Difference between revisions of "2009 AMC 10A Problems"
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== Problem 1 == | == Problem 1 == | ||
− | One can | + | One can holds <math>12</math> ounces of soda. What is the minimum number of cans needed to provide a gallon (128 ounces) of soda? |
<math> | <math> |
Revision as of 18:41, 4 December 2019
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 See also
Problem 1
One can holds ounces of soda. What is the minimum number of cans needed to provide a gallon (128 ounces) of soda?
Problem 2
Four coins are picked out of a piggy bank that contains a collection of pennies, nickels, dimes and quarters. Which of the following could not be the total value of the four coins, in cents?
Problem 3
Which of the following is equal to ?
Problem 4
Eric plans to compete in a triathlon. He can average miles per hour in the
-mile swim and
miles per hour in the
-mile run. His goal is to finish the triathlon in
hours. To accomplish his goal what must his average speed in miles per hour, be for the
-mile bicycle ride?
Problem 5
What is the sum of the digits of the square of ?
Problem 6
A circle of radius is inscribed in a semicircle, as shown. The area inside the semicircle but outside the circle is shaded. What fraction of the semicircle's area is shaded?
Problem 7
A carton contains milk that is % fat, an amount that is
% less fat than the amount contained in a carton of whole milk. What is the percentage of fat in whole milk?
Problem 8
Three generations of the Wen family are going to the movies, two from each generation. The two members of the youngest generation receive a % discount as children. The two members of the oldest generation receive a
discount as senior citizens. The two members of the middle generation receive no discount. Grandfather Wen, whose senior ticket costs
, is paying for everyone. How many dollars must he pay?
Problem 9
Positive integers ,
, and
, with
, form a geometric sequence with an integer ratio. What is
?
Problem 10
Triangle has a right angle at
. Point
is the foot of the altitude from
,
, and
. What is the area of
?
Problem 11
One dimension of a cube is increased by , another is decreased by
, and the third is left unchanged. The volume of the new rectangular solid is
less than that of the cube. What was the volume of the cube?
Problem 12
In quadrilateral ,
,
,
,
, and
is an integer. What is
?
![[asy] unitsize(4mm); defaultpen(linewidth(.8pt)+fontsize(8pt)); dotfactor=4; pair C=(0,0), B=(17,0); pair D=intersectionpoints(Circle(C,5),Circle(B,13))[0]; pair A=intersectionpoints(Circle(D,9),Circle(B,5))[0]; pair[] dotted={A,B,C,D}; draw(D--A--B--C--D--B); dot(dotted); label("$D$",D,NW); label("$C$",C,W); label("$B$",B,E); label("$A$",A,NE); [/asy]](http://latex.artofproblemsolving.com/5/7/0/570b698fea7c6d66beab04dd6aee44dd24f2b6c1.png)
Problem 13
Suppose that and
. Which of the following is equal to
for every pair of integers
?
Problem 14
Four congruent rectangles are placed as shown. The area of the outer square is times that of the inner square. What is the ratio of the length of the longer side of each rectangle to the length of its shorter side?
![[asy] unitsize(6mm); defaultpen(linewidth(.8pt)); path p=(1,1)--(-2,1)--(-2,2)--(1,2); draw(p); draw(rotate(90)*p); draw(rotate(180)*p); draw(rotate(270)*p); [/asy]](http://latex.artofproblemsolving.com/4/f/8/4f8d08abd119d663ad6630516ebc1b3bd784727e.png)
Problem 15
The figures ,
,
, and
shown are the first in a sequence of figures. For
,
is constructed from
by surrounding it with a square and placing one more diamond on each side of the new square than
had on each side of its outside square. For example, figure
has
diamonds. How many diamonds are there in figure
?
![[asy] unitsize(3mm); defaultpen(linewidth(.8pt)+fontsize(8pt)); path d=(1/2,0)--(0,sqrt(3)/2)--(-1/2,0)--(0,-sqrt(3)/2)--cycle; marker m=marker(scale(5)*d,Fill); path f1=(0,0); path f2=(0,0)--(-1,1)--(1,1)--(1,-1)--(-1,-1); path[] g2=(-1,1)--(-1,-1)--(0,0)^^(1,-1)--(0,0)--(1,1); path f3=f2--(-2,-2)--(-2,0)--(-2,2)--(0,2)--(2,2)--(2,0)--(2,-2)--(0,-2); path[] g3=g2^^(-2,-2)--(0,-2)^^(2,-2)--(1,-1)^^(1,1)--(2,2)^^(-1,1)--(-2,2); path[] f4=f3^^(-3,-3)--(-3,-1)--(-3,1)--(-3,3)--(-1,3)--(1,3)--(3,3)-- (3,1)--(3,-1)--(3,-3)--(1,-3)--(-1,-3); path[] g4=g3^^(-2,-2)--(-3,-3)--(-1,-3)^^(3,-3)--(2,-2)^^(2,2)--(3,3)^^ (-2,2)--(-3,3); draw(f1,m); draw(shift(5,0)*f2,m); draw(shift(5,0)*g2); draw(shift(12,0)*f3,m); draw(shift(12,0)*g3); draw(shift(21,0)*f4,m); draw(shift(21,0)*g4); label("$F_1$",(0,-4)); label("$F_2$",(5,-4)); label("$F_3$",(12,-4)); label("$F_4$",(21,-4)); [/asy]](http://latex.artofproblemsolving.com/d/2/4/d24a6830ee3382e3945990d8af2e9e00ba14c85e.png)
Problem 16
Let ,
,
, and
be real numbers with
,
, and
. What is the sum of all possible values of
?
Problem 17
Rectangle has
and
. Segment
is constructed through
so that
is perpendicular to
, and
and
lie on
and
, respectively. What is
?
Problem 18
At Jefferson Summer Camp, of the children play soccer,
of the children swim, and
of the soccer players swim. To the nearest whole percent, what percent of the non-swimmers play soccer?
Problem 19
Circle has radius
. Circle
has an integer radius
and remains internally tangent to circle
as it rolls once around the circumference of circle
. The two circles have the same points of tangency at the beginning and end of circle
's trip. How many possible values can
have?
Problem 20
Andrea and Lauren are kilometers apart. They bike toward one another with Andrea traveling three times as fast as Lauren, and the distance between them decreasing at a rate of
kilometer per minute. After
minutes, Andrea stops biking because of a flat tire and waits for Lauren. After how many minutes from the time they started to bike does Lauren reach Andrea?
Problem 21
Many Gothic cathedrals have windows with portions containing a ring of congruent circles that are circumscribed by a larger circle. In the figure shown, the number of smaller circles is four. What is the ratio of the sum of the areas of the four smaller circles to the area of the larger circle?
Problem 22
Two cubical dice each have removable numbers through
. The twelve numbers on the two dice are removed, put into a bag, then drawn one at a time and randomly reattached to the faces of the cubes, one number to each face. The dice are then rolled and the numbers on the two top faces are added. What is the probability that the sum is
?
Problem 23
Convex quadrilateral has
and
. Diagonals
and
intersect at
,
, and
and
have equal areas. What is
?
Problem 24
Three distinct vertices of a cube are chosen at random. What is the probability that the plane determined by these three vertices contains points inside the cube?
Problem 25
For , let
, where there are
zeros between the
and the
. Let
be the number of factors of
in the prime factorization of
. What is the maximum value of
?
See also
2009 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by 2008 AMC 10B Problems |
Followed by 2009 AMC 10B Problems | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.