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USA AMC 10 2000

1

In the year , the United States will host the International Mathematical Olympiad. Let , , and be distinct positive integers such that the product . What's the largest possible value of the sum ?

Solution

The sum is the highest if two factors are the lowest. So, 2

and

.

Solution

.

3

Each day, Jenny ate of the jellybeans that were in her jar at the beginning of the day. At the end of the second day, remained. How many jellybeans were in the jar originally?

Solution

;..

.

4

Chandra pays an online service provider a fixed monthly fee plus an hourly charge for connect time. Her December bill was , but in January her bill was because she used twice as much connect time as in December. What is the fixxed monthly fee?

Solution

Let be the fixed fee, and be the amount she pays for the minutes she used in the first month.

We want the fixed fee, which is 5

Points and are the midpoints of sides moves along a line that is parallel to side quantities listed below change? (a) the length of the segment (b) the perimeter of (c) the area of

and of . As , how many of the four

(d) the area of trapezoid

;..

.

Solution (a) Clearly

does not change, and

, so

doesn't

change either.

(b) Obviously, the perimeter changes.

(c) The area clearly doesn't change, as both the base corresponding height remain the same.

and its

(d) The bases and do not change, and neither does the height, so the trapezoid remains the same.

Only quantity changes, so the correct answer is

.

6

The Fibonacci Sequence starts with two 1s and each term afterwards is the sum of its predecessors. Which one of the ten digits is the last to appear in thet units position of a number in the Fibonacci Sequence?

Solution

The pattern of the units digits are

In order of appearance:

.

is the last. 7

In rectangle , , is on , and

. What is the perimeter of ?

and

trisect

;..