高中数学集中培训材料-2017吉林高中数学初赛
Problem 1
Each third-grade
classroom at Pearl Creek Elementary has 18
students and 2 pet
rabbits. How many more
students than rabbits are there in all 4 of the
third-grade
classrooms?
Solution
Problem 2
A circle of radius 5 is
inscribed in a rectangle as shown. The ratio of
the length of the
rectangle to its width is
2:1. What is the area of the rectangle?
Solution
Problem 3
The point in the
xy-plane with coordinates (1000, 2012) is
reflected across the line
y=2000. What are the
coordinates of the reflected point?
Solution
Problem 4
When Ringo places
his marbles into bags with 6 marbles per bag, he
has 4 marbles
left over. When Paul does the
same with his marbles, he has 3 marbles left over.
Ringo and Paul pool their marbles and place
them into as many bags as possible,
with 6
marbles per bag. How many marbles will be left
over?
Solution
Problem 5
Anna enjoys dinner at a restaurant in
Washington, D.C., where the sales tax on
meals
is 10%. She leaves a 15% tip on the price of her
meal before the sales tax is
added, and the
tax is calculated on the pre-tip amount. She
spends a total of 27.50
dollars for dinner.
What is the cost of her dinner without tax or tip
in dollars?
Solution
Problem 6
In order to estimate the value of x-y where x
and y are real numbers with x > y >
0, Xiaoli
rounded x up by a small amount, rounded y down by
the same amount, and
then subtracted her
rounded values. Which of the following statements
is
necessarily correct?
A) Her estimate
is larger than x-y B) Her estimate is smaller than
x-y C) Her
estimate equals x-y D) Her estimate
equals y - x E) Her estimate is 0
Solution
Problem 7
For a science project, Sammy
observed a chipmunk and a squirrel stashing acorns
in holes. The chipmunk hid 3 acorns in each of
the holes it dug. The squirrel hid 4
acorns in
each of the holes it dug. They each hid the same
number of acorns,
although the squirrel needed
4 fewer holes. How many acorns did the chipmunk
hide?
Solution
Problem 8
What is the sum of all integer solutions to
Solution
?
Problem 9
Two integers have a sum of 26. When two more
integers are added to the first two
integers
the sum is 41. Finally when two more integers are
added to the sum of the
previous four integers
the sum is 57. What is the minimum number of even
integers
among the 6 integers?
Solution
Problem 10
How many
ordered pairs of positive integers (M,N) satisfy
the equation
Solution
=
Problem 11
A dessert chef prepares the
dessert for every day of a week starting with
Sunday.
The dessert each day is either cake,
pie, ice cream, or pudding. The same dessert
may not be served two days in a row. There
must be cake on Friday because of a
birthday.
How many different dessert menus for the week are
possible?
Solution
Problem 12
Point B is due east of point A. Point C is due
north of point B. The distance between
points
A and C is , and = 45 degrees. Point D is 20
meters due North
of point C. The distance AD
is between which two integers?
Solution
Problem 13
It takes Clea 60
seconds to walk down an escalator when it is not
operating, and
only 24 seconds to walk down
the escalator when it is operating. How many
seconds
does it take Clea to ride down the
operating escalator when she just stands on it?
Solution
Problem 14
Two
equilateral triangles are contained in square
whose side length is
a rhombus. What is the
area of the rhombus?
. The
bases of these
triangles are the opposite side of the square, and
their intersection is
Solution
Problem 15
In a round-robin tournament
with 6 teams, each team plays one game against
each
other team, and each game results in one
team winning and one team losing. At the
end
of the tournament, the teams are ranked by the
number of games won. What is
the maximum
number of teams that could be tied for the most
wins at the end on
the tournament?
Solution
Problem 16
Three
circles with radius 2 are mutually tangent. What
is the total area of the circles
and the
region bounded by them, as shown in the figure?
Solution
Problem 17
Jesse cuts a
circular paper disk of radius 12 along two radii
to form two sectors, the
smaller having a
central angle of 120 degrees. He makes two
circular cones, using
each sector to form the
lateral surface of a cone. What is the ratio of
the volume of
the smaller cone to that of the
larger?
Solution
Problem 18
Suppose that one of every 500 people in a
certain population has a particular
disease,
which displays no symptoms. A blood test is
available for screening for this
disease. For
a person who has this disease, the test always
turns out positive. For a
person who does not
have the disease, however, there is a
other
words, for such people,
false positive rate--
in
of the time the test will turn out
negative, but
of the time the test will turn
out positive and will incorrectly indicate that
the person
has the disease. Let be the
probability that a person who is chosen at random
from
this population and gets a positive test
result actually has the disease. Which of the
following is closest to ?
Solution
Problem 19
In rectangle , ,
to point
?
, and
, and
is the midpoint of
. Segment
and is extended 2 units beyond
.
What is the area of
is the intersection of
Solution
Problem 20
Bernardo and Silvia play the following game.
An integer between 0 and 999,
inclusive, is
selected and given to Bernardo. Whenever Bernardo
receives a number,
he doubles it and passes
the result to Silvia. Whenever Silvia receives a
number, she
adds 50 to it and passes the
result to Bernardo. The winner is the last person
who
produces a number less than 1000. Let be
the smallest initial number that results
? in
a win for Bernardo. What is the sum of the digits
of
Solution
Problem 21
Four
distinct points are arranged on a plane so that
the segments connecting them
have lengths , ,
, , , and . What is the ratio of to ?
Solution
Problem 22
Let
either
be a list of the first 10 positive integers
such that for each
or or both appear somewhere
before in the list.
How many such lists are
there?
Solution
Problem 23
A
solid tetrahedron is sliced off a wooden unit cube
by a plane passing through two
nonadjacent
vertices on one face and one vertex on the
opposite face not adjacent
to either of the
first two vertices. The tetrahedron is discarded
and the remaining
portion of the cube is
placed on a table with the cut surface face down.
What is the
height of this object?
Solution
Problem 23
A
solid tetrahedron is sliced off a wooden unit cube
by a plane passing through two
nonadjacent
vertices on one face and one vertex on the
opposite face not adjacent
to either of the
first two vertices. The tetrahedron is discarded
and the remaining
portion of the cube is
placed on a table with the cut surface face down.
What is the
height of this object?
Solution
Problem 24
Amy, Beth, and
Jo listen to four different songs and discuss
which ones they like. No
song is liked by all
three. Furthermore, for each of the three pairs of
the girls, there
is at least one song liked by
those girls but disliked by the third. In how many
different ways is this possible?
Solution
Problem 25
A bug travels
from to along the segments in the hexagonal
lattice pictured
below. The segments marked
with an arrow can be traveled only in the
direction of
the arrow, and the bug never
travels the same segment more than once. How many
different paths are there?
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