5

Part of 2007 AIME Problems

Problems(2)

Converting Temperatures

Source: AIME I 2007 #5

The formula for converting a Fahrenheit temperature

F

to the corresponding Celsius temperature

C

C=\frac{5}{9}(F-32)

. An integer Fahrenheit temperature is converted to Celsius and rounded to the nearest integer; the resulting integer Celsius temperature is converted back to Fahrenheit and rounded to the nearest integer. For how many integer Fahrenheit temperatures

T

32 \leq T \leq 1000

does the original temperature equal the final temperature?

AMCAIMEnumber theory

Source: AIME II 2007 #5

The graph of the equation

9x+223y=2007

is drawn on graph paper with each square representing one unit in each direction. How many of the

1

1

graph paper squares have interiors lying entirely below the graph and entirely in the first quadrant?

geometryrectanglefloor functionfunctionAMCAIME3D geometry