Difference between revisions of "1997 AHSME Problems"
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==Problem 20== | ==Problem 20== | ||
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+ | Which one of the following integers can be expressed as the sum of <math>100</math> consecutive positive integers? | ||
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+ | <math> \textbf{(A)}\ 1,\!627,\!384,\!950\qquad\textbf{(B)}\ 2,\!345,\!678,\!910\qquad\textbf{(C)}\ 3,\!579,\!111,\!300\qquad\textbf{(D)}\ 4,\!692,\!581,\!470\qquad\textbf{(E)}\ 5,\!815,\!937,\!260 </math> | ||
[[1997 AHSME Problems/Problem 20|Solution]] | [[1997 AHSME Problems/Problem 20|Solution]] |
Revision as of 10:07, 9 August 2011
Contents
- 1 Problem 1
- 2 Problem 2
- 3 Problem 3
- 4 Problem 4
- 5 Problem 5
- 6 Problem 6
- 7 Problem 7
- 8 Problem 8
- 9 Problem 9
- 10 Problem 10
- 11 Problem 11
- 12 Problem 12
- 13 Problem 13
- 14 Problem 14
- 15 Problem 15
- 16 Problem 16
- 17 Problem 17
- 18 Problem 18
- 19 Problem 19
- 20 Problem 20
- 21 Problem 21
- 22 Problem 22
- 23 Problem 23
- 24 Problem 24
- 25 Problem 25
- 26 Problem 26
- 27 Problem 27
- 28 Problem 28
- 29 Problem 29
Problem 1
If and
are digits for which
$\begin{tabular}{ccc}& 2 & a\\ \times & b & 3\\ \hline & 6 & 9\\ 9 & 2\\ \hline 9 & 8 & 9\end{tabular}$ (Error compiling LaTeX. Unknown error_msg)
then
Problem 2
The adjacent sides of the decagon shown meet at right angles. What is its perimeter?
Problem 3
If ,
, and
are real numbers such that
![$(x-3)^2 + (y-4)^2 + (z-5)^2 = 0$](http://latex.artofproblemsolving.com/b/5/a/b5a9f6e5a95f76eda476bf20df06fc2bdb1e8fc9.png)
then
Problem 4
If is
larger than
, and
is
larger than
, then
is what percent larger than
?
Problem 5
A rectangle with perimeter is divided into five congruent rectangles as shown in the diagram. What is the perimeter of one of the five congruent rectangles?
Problem 6
Consider the sequence
whose th term is
. What is the average of the first
terms of the sequence?
Problem 7
The sum of seven integers is . What is the maximum number of the seven integers that can be larger than
?
Problem 8
Mientka Publishing Company prices its bestseller Where's Walter? as follows:
where is the number of books ordered, and
is the cost in dollars of
books. Notice that
books cost less than
books. For how many values of
is it cheaper to buy more than
books than to buy exactly
books?
Problem 9
In the figure, is a
square,
is the midpoint of
, and
is on
. If
is perpendicular to
, then the area of quadrilateral
is
Problem 10
Two six-sided dice are fair in the sense that each face is equally likely to turn up. However, one of the dice has the replaced by
and the other die has the
replaced by
. When these dice are rolled, what is the probability that the sum is an odd number?
Problem 11
In the sixth, seventh, eighth, and ninth basketball games of the season, a player scored ,
,
, and
points, respectively. Her points-per-game average was higher after nine games than it was after the first five games. If her average after ten games was greater than
, what is the least number of points she could have scored in the tenth game?
Problem 12
If and
are real numbers and
, then the line whose equation is
cannot contain the point
Problem 13
How many two-digit positive integers have the property that the sum of
and the number obtained by reversing the order of the digits of is a perfect square?
Problem 14
The number of geese in a flock increases so that the difference between the populations in year and year
is directly proportional to the population in year
. If the populations in the years
,
, and
were
,
, and
, respectively, then the population in
was
Problem 15
Medians and
of triangle
are perpendicular,
, and
. The area of triangle
is
Problem 16
The three row sums and the three column sums of the array
are the same. What is the least number of entries that must be altered to make all six sums different from one another?
Problem 17
A line intersects the graph of
and the graph of
. The distance between the points of intersection is
. Given that
, where
and
are integers, what is
?
Problem 18
A list of integers has mode and mean
. The smallest number in the list is
. The median
of the list is a member of the list. If the list member
were replaced by
, the mean and median of the new list would be
and
, respectively. If were
instead replaced by
, the median of the new list would be
. What is
?
Problem 19
A circle with center is tangent to the coordinate axes and to the hypotenuse of the
-
-
triangle
as shown, where
. To the nearest hundredth, what is the radius of the circle?
Problem 20
Which one of the following integers can be expressed as the sum of consecutive positive integers?