In the following picture, rectangle ABCD is inscribed in a quarter-circle. $DC = 5$, and $CE = 4$. Can you figure out the length of diagonal AC?
Often a problem that can seem particularly perplexing can be solved by looking at it in a different way. Sometimes the best way to keep track of the different information and variables in a problem is to draw a picture of some sort, to visualize the information so that it is in a form that is easier to understand. You saw this last year as you solved a variety of problems, at times by constructing models, and at times by finding clever ways to visually represent things that at first seemed quite nonvisual.
Four trees are planted at each corner of a square park. The city wants to expand the park to twice its current area, but in such a way that the park is still a square, and none of the four trees is in the interior of the park. (The trees cannot be transplanted.) Draw a plan for a new park that meets these criteria.
The shaded parts of the spheres below are hemispheres.
You throw three darts onto the surface of a globe, each from a randomly chosen direction. What is the probability that all three darts lie in one hemisphere?
The circles in the diagram each have radius 1 cm, are tangent to each other and also to the square PARK. Their centers are on the line PR. Find the area of the square PARK.
There are many different triangles with side lengths 12 and 13 cm. Which of these triangles has the greatest area?
A rectangle and a square are inscribed in congruent circles. The rectangle has a width of 6 and a length of 8. What is the area of the square?
Three identical, spherical oranges are placed in a bin as part of a supermarket display. The bin is exactly long and wide enough to have two oranges fit snugly in the bottom, but there’s plenty of room to place the third orange on top of these two. If the radius of an orange is 2 inches, find the height of this small stack.
A street has parallel curbs 40 ft apart. A crosswalk bounded by two parallel stripes crosses the street at an angle. The length of the curb between the stripes is 15 feet, and each stripe is 50 feet long. Find the distance between the stripes.
Let A and B be any two points in a plane.
How many different circles can you draw that go through points A and B? Can you give the radius of the smallest possible circle? Of the biggest?
How many different rectangles can you draw with opposite vertices on points A and B?
PQ and QR are diagonals of two faces of a cube. Find the measure of $\angle PQR$.
A trapezoid is inscribed in a circle of radius 5 cm so that one base is a diameter of the circle, and the other base has length 5 cm. What is the perimeter of the trapezoid?
If you start with $\frac{1}{2}$ then add $\frac{1}{4}$, then $\frac{1}{8}$, then $\frac{1}{{16}}$, and so on, ad infinitum, what do you suppose the answer would be? Draw a diagram that would justify your response.
How many sides does a cube have? How about a pyramid?
Can you build a closed 3-dimensional shape out of 4 flat sides? How about out of 3 flat sides? Give examples, or explain why not.
Suppose you have a box that has a base of 1 inch by 5 inches and that stands 8 inches tall. How many ½ inch radius spherical balls can you get into this box if you can’t let any ball protrude above the top of the box?
Craziola, the wacky pizza guy, has decided he wants to cut a pizza into as many pieces as possible, with as few straight cuts as possible. He doesn’t care at all if the pieces are of equal size, he just wants to make the most number of distinct pieces. With 1 cut, he produces 2 pieces. With 2 cuts, he creates a maximum of 4 pieces, no matter how crazy the 2 cuts he makes are. How many pieces can he possibly make with 3 cuts? 4 cuts? What about $n$ cuts? Can you find the pattern?
Is it possible to arrange six unsharpened pencils so that they all touch each other?
Mr. Shimano gives an extremely difficult Japanese test. The highest score was 74% and the lowest 31%. Rather than give a retest, Mr. Shimano decides to raise the 74% to 93% and the 31% to 61%. For a student who scored 57 on the original test, what score do you think Mr. Shimano should give him after the adjustment?
A circle is sitting inside an equilateral triangle so that it’s tangent to the sides of the triangle in three places. Another equilateral triangle is inside the circle with its three vertices on the circle. If the length of a side of the smaller triangle is 1 cm, find the length of a side of the bigger triangle.
Two sides of a triangle measure 6 and 11 cm respectively. If the length of the third side is also an integer, what possible lengths can the third side have?
What is the maximum number of acute angles a convex polygon can have?
A hungry spider and a fly are in a room 30 feet long, 12 feet wide, and 12 feet high. The spider is on one of the smaller walls, 6 feet from each side and 1 foot from the ceiling. The fly is on the opposite wall, 6 feet from each side and 1 foot from the floor. Assume that the fly does not move (it is paralyzed by fear!) and find the shortest path that the spider can take to eat the fly.
