SAT数学难题 hard math problem
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I. (a + b)∗ is even. II. (a + b)∗ is odd. III. (a − b)∗ is negative.
3. The graphs of two linear functions, f and g, are perpendicular. If f (2) = 4 and f (−4) = 1, which of the following could define g?
I. ab · a−1
11. For all x and y, let the operation x y be equal to the 2x
whole number remainder obtained when finding . y
If s is an even integer and the value of 7 s = 2,
B
4. If (x − h)(x + k) = x2 − 16, what is the value of h + k?
(A) −8 (B) −4 (C) 0 (D) 4 (E) 8
A (–1,0)
C (3,0)
2. For all n, let n∗ be defined as n∗ = n2 − n − 2. If a and b are positive integers and a b, which of the following could be true?
B xº
(A) −5 (B) −4 (C) −3 (D) 1 (E) 3
Stop
22. A small circle, A, is rolled around the circumference of a larger circle, B, as shown in the figure above. Circle A starts at the position indicated “Start” and makes two complete rotations, coming to rest at the position marked “Stop”. If the ratio of the areas of A to B is 1 to 25, what is the value of x?
14. A student is weighing large and small marbles. While weighing one large marble with 8 small marbles, the student discovers that the large marble’s weight is 5 times the average (arithmetic mean) of the weight of the small marbles. The large marble’s weight is what fraction of the total weight of the 9 marbles?
(B) 6 √√
(C) 6 3 − 3 2 √
(D) 6 3 √
(E) 6 3 + 6
3
17. The function f is defined as f (x) = (x − n)(x + m) for all values of x. If the function g is defined as g(x) = f (x) + 3, what is the y-intercept of the graph of g in terms of n and m?
what is a possible value of s?
II. 0
III. 1
2
O 30º
P
S
A
B
E
R
Q
12. In the figure above, OPQR is a square, and PS is the arc of a circle with center O. If the length of arc PS is π, the area of the shaded sector is what fraction of the area of OPQR?
(A) g(x) = − 1 x + 4 2
(B) g(x) = −x + 1 (C) g(x) = −2x + 4 (D) g(x) = 2x − 3 (E) g(x) = x + 5
7 6 5 4 3 2 1
-3 -2 -1 -1 -2
1234567
5. A portion of the graph of the function h is shown in the xy-plane above. If h(4) − h(s) = 4, which of the following could be the value of s?
1. In the figure below, the graph of y = kx2 intersects triangle ABC at B. If AB = BC and the area of triangle ABC is 6, what is the value of k?
y = kx2
(A) 3n + 3m (B) 3n − 3m (C) 3nm (D) 3 − nm
nm (E)
3
(A) −10 (B) −8 (C) −6 (D) 0 (E) 5
a, b, b − a, −a, . . .
18. In the sequence above, each term after the first two is obtained by finding the difference between the two previous terms. If a = 3 and b = 4, what is the sum of the first 200 terms of this sequence?
Q
beginning of the week?
O
P
6. In the figure above, three identical circles of radius 8 with centers O, P and Q are tangent to each other. What is the area of the shaded region bound by the circles?
21. An antique coin collector has a number of coins and several display cases. If he puts one coin in each case, there are four coins left out of the cases. If instead he puts three coins in each case, all coins are placed, and there are six empty cases left over. How many coins does the collector have?
g(x) = x2 + c h(x) = g(x − 5) + 3
A Start
19. The equations of two functions, g and h, are shown above, where c is a constant integer. If the graph of h has two solutions, what is the maximum possible value of c?
what
is
the
value
of
h ?
k
8. For integers a and b where a > 0 and b ≥ 0, let the
operation
பைடு நூலகம்be defined as a
b
=
ab .
Which
of
the
a
following could be the value of a b?
B 105º
C
D
A
150º
F
E
Note: Figure not drawn to scale.
16. In the figure above, triangle ABC and parallelogram CDEF are constructed with B, C, and F along the same lin√e. If the ratio of CF : BC = 1 : 2 and DE = 2 3, what is the area of triangle ABC? √ (A) 3 2
-3 -2 -1 -1
-2
1234567
10. The figure above shows the graphs of the functions f
and g. If g is defined in terms of a transformation of
f
such
that
g(x)
=
f (x+h)+k,
(–1,0) D
C
15. In the figure above, rectangle ABCD lies on the coordinate plane with point D located at the origin. If AB = AE = ED, what is the area of quadrilateral BC DE ?
π (A)
2 π (B) 4 π (C) 8 π (D) 12 π (E) 15
13. Jack drove to work in the morning at an average speed of 45 miles per hour. He returned home in the evening along the same route and averaged 30 miles per hour. If Jack spent a total of one hour commuting to and from work, how many miles did he drive to work in the morning?
20. Triangle ABC has side lengths 3, 8, and a. Triangle DEF has side lengths 7, 10, and d. If a and d are both integers, what is the smallest possible value of (d − a)?
7 6 5 4 3 y = f (x) 2 1
y = g(x)
7. The participants in a certain community service club come from all four grade levels in Smithville High School. There are equal numbers of sophomores and juniors. The number of seniors is twenty more than twice the number of sophomores and juniors combined, and there are 20 freshmen. If the number of sophomores, juniors and seniors combined accounts for 80 percent of the club’s participants, how many seniors are there in the club?
(A) −2 (B) 0 (C) 1 (D) 4 (E) 7
1
9. The value of an investment increased by 20 percent
to $800 by the end of the first week. To the nearest
dollar, what was the value of the investment at the
3. The graphs of two linear functions, f and g, are perpendicular. If f (2) = 4 and f (−4) = 1, which of the following could define g?
I. ab · a−1
11. For all x and y, let the operation x y be equal to the 2x
whole number remainder obtained when finding . y
If s is an even integer and the value of 7 s = 2,
B
4. If (x − h)(x + k) = x2 − 16, what is the value of h + k?
(A) −8 (B) −4 (C) 0 (D) 4 (E) 8
A (–1,0)
C (3,0)
2. For all n, let n∗ be defined as n∗ = n2 − n − 2. If a and b are positive integers and a b, which of the following could be true?
B xº
(A) −5 (B) −4 (C) −3 (D) 1 (E) 3
Stop
22. A small circle, A, is rolled around the circumference of a larger circle, B, as shown in the figure above. Circle A starts at the position indicated “Start” and makes two complete rotations, coming to rest at the position marked “Stop”. If the ratio of the areas of A to B is 1 to 25, what is the value of x?
14. A student is weighing large and small marbles. While weighing one large marble with 8 small marbles, the student discovers that the large marble’s weight is 5 times the average (arithmetic mean) of the weight of the small marbles. The large marble’s weight is what fraction of the total weight of the 9 marbles?
(B) 6 √√
(C) 6 3 − 3 2 √
(D) 6 3 √
(E) 6 3 + 6
3
17. The function f is defined as f (x) = (x − n)(x + m) for all values of x. If the function g is defined as g(x) = f (x) + 3, what is the y-intercept of the graph of g in terms of n and m?
what is a possible value of s?
II. 0
III. 1
2
O 30º
P
S
A
B
E
R
Q
12. In the figure above, OPQR is a square, and PS is the arc of a circle with center O. If the length of arc PS is π, the area of the shaded sector is what fraction of the area of OPQR?
(A) g(x) = − 1 x + 4 2
(B) g(x) = −x + 1 (C) g(x) = −2x + 4 (D) g(x) = 2x − 3 (E) g(x) = x + 5
7 6 5 4 3 2 1
-3 -2 -1 -1 -2
1234567
5. A portion of the graph of the function h is shown in the xy-plane above. If h(4) − h(s) = 4, which of the following could be the value of s?
1. In the figure below, the graph of y = kx2 intersects triangle ABC at B. If AB = BC and the area of triangle ABC is 6, what is the value of k?
y = kx2
(A) 3n + 3m (B) 3n − 3m (C) 3nm (D) 3 − nm
nm (E)
3
(A) −10 (B) −8 (C) −6 (D) 0 (E) 5
a, b, b − a, −a, . . .
18. In the sequence above, each term after the first two is obtained by finding the difference between the two previous terms. If a = 3 and b = 4, what is the sum of the first 200 terms of this sequence?
Q
beginning of the week?
O
P
6. In the figure above, three identical circles of radius 8 with centers O, P and Q are tangent to each other. What is the area of the shaded region bound by the circles?
21. An antique coin collector has a number of coins and several display cases. If he puts one coin in each case, there are four coins left out of the cases. If instead he puts three coins in each case, all coins are placed, and there are six empty cases left over. How many coins does the collector have?
g(x) = x2 + c h(x) = g(x − 5) + 3
A Start
19. The equations of two functions, g and h, are shown above, where c is a constant integer. If the graph of h has two solutions, what is the maximum possible value of c?
what
is
the
value
of
h ?
k
8. For integers a and b where a > 0 and b ≥ 0, let the
operation
பைடு நூலகம்be defined as a
b
=
ab .
Which
of
the
a
following could be the value of a b?
B 105º
C
D
A
150º
F
E
Note: Figure not drawn to scale.
16. In the figure above, triangle ABC and parallelogram CDEF are constructed with B, C, and F along the same lin√e. If the ratio of CF : BC = 1 : 2 and DE = 2 3, what is the area of triangle ABC? √ (A) 3 2
-3 -2 -1 -1
-2
1234567
10. The figure above shows the graphs of the functions f
and g. If g is defined in terms of a transformation of
f
such
that
g(x)
=
f (x+h)+k,
(–1,0) D
C
15. In the figure above, rectangle ABCD lies on the coordinate plane with point D located at the origin. If AB = AE = ED, what is the area of quadrilateral BC DE ?
π (A)
2 π (B) 4 π (C) 8 π (D) 12 π (E) 15
13. Jack drove to work in the morning at an average speed of 45 miles per hour. He returned home in the evening along the same route and averaged 30 miles per hour. If Jack spent a total of one hour commuting to and from work, how many miles did he drive to work in the morning?
20. Triangle ABC has side lengths 3, 8, and a. Triangle DEF has side lengths 7, 10, and d. If a and d are both integers, what is the smallest possible value of (d − a)?
7 6 5 4 3 y = f (x) 2 1
y = g(x)
7. The participants in a certain community service club come from all four grade levels in Smithville High School. There are equal numbers of sophomores and juniors. The number of seniors is twenty more than twice the number of sophomores and juniors combined, and there are 20 freshmen. If the number of sophomores, juniors and seniors combined accounts for 80 percent of the club’s participants, how many seniors are there in the club?
(A) −2 (B) 0 (C) 1 (D) 4 (E) 7
1
9. The value of an investment increased by 20 percent
to $800 by the end of the first week. To the nearest
dollar, what was the value of the investment at the