Download - 2010 AMC 10A Problems
La AMC – 10 participa elevi din clasele a IX-a si a X-a, iar la AMC – 12, elevi
din clasele a XI-a si a XII-a. Concursul s-a desfasurat in 9 februarie, la
Universitatea de Nord Baia Mare, in organizarea Centrului de Pregatire a
Concursurilor :colare de la Departamentul de Matematica si Informatica, in
colaborare cu Inspectoratul Scolar Judetean (inspector de specialitate
profesor Gheorge Maiorescu).
In acest an au fost admisi, pe baza rezultatelor la concursurile de matematica
din anul precedent si a rezultatelor din anii anteriori la concursurile AMC, 50
de elevi din clasele VIII-XII, provenind de la mai multe licee din judet.
Profesor dr. Vasile Berinde a reamintit faptul ca fiecare dintre competitii cere
rezolvarea in 75 minute a 25 de probleme tip grila, formulate in limba
engleza. La fiecare problema elevii trebuie sa aleaga raspunsul corect din 5
variante posibile. Pentru fiecare raspuns corect se acorda cate 6 puncte,
pentru fiecare problema la care nu a fost ales un raspuns se acorda 2,5
puncte, iar pentru un raspuns gresit se acorda 0 puncte. Un elev poate obtine
asadar cel mult 150 puncte.
2010 AMC 10A ProblemsProblem 1
Mary’s top book shelf holds five books with the following widths, in centimeters: , ,
, , and . What is the average book width, in centimeters?
SolutionTo find the average, we add up the widths , , , , and , to get a total sum of .
Since there are books, the average book width is The answer is .
Problem 2
Four identical squares and one rectangle are placed together to form one large
square as shown. The length of the rectangle is how many times as large as its width?
Solution
Let the length of the small square be , intuitively, the length of the big square is .
It can be seen that the width of the rectangle is . Thus, the length of the rectangle is times large as the width. The answer is .
Problem 3
Tyrone had marbles and Eric had marbles. Tyrone then gave some of his marbles
to Eric so that Tyrone ended with twice as many marbles as Eric. How many marbles
did Tyrone give to Eric?
Solution
Let be the number of marbles Tyrone gave to Eric. Then, . Solving for yields and . The answer is .
Problem 4
A book that is to be recorded onto compact discs takes minutes to read aloud.
Each disc can hold up to minutes of reading. Assume that the smallest possible
number of discs is used and that each disc contains the same length of reading. How
many minutes of reading will each disc contain?
Solution
Assuming that there were fractions of compact discs, it would take CDs
to have equal reading time. However, since the number of discs can only be a whole number, there are at least 8 CDs, in which case it would have minutes on each of the 8 discs. The answer is
Problem 5
The area of a circle whose circumference is is . What is the value of ?
Solution
If the circumference of a circle is , the radius would be . Since the area of a circle is , the area is . The answer is .
Problem 6
For positive numbers and the operation is defined as
What is ?
Solution. Then, is The answer is
Problem 7
Crystal has a running course marked out for her daily run. She starts this run by
heading due north for one mile. She then runs northeast for one mile, then southeast
for one mile. The last portion of her run takes her on a straight line back to where she
started. How far, in miles is this last portion of her run?
Solution
Crystal first runs North for one mile. Changing directions, she runs Northeast for
another mile. The angle difference between North and Northeast is 45 degrees. She
then switches directions to Southeast, meaning a 90 degree angle change. The distance now from travelling North for one mile, and her current destination is
miles, because it is the hypotenuse of a 45-45-90 triangle with side length one (mile). Therefore, Crystal's distance from her starting position, x, is equal to ,
which is equal to . The answer is
Problem 8
Tony works hours a day and is paid $ per hour for each full year of his age.
During a six month period Tony worked days and earned $ . How old was Tony at
the end of the six month period?
Solution
Tony worked hours a day and is paid dollars per hour for each full year of his
age. This basically says that he gets a dollar for each year of his age. So if he is
years old, he gets dollars a day. We also know that he worked days and
earned dollars. If he was years old at the beginning of his working period, he
would have earned dollars. If he was years old at the beginning of his
working period, he would have earned dollars. Because he earned
dollars, we know that he was for some period of time, but not the whole time,
because then the money earned would be greater than or equal to . This is why he
was when he began, but turned sometime in the middle and earned dollars in
total. So the answer is .The answer is . We could find out for how long he was
and . . Then is and we know that he was for days,
and for days. Thus, the answer is .
Problem 9
A palindrome, such as , is a number that remains the same when its digits are
reversed. The numbers and are three-digit and four-digit palindromes,
respectively. What is the sum of the digits of ?
Solution
is at most , so is at most . The minimum value of is .
However, the only palindrome between and is , which means that
must be .
It follows that is , so the sum of the digits is .
See also
(E) 2017
There are 365 days in a non-leap year. There are 7 days in a week. Since 365 = 52 * 7
+ 1 (or 365 is congruent to 1 mod 7), the same date (after February) moves "forward"
one day in the subsequent year, if that year is not a leap year.
For example: 5/27/08 Tue 5/27/09 Wed
However, a leap year has 366 days, and 366 = 52 * 7 + 2. So the same date (after
February) moves "forward"two days in the subsequent year, if that year is a leap
year.
For example: 5/27/11 Fri 5/27/12 Sun
You can keep count forward to find that the first time this date falls on a Saturday is in
2017:
-------------------------------------------------------------------------------
5/27/13 Mon 5/27/14 Tue 5/27/15 Wed 5/27/16 Fri 5/27/17 Sat
Since we are given the range of the solutions, we must re-write the inequalities so
that we have in terms of and .
Subtract from all of the quantities:
Divide all of the quantities by .
Since we have the range of the solutions, we can make them equal to .
Multiply both sides by 2.
Re-write without using parentheses.
Simplify.
We need to find for the problem, so the answer is
Problem 12
Logan is constructing a scaled model of his town. The city's water tower stands 40
meters high, and the top portion is a sphere that holds 100,000 liters of water.
Logan's miniature water tower holds 0.1 liters. How tall, in meters, should Logan
make his tower?
SolutionThe water tower holds times more water than Logan's miniature.
Therefore, Logan should make his tower times shorter than the actual tower. This is meters high, or choice .
See also
Problem13
Angelina drove at an average rate of kph and then stopped minutes for gas.
After the stop, she drove at an average rate of kph. Altogether she drove km in
a total trip time of hours including the stop. Which equation could be used to solve
for the time in hours that she drove before her stop?
The answer is because she drove at kmh for hours (the amount of time before the stop), and 100 kmh for because she wasn't driving for minutes, or hours.
Multiplying by gives the total distance, which is kms. Therefore, the answer is
Problem 14
Triangle has . Let and be on and , respectively, such
that . Let be the intersection of segments and , and suppose
that is equilateral. What is ?
Solution
Let .
Since ,
See also
Problem15
In a magical swamp there are two species of talking amphibians: toads, whose
statements are always true, and frogs, whose statements are always false. Four
amphibians, Brian, Chris, LeRoy, and Mike live together in this swamp, and they make
the following statements.
Brian: "Mike and I are different species."
Chris: "LeRoy is a frog."
LeRoy: "Chris is a frog."
Mike: "Of the four of us, at least two are toads."
How many of these amphibians are frogs?
Solution
Solution 1
We can begin by first looking at Chris and LeRoy.
Suppose Chris and LeRoy are the same species. If Chris is a toad, then what he says is
true, so LeRoy is a frog. However, if LeRoy is a frog, then he is lying, but clearly Chris
is not a frog, and we have a contradiction. The same applies if Chris is a frog.
Clearly, Chris and LeRoy are different species, and so we have at least frog out of
the two of them.
Now suppose Mike is a toad. Then what he says is true because we already have
toads. However, if Brian is a frog, then he is lying, yet his statement is true, a
contradiction. If Brian is a toad, then what he says is true, but once again it conflicts
with his statement, resulting in contradiction.
Therefore, Mike must be a frog. His statement must be false, which means that there
is at most toad. Since either Chris or LeRoy is already a toad, Brain must be a frog.
We can also verify that his statement is indeed false.
Both Mike and Brian are frogs, and one of either Chris or LeRoy is a frog, so we have frogs total.
Solution 2
Start with Brian. If he is a toad, he tells the truth, hence Mike is a frog. If Brian is a
frog, he lies, hence Mike is a frog, too. Thus Mike must be a frog.
As Mike is a frog, his statement is false, hence there is at most one toad.
As there is at most one toad, at least one of Chris and LeRoy is a frog. But then the
other one tells the truth, and therefore is a toad.
Hence we must have one toad and three frogs.
See also
Problem 16
Nondegenerate has integer side lengths, is an angle bisector, ,
and . What is the smallest possible value of the perimeter?
SolutionBy the Angle Bisector Theorem, we know that . If we use the lowest possible
integer values for AB and BC (the measures of AD and DC, respectively),
then , contradicting theTriangle Inequality. If we use the
next lowest values ( and ), the Triangle Inequality is satisfied. Therefore, our answer is , or choice
Problem 17
A solid cube has side length 3 inches. A 2-inch by 2-inch square hole is cut into the
center of each face. The edges of each cut are parallel to the edges of the cube, and
each hole goes all the way through the cube. What is the volume, in cubic inches, of
the remaining solid?
Solution
Solution 1
Imagine making the cuts one at a time. The first cut removes a box . The
second cut removes two boxes, each of dimensions , and the third cut does
the same as the second cut, on the last two faces. Hence the total volume of all cuts
is .
Therefore the volume of the rest of the cube is .
Solution 2
We can use Principle of Inclusion-Exclusion to find the final volume of the cube.
There are 3 "cuts" through the cube that go from one end to the other. Each of these
"cuts" has cubic inches. However, we can not just sum their volumes, as
the central cube is included in each of these three cuts. To get the correct
result, we can take the sum of the volumes of the three cuts, and subtract the volume
of the central cube twice.
Hence the total volume of the cuts is .
Therefore the volume of the rest of the cube is .
Solution 3
We can visualize the final figure and see a cubic frame. We can find the volume of the
figure by adding up the volumes of the edges and corners.
Each edge can be seen as a box, and each corner can be seen as
a box.
.
Problem18
Bernardo randomly picks 3 distinct numbers from the set and
arranges them in descending order to form a 3-digit number. Silvia randomly picks 3 distinct numbers from the set and also arranges them in descending
order to form a 3-digit number. What is the probability that Bernardo's number is
larger than Silvia's number?
Solution
We can solve this by breaking the problem down into cases and adding up the
probabilities.
Case : Bernardo picks . If Bernardo picks a then it is guaranteed that his number will be larger than Silvia's. The probability that he will pick a is .
Case : Bernardo does not pick . Since the chance of Bernardo picking is , the
probability of not picking is .
If Bernardo does not pick 9, then he can pick any number from to . Since Bernardo
is picking from the same set of numbers as Silvia, the probability that Bernardo's
number is larger is equal to the probability that Silvia's number is larger.
Ignoring the for now, the probability that they will pick the same number is the
number of ways to pick Bernardo's 3 numbers divided by the number of ways to pick
any 3 numbers.
We get this probability to be
Probability of Bernardo's number being greater is
Factoring the fact that Bernardo could've picked a but didn't:
Adding up the two cases we get
Problem19
Equiangular hexagon has side lengths
and . The area of is of the area of the hexagon. What is
the sum of all possible values of ?
Solution
It is clear that is an equilateral triangle. From the Law of Cosines, we get that . Therefore, the area of
is .
If we extend , and so that and meet at , and meet at ,
and and meet at , we find that hexagon is formed by taking
equilateral triangle of side length and removing three equilateral
triangles, , and , of side length . The area of is therefore
.
Based on the initial conditions,
Simplifying this gives us . By Vieta's Formulas we know that the sum of the possible value of is .
See also
Problem20
A fly trapped inside a cubical box with side length meter decides to relieve its
boredom by visiting each corner of the box. It will begin and end in the same corner
and visit each of the other corners exactly once. To get from a corner to any other
corner, it will either fly or crawl in a straight line. What is the maximum possible
length, in meters, of its path?
Problem21
The polynomial has three positive integer zeros. What is the
smallest possible value of ?
Solution
By Vieta's Formulas, we know that is the sum of the three roots of the polynomial . Also, 2010 factors into . But, since there
are only three roots to the polynomial, two of the four prime factors must be
multiplied so that we are left with three roots. To minimize , and should be multiplied, which means will be and the answer is .
Problem22
Eight points are chosen on a circle, and chords are drawn connecting every pair of
points. No three chords intersect in a single point inside the circle. How many
triangles with all three vertices in the interior of the circle are created?
Solution
To choose a chord, we know that two points must be chosen. This implies that for
three chords to create a triangle and not intersect at a single point, six points need to be chosen. Therefore, the answer is which is equivalent to 28,
Problem23
Each of boxes in a line contains a single red marble, and for , the box
in the position also contains white marbles. Isabella begins at the first box and
successively draws a single marble at random from each box, in order. She stops when she first draws a red marble. Let be the probability that Isabella stops after drawing exactly marbles. What is the smallest value of for which ?
SolutionThe probability of drawing a white marble from box is . The probability of
drawing a red marble from box is .
The probability of drawing a red marble at box is therefore
It is then easy to see that the lowest integer value of that satisfies the inequality is .
See also
Problem24
The number obtained from the last two nonzero digits of is equal to . What is ?
Solution
We will use the fact that for any integer ,
First, we find that the number of factors of in is equal to . Let . The we want is therefore the last two
digits of , or . Since there is clearly an excess of factors of 2, we know
that , so it remains to find .
If we divide by by taking out all the factors of in , we can write as
where where every multiple of 5 is replaced by
the number with all its factors of 5 removed. Specifically, every number in the form
is replaced by , and every number in the form is replaced by .
The number can be grouped as follows:
Using the identity at the beginning of the solution, we can reduce to
Using the fact that (or simply the fact that if you
have your powers of 2 memorized), we can deduce that . Therefore .
Finally, combining with the fact that yields .
See also