Two congruent squares are placed so that their overlap is a regular octagon. The squares each have area 1 sq. meter.
What is the area of the octagon?
Pre-Calculus
City Hall Mouse, Numerical
City Hall has a splendid rectangular black-and-white tiled floor. The tiles are squares. There are 93 of them in one direction and 231 in the other. A mouse runs in a straight line diagonally from one corner of the floor to the opposite corner. How many tiles does it cross?
Trapezoid, Find Angle Measure
Here is a nice trapezoid, ABCD, with AB parallel to CD. Diagonal BD is drawn, and BD = AD. ∠DCB = 110° and ∠CBD = 30°. What is ∠ADB?
- 80°
- 90°
- 100°
- 110°
- 120°
Sum of Interior Angles
The sum of all but one of the interior angles of a convex polygon equals 2570°. The remaining angle is:
- 90°
- 105°
- 120°
- 130°
- 144°
Diagonals of Two Regular Polygons
The total number of angles in two regular polygons is 13, and the total number of diagonals is 25. How many angles are in each polygon?
Find Altitude, Given Sides
In triangle ABC, a = 7, b = 8, and c = 9. Find the altitude of the triangle to side c.
Width of Alley
In a narrow, sinister alley of width w a ladder of length a is placed with its foot at point P between the walls. Resting against one wall at Q, a distance k above the ground, the ladder makes a 45° angle with the ground. Resting against the other wall at R, a distance h above the ground, the ladder makes a 75° angle with the ground. The width of the alley is equal to
- a
- RQ
- k
- (h + k)/2
- h
Seventeen People, Love, Etc.
Seventeen intelligent people all correspond with one another. Each person writes “letters” to each of the 16 others (emails, texts, facebook messages, carrier pigeons, whatever). In their letters, one of only three topics is discussed: love, death, or dragons. Each pair of correspondents always writes about the same one of these three topics. Prove that there is a group of at least three people who write to each other about the same topic.
Walking to Make Triangle
Nigel walks x miles due west, turns 150° to his left and walks 3 miles in the new direction. If he finishes at a point 2 miles from his starting point, then x is:
- 13/10
- 39/10
- 1.5√3
- not uniquely determined by the given information.
Sums of Series
The initial term of an arithemetic series is 1. If the sum of the first twenty terms is four times the sum of the first twelve terms, then the second term is:
