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目录
一、平面几何————————————————————————————————1
二、代数——————————————————————————————————10
三、概率 数学期望—————————————————————————————36
四、计数——————————————————————————————————45
五、立体几何————————————————————————————————62
六、数论——————————————————————————————————66
1
一、平面几何
公式与定理:
垂直的等价公式 正弦定理 余弦定理 塞瓦定理 梅涅劳斯定理 中线
长定理 角平分线长定理 托勒密定理 海伦公式 圆内接四边形面积
公式 费马点 三角形的五心
直线型
1. In triangle ABC, AB=13, BC=14, and AC=15. Let D denote the
midpoint of
and let E denote the intersection of
with the
bisector of angle BAC. Which of the following is closest to the area
of the triangle ADE?
(A)2 (B)2.5 (C)3 (D)3.5 (E)4
2. A round table has radius 4. Six rectangular place mats are placed on
the table. Each place mat has width 1 and length as shown. Then
are positioned so that each mat has two corners on the edge of the
table, these two corners being end points of the same side of length .
Further, the mats are positioned so that the inner corners each touch
an inner corner of an adjacent mat. What is ?
(A)
(B)3 (C)
(D)
(E)
2
3. Quadrilateral ABCD has AB=BC=CD, ∠ABC=70°, and
∠BCD=170°. What is the degree measure of ∠BAD?
(A)75 (B)80 (C)85 (D)90 (E)95
4. In ABC, ∠ABC=45°. Point D is on
so that and
∠DAB=15°. Find ∠ACB.
(A)54° (B)60° (C)72° (D)75° (E)90°
5. Two particles move along the edges of equilateral triangle ABC in
the direction starting simultaneously and moving at
the same speed. One starts at A, and the other starts at the midpoint of
. The midpoint of the line segment joining the two particles traces
out a path that encloses a region R. What is the ratio of the area of R
to the area of ABC?
(A)
(B)
(C)
(D)
(E)
6. Isosceles ABC has a right angle at C. Point P is inside ABC,
such that PA=11, PB=7, and PC=6. Legs
and
have length
, where and are positive integers. What is ?
(A)85 (B)91 (C)108 (D)121 (E)127
3
7. Triangle ABC has AC=3, BC=4, and AB=5. Point D is on
, and
bisects the right angle. The inscribed circles of ADC and
BCD have radii
and
, respectively. What is
?
(A)
(B)
(C)
(D)
(E)
8. Triangle ABC has ∠C=60° and BC=4. Point D is the midpoint of BC.
What is the largest possible value of tan ∠BAD?
(A)
(B)
(C)
(D)
(E)1
9. Consider all quadrilaterals ABCD such that AB=14, BC=9, CD=7,
DA=12. What is the radius of the largest possible circle that fits inside
or on the boundary of such a quadrilateral?
(A)
(B)
(C)
(D)5 (E)
10. Triangle ABC has ∠BAC=60°, ∠CBA≤90°, BC=1, and AC≥AB.
Let H, I, and O be the orthocenter, incenter, and circumcenter of
ABC, respectively. Assume that the area of the pentagon BCOIH is
the maximum possible. What is ∠CBA?
(A)60° (B)72° (C)75° (D)80° (E)90°
4
11. Triangle ABC has AB=13, BC=14, and AC=15. The points D, E, and
F are the mid-points of
, and
respectively. Let be
the intersection of the circumcircles of BDE and CEF. What is
XA+XB+XC?
(A)24 (B)
(C)
(D)
(E)
12. Let
be a triangle with sides 2011, 2012, and 2013. For , if
and D, E, and F are the points of tangency of the incircle
of to the sides AB, BC and AC, respectively, then
is a
triangle with side lengths AD, BE, and CF, if it exists. What is the
perimeter of the last triangle in the sequence
?
(A)
(B)
(C)
(D)
(E)
13. Square PQRS lies in the first quadrant. Points (3,0), (5,0), (7,0), and
(13,0) lie on lines SP, RQ, PQ, and SR, respectively. What is the sum
of the coordinates of the center of the square PQRS?
(A)6 (B)6.2 (C)6.4 (D)6.6 (E)6.8
14. In BAC, ∠BAC=40°, AB=10, and AC=6. Point D and E lie on
and
respectively. What is the minimum possible value of
BE+DE+CD?
(A)
(B)
(C)
(D)14 (E)
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