisector

Consider the line *y* = *x* and the line *y* = 0, both of which go through the origin. They form a 45° angle in the first quadrant. The bisector of this angle is therefore a line with *y* = *mx* for its equation. So what’s the value of *m*? It is tempting to think that it’s ½, but this is *not* the case.

## Cylinders, Pouring Contents

When the contents of a cylinder are poured into a second cylinder whose radius is 2 inches greater, the height reached in the second cylinder is one half of that reached in the first. Find to the nearest tenth of an inch the radii of the two cylinders.

## Grasshopper Jump

A grasshopper jumps from its position at (*x*,*y*) over the point (*a*,*b*) to the point that is diametrically opposite to (*x*,*y*), i.e., the three points are collinear, and the grasshopper’s new position is just as far from (*a*,*b*) as it was before it jumped. What are the coordinates of this new point, in terms of *a*, *b*, *x*, and *y*?

## Volume of a Shell and Solid

Imagine a rectangular solid that is made up of smaller unit cubes. The solid measures *x* by *y* by *z*. Now imagine building a shell of unit cubes around the entire solid.

First question: what is an equation for the number of unit cubes in the shell, in terms of *x*, *y*, and *z*?

Second question: for what *x*, *y*, and *z* is the number of cubes in the rectangular solid equal to the number of cubes in the shell?

## Water Poured Into Liter Cube

Water is poured at a constant rate into a one-liter cube near its corner. The water rises to the halfway mark, stops for a moment, and then continues to rise at exactly half its previous rate. The reason is that inside the cube is a hollow cylindrical can that has been attached to the bottom. What are the height and radius of the can? (Recall that a one-liter cube measures 10 cm on each edge.)