Tài liệu Collection of all geometric world champion P3 pptx

Chapter 9. GEOMETRIC INEQUALITIES
Background
1) For elements of a triangle the following notations are used:
a, b, c are the lengths of sides BC, CA, AB, respectively;
α, β, γ the values of the angles at vertices A, B, C, respectively;
m
a
, m
b
, m
c
are the lengths of the medians drawn from vertices A, B, C, respectively;
h
a
, h
b
, h
c
are the lengths of the heights dropped from vertices A, B, C, respectively;
l
a
, l
b
, l
c
are the lengths of the bisectors drawn from vertices A, B, C, respectively;
r and R are the radii of the inscribed and circumscribed circles, respectively.
2) If A, B, C are arbitrary points, then AB ≤ AC + CB and the equality takes place
only if point C lies on segment AB (the triangle inequality).
3) The median of a triangle is shorter than a half sum of the sides that confine it:
m
a
<
1
2(b+c)
(Problem 9.1).
4) If one convex polygon lies inside another one, then the perimeter of the outer polygon
is greater than the perimeter of the inner one (Problem 9.27 b).
5) The sum of the lengths of the diagonals of a convex quadrilateral is greater than the
sum of the length of any pair of the opposite sides of the quadrilateral (Problem 9.14).
6) The longer side of a triangle subtends the greater angle (Problem 10.59).
7) The length of the segment that lies inside a convex polygon does not exceed either
that of its longest side or that of its longest diagonal (Problem 10.64).
Remark. While solving certain problems of this chapter we have to know various alge-
braic inequalities. The data on these inequalities and their proof are given in an appendix to
this chapter; one should acquaint oneself with them but it should be taken into account that
these inequalities are only needed in the solution of comparatively complicated problems;
in order to solve simple problems we will only need the inequality
√
ab ≤
1
2
a + b and its
corollaries.
Introductory problems
1. Prove that S
ABC
≤
1
2
AB · BC.
2. Prove that S
ABCD
≤
1
2
(AB · BC + AD ·DC).
3. Prove that ∠ABC > 90
◦
if and only if point B lies inside the circle with diameter
AC.
4. The radii of two circles are equal to R and r and the distance between the centers of
the circles is equal to d. Prove that these circles intersect if and only if |R −r| < d < R + r.
5. Prove that any diagonal of a quadrilateral is shorter than the quadrilateral’s semiperime-
ter.
§1. A median of a triangle
9.1. Prove that
1
2
(a + b − c) < m
c
<
1
2
(a + b).
205
206 CHAPTER 9. GEOMETRIC INEQUALITIES
9.2. Prove that in any triangle the sum of the medians is greater than
3
4
of the perimeter
but less than the perimeter.
9.3. Given n points A
1
, . . . , A
n
and a unit circle, prove that it is possible to find a point
M on the circle so that MA
1
+ ···+ MA
n
≥ n.
9.4. Points A
1
, . . . , A
n
do not lie on one line. Let two distinct points P and Q have the
following property
A
1
P + ···+ A
n
P = A
1
Q + ···+ A
n
Q = s.
Prove that A
1
K + ··· + A
n
K < s for a point K.
9.5. On a table lies 50 working watches (old style, with hands); all work correctly.
Prove that at a certain moment the sum of the distances from the center of the table to the
endpoints of the minute’s hands becomes greater than the sum of the distances from the
center of the table to the centers of watches. (We assume that each watch is of the form of
a disk.)
§2. Algebraic problems on the triangle inequality
In problems of this section a, b and c are the lengths of the sides of an arbitrary triangle.
9.6. Prove that a = y + z, b = x + z and c = x + y, where x, y and z are positive
numbers.
9.7. Prove that a
2
+ b
2
+ c
2
< 2(ab + bc + ca).
9.8. For any positive integer n, a triangle can be composed of segments whose lengths
are a
n
, b
n
and c
n
. Prove that among numbers a, b and c two are equal.
9.9. Prove that
a(b − c)
2
+ b(c − a)
2
+ c(a − b)
2
+ 4abc > a
3
+ b
3
+ c
3
.
9.10. Let p =
a
b
+
b
c
+
c
a
and q =
a
c
+
c
b
+
b
a
. Prove that |p −q| < 1.
9.11. Five segments are such that from any three of them a triangle can be constructed.
Prove that at least one of these triangles is an acute one.
9.12. Prove that
(a + b − c)(a − b + c)(−a + b + c) ≤ abc.
9.13. Prove that
a
2
b(a − b) + b
2
c(b − c) + c
2
a(c − a) ≥ 0.
§3. The sum of the lengths of quadrilateral’s diagonals
9.14. Let ABCD be a convex quadrilateral. Prove that AB + CD < AC + BD.
9.15. Let ABCD be a convex quadrilateral and AB + BD ≤ AC + CD. Prove that
AB < AC.
9.16. Inside a convex quadrilateral the sum of lengths of whose diagonals is equal to d,
a convex quadrilateral the sum of lengths of whose diagonals is equal to d
′
is placed. Prove
that d
′
< 2d.
9.17. Given closed broken line has the property that any other closed broken line with
the same vertices (?) is longer. Prove that the given broken line is not a self-intersecting
one.
9.18. How many sides can a convex polygon have if all its diagonals are of equal length?
9.19. In plane, there are n red and n blue dots no three of which lie on one line. Prove
that it is possible to draw n segments with the endpoints of distinct colours without common
points.
9.20. Prove that the mean arithmetic of the lengths of sides of an arbitrary convex
polygon is less than the mean arithmetic of the lengths of all its diagonals.
THE AREA OF A TRIANGLE 207
9.21. A convex (2n + 1)-gon A
1
A
3
A
5
. . . A
2n+1
A
2
. . . A
2n
is given. Prove that among all
the closed broken lines with the vertices in the vertices of the given (2n + 1)-gon the broken
line A
1
A
2
A
3
. . . A
2n+1
A
1
is the longest.
§4. Miscellaneous problems on the triangle inequality
9.22. In a triangle, the lengths of two sides are equal to 3.14 and 0.67. Find the length
of the third side if it is known that it is an integer.
9.23. Prove that the sum of lengths of diagonals of convex pentagon ABCDE is greater
than its perimeter but less than the doubled perimeter.
9.24. Prove that if the lengths of a triangle’s sides satisfy the inequality a
2
+ b
2
> 5c
2
,
then c is the length of the shortest side.
9.25. The lengths of two heights of a triangle are equal to 12 and 20. Prove that the
third height is shorter than 30.
9.26. On sides AB, BC, CA of triangle ABC, points C
1
, A
1
, B
1
, respectively, are taken
so that BA
1
= λ ·BC, CB
1
= λ ·CA and AC
1
= λ · AB, where
1
2
< λ < 1. Prove that the
perimeter P of triangle ABC and the perimeter P
1
of triangle A
1
B
1
C
1
satisfy the following
inequality: (2λ −1)P < P
1
< λP .
* * *
9.27. a) Prove that under the passage from a nonconvex polygon to its convex hull the
perimeter diminishes. (The convex hull of a polygon is the smallest convex polygon that
contains the given one.)
b) Inside a convex polygon there lies another convex polygon. Prove that the perimeter
of the outer polygon is not less than the perimeter of the inner one.
9.28. Inside triangle ABC of perimeter P, a point O is taken. Prove that
1
2
P <
AO + BO + CO < P .
9.29. On base AD of trapezoid ABCD, a point E is taken such that the perimeters of
triangles ABE, BCE and CDE are equal. Prove that BC =
1
2
AD.
See also Problems 13.40, 20.11.
§5. The area of a triangle does not exceed a half product of two sides
9.30. Given a triangle of area 1 the lengths of whose sides satisfy a ≤ b ≤ c. Prove that
b ≥
√
2.
9.31. Let E, F , G and H be the midpoints of sides AB, BC, CD and DA of quadrilateral
ABCD. Prove that
S
ABCD
≤ EG · HF ≤
(AB + CD)(AD + BC)
4
.
9.32. The perimeter of a convex quadrilateral is equal to 4. Prove that its area does not
exceed 1.
9.33. Inside triangle ABC a point M is taken. Prove that
4S ≤ AM · BC + BM · AC + CM ·AB,
where S is the area of triangle ABC.
9.34. In a circle of radius R a polygon of area S is inscribed; the polygon contains the
center of the circle and on each of its sides a point is chosen. Prove that the perimeter of
the convex polygon with vertices in the chosen points is not less than
2S
R
.
9.35. Inside a convex quadrilateral ABCD of area S point O is taken such that AO
2
+
BO
2
+ CO
2
+ DO
2
= 2S. Prove that ABCD is a square and O is its center.
208 CHAPTER 9. GEOMETRIC INEQUALITIES
§6. Inequalities of areas
9.36. Points M and N lie on sides AB and AC, respectively, of triangle ABC, where
AM = CN and AN = BM. Prove that the area of quadrilateral BMNC is at least three
times that of triangle AMN.
9.37. Areas of triangles ABC, A
1
B
1
C
1
, A
2
B
2
C
2
are equal to S, S
1
, S
2
, respectively, and
AB = A
1
B
1
+ A
2
B
2
, AC = A
1
C
1
+ A
2
B
2
, BC = B
1
C
1
+ B
2
C
2
. Prove that S ≤ 4
√
S
1
S
2
.
9.38. Let ABCD be a convex quadrilateral of area S. The angle between lines AB and
CD is equal to α and the angle between AD and BC is equal to β. Prove that
AB · CD sin α + AD ·BC sin β ≤ 2S ≤ AB · CD + AD ·BC.
9.39. Through a point inside a triangle three lines parallel to the triangle’s sides are
drawn.
Figure 94 (9.39)
Denote the areas of the parts into which these lines divide the triangle as plotted on Fig.
94. Prove that
a
α
+
b
β
+
c
γ
≥
3
2
.
9.40. The areas of triangles ABC and A
1
B
1
C
1
are equal to S and S
1
, respectively, and
we know that triangle ABC is not an obtuse one. The greatest of the ratios
a
1
a
,
b
1
b
and
c
1
c
is
equal to k. Prove that S
1
≤ k
2
S.
9.41. a) Points B, C and D divide the (smaller) arc ⌣ AE of a circle into four equal
parts. Prove that S
ACE
< 8S
BCD
.
b) From point A tangents AB and AC to a circle are drawn. Through the midpoint D
of the (lesser) arc ⌣ BC the tangent that intersects segments AB and AC at points M and
N, respectively is drawn. Prove that S
BCD
< 2S
MAN
.
9.42. All sides of a convex polygon are moved outwards at distance h and extended
to form a new polygon. Prove that the d ifference of areas of the polygons is more than
P h + π h
2
, where P is the perimeter.
9.43. A square is cut into rectangles. Prove that the sum of areas of the disks circum-
scribed about all these rectangles is not less than the area of the disk circumscribed about
the initial square.
9.44. Prove that the sum of areas of five triangles formed by the pairs of neighbouring
sides and the corresponding diagonals of a convex pentagon is greater than the area of the
pentagon itself.
9.45. a) Prove that in any convex hexagon of area S there exists a diagonal that cuts
off the hexagon a triangle whose area does not exceed
1
6
S.
b) Prove that in any convex 8-gon of area S there exists a diagonal that cuts off it a
triangle of area not greater than
1
8
S.
See also Problem 17.19.
§8. BROKEN LINES INSIDE A SQUARE 209
§7. Area. One figure lies inside another
9.46. A convex polygon whose area is greater than 0.5 is placed in a unit square. Prove
that inside the polygon one can place a segment of length 0.5 parallel to a side of the square.
9.47. Inside a unit square n points are given. Prove that:
a) the area of one of the triangles some of whose vertices are in these points and some in
vertices of the square does not exceed
1
2(n+1)
;
b) the area of one of the triangles with the vertices in these points does not exceed
1
n−2
.
9.48. a) In a disk of area S a regular n-gon of area S
1
is inscribed and a regular n-gon
of area S
2
is circumscribed about the disk. Prove that S
2
> S
1
S
2
.
b) In a circle of length L a regular n-gon of perimeter P
1
is inscribed and another regular
n-gon of perimeter P
2
is circumscribed about the circle. Prove that L
2
< P
1
P
2
.
9.49. A polygon of area B is inscribed in a circle of area A and circumscribed about a
circle of area C. Prove that 2B ≤ A + C.
9.50. In a unit disk two triangles the area of each of which is greater t han 1 are placed.
Prove that these triangles intersect.
9.51. a) Prove that inside a convex polygon of area S and perimeter P one can place a
disk of radius
S
P
.
b) Inside a convex polygon of area S
1
and perimeter P
1
a convex polygon of area S
2
and
perimeter P
2
is placed. Prove that
2S
1
P
1
>
S
2
P
2
.
9.52. Prove that the area of a parallelogram that lies inside a triangle does not exceed
a half area of the triangle.
9.53. Prove that the area of a triangle whose vertices lie on sides of a parallelogram does
not exceed a half area of the parallelogram.
* * *
9.54. Prove that any acute triangle of area 1 can be placed in a right triangle of area
√
3.
9.55. a) Prove that a convex polygon of area S can be placed in a rectangle of area not
greater than 2S.
b) Prove that in a convex polygon of area S a parallelogram of area not less than
1
2
S can
be inscribed.
9.56. Prove that in any convex polygon of area 1 a triangle whose area is not less than
a)
1
4
; b)
3
8
can be placed.
9.57. A convex n-gon is placed in a unit square. Prove that there are three vertices A, B
and C of this n-gon, such that the area of triangle ABC does not exceed a)
8
n
2
; b)
16π
n
3
.
See also Problem 15.6.
§8. Broken lines inside a square
9.58. Inside a unit square a non-self-intersecting broken line of length 1000 is placed.
Prove that there exists a line parallel to one of the sides of the square that intersects this
broken line in at least 500 points.
9.59. In a unit square a broken line of length L is placed. It is known that each point of
the square is distant from a point of this broken line less than by ε. Prove that L ≥
1
2ε
−
1
2
πε.
9.60. Inside a unit square n
2
points are placed. Prove that there exists a broken line
that passes through all these points and whose length does not exceed 2n.
9.61. Inside a square of side 100 a broken line L is placed. This broken line has the
following property: the distance from any point of the square to L does not exceed 0.5.






Collection of all geometric
world champion P3






210 CHAPTER 9. GEOMETRIC INEQUALITIES
Prove that on L there are two points the distance between which does not exceed 1 and the
distance between which along L is not less than 198.
§9. The quadrilateral
9.62. In quadrilateral ABCD angles ∠A and ∠B are equal and ∠D > ∠C. Prove that
AD < BC.
9.63. In trapezoid ABCD, the angles at base AD satisfy inequalities ∠A < ∠D < 90
◦
.
Prove that AC > BD.
9.64. Prove that if two opposite angles of a quadrilateral are obtuse ones, then the
diagonal that connects the vertices of these angles is shorter than the other diagonal.
9.65. Prove that the sum of distances from an arbitrary point to three vertices of an
isosceles trapezoid is greater than the distance from this point to the fourth vertex.
9.66. Angle ∠A of quadrilateral ABCD is an obtuse one; F is the midpoint of side BC.
Prove that 2F A < BD + CD.
9.67. Quadrilateral ABCD is given. Prove that AC · BD ≤ AB · CD + BC · AD.
(Ptolemy’s inequality.)
9.68. Let M and N be the midpoints of sides BC and CD, respectively, of a convex
quadrilateral ABCD. Prove that S
ABCD
< 4S
AMN
.
9.69. Point P lies inside convex quadrilateral ABCD. Prove that the sum of distances
from point P to the vertices of the quadrilateral is less than the sum of pairwise distances
between the vertices of the quadrilateral.
9.70. The diagonals divide a convex quadrilateral ABCD into four triangles. Let P be
the perimeter of ABCD and Q the perimeter of the quadrilateral formed by the centers of
the inscribed circles of the obtained triangles. Prove that PQ > 4S
ABCD
.
9.71. Prove that the distance from one of the vertices of a convex quadrilateral to the
opposite diagonal does not exceed a half length of this diagonal.
9.72. Segment KL passes through the intersection point of diagonals of quadrilateral
ABCD and the endpoints of KL lie on sides AB and CD of the quadrilateral. Prove that the
length of segment KL does not exceed the length of one of the diagonals of the quadrilateral.
9.73. Parallelogram P
2
is inscribed in parallelogram P
1
and parallelogram P
3
whose sides
are parallel to the corresponding sides of P
1
is inscribed in parallelogram P
2
. Prove that the
length of at least one of the sides of P
1
does not exceed the doub led length of a parallel to
it side of P
3
.
See also Problems 13.19, 15.3 a).
§10. Polygons
9.74. Prove that if the angles of a convex pentagon form an arithm etic progression, then
each of them is greater than 36
◦
.
9.75. Let ABCDE is a convex pentagon inscribed in a circle of radius 1 so that AB = A,
BC = b, CD = c, DE = d, AE = 2. Prove that
a
2
+ b
2
+ c
2
+ d
2
+ abc + bcd < 4.
9.76. Inside a regular hexagon with side 1 point P is taken. Prove that the sum of the
distances from point P to certain three vertices of the hexagon is not less than 1.
9.77. Prove that if the sides of convex hexagon ABCDEF are equal to 1, then the
radius of the circumscribed circle of one of triangles ACE and BDF does not exceed 1.
9.78. Each side of convex hexagon ABCDEF is shorter than 1. Prove that one of the
diagonals AD, BE, CF is shorter than 2.
* * * 211
9.79. Heptagon A
1
. . . A
7
is inscribed in a circle. Prove that if the center of this circle
lies inside it, then the value of any angle at vertices A
1
, A
3
, A
5
is less than 450
◦
.
* * *
9.80. a) Prove that if the lengths of the projections of a segment to two perpendicular
lines are equal to a and b, then the segment’s length is not less than
a+b
√
2
.
b) The lengths of the projections of a polygon to coordinate axes are equal to a and b.
Prove that its perimeter is not less than
√
2(a + b).
9.81. Prove that from the sides of a convex polygon of perimeter P two segments whose
lengths differ not more than by
1
3
P can be constructed.
9.82. Inside a convex polygon A
1
. . . A
n
a p oint O is taken. Let α
k
be the value of the
angle at vertex A
k
, x
k
= OA
k
and d
k
the distance from point O to line A
k
A
k+1
. Prove that

x
k
sin
α
k
2
≥

d
k
and

x
k
cos
α
k
2
≥ p, where p is the semiperimeter of the polygon.
9.83. Regular 2n-gon M
1
with side a lies inside regular 2n-gon M
2
with side 2a. Prove
that M
1
contains the center of M
2
.
9.84. Inside regular polygon A
1
. . . A
n
point O is taken.
Prove that at least one of the angles ∠A
i
OA
j
satisfies the inequalities π

1 −
1
n

≤
∠A
i
OA
j
≤ π.
9.85. Prove that for n ≥ 7 inside a convex n-gon there is a point the sum of distances
from which to the vertices is greater than the semiperimeter of the n-gon.
9.86. a) Convex polygons A
1
. . . A
n
and B
1
. . . B
n
are such that all their corresponding
sides except for A
1
A
n
and B
1
B
n
are equal and ∠A
2
≥ ∠B
2
, . . . , ∠A
n−1
≥ ∠B
n−1
, where at
least one of the inequalities is a strict one. Prove that A
1
A
n
> B
1
B
n
.
b) The corresponding sides of nonequal polygons A
1
. . . A
n
and B
1
. . . B
n
are equal.
Let us write beside each vertex of polygon A
1
. . . A
n
the sign of the difference ∠A
i
−∠B
i
.
Prove that for n ≥ 4 there are at least four pairs of neighbour ing vertices with distinct signs.
(The vertices with the zero difference are disregarded: two vertices between which there only
stand vertices with the zero difference are considered to be neighbouring ones.)
See also Problems 4.37, 4.53, 13.42.
§11. Miscellaneous problems
9.87. On a segment of length 1 there are given n points. Pr ove that the sum of distances
from a point of the segment to these points is not less than
1
2
n.
9.88. In a forest, trees of cylindrical form grow. A communication service person has to
connect a line from point A to point B through this forest the distance between the points
being equal to l. Prove that to acheave the goal a piece of wire of length 1.6l will be sufficient.
9.89. In a forest, the distance between any two trees does not exceed the difference of
their heights. Any tree is shorter than 100 m. Prove that this forest can be fenced by a
fence of length 200 m.
9.90. A (not necessarily convex) paper polygon is folded along a line and both halves
are glued together. Can the perimeter of the obtained lamina be greater than the perimeter
of the initial polygon?
* * *
9.91. Prove that a closed broken line of length 1 can be placed in a disk of radius 0.25.
9.92. An acute triangle is placed inside a circumscribed circle. Prove that the radius of
the circle is not less than the radius of the circumscribed circle of the triangle.
212 CHAPTER 9. GEOMETRIC INEQUALITIES
Is a similar statement true for an obtuse triangle?
9.93. Prove that the perimeter of an acute triangle is not less than 4R.
See also problems 14.23, 20.4.
Problems for independent study
9.94. Two circles divide rectangle ABCD into four rectangles. Prove that the area of
one of the rectangles, the one adjacent to vertices A and C, does not exceed a quarter of the
area of ABCD.
9.95. Prove that if AB + BD = AC + CD, then the midperpendicular to side BC of
quadrilateral ABCD intersects segment AD.
9.96. Prove that if diagonal BD of convex quadrilateral ABCD divides diagonal AC in
halves and AB > BC, then AD < DC.
9.97. The lengths of bases of a circumscribed trapezoid are equal to 2 and 11. Prove
that the angle between the extensions of its lateral sides is an acute one.
9.98. The bases of a trapezoid are equal to a and b and its height is equal to h. Prove
that the length of one of its diagonals is not less than

h
2
+(b+a)
2
4
.
9.99. The vertices of an n-gon M
1
are the midpoints of sides of a convex n-gon M. Prove
that for n ≥ 3 the perimeter of M
1
is not less than the semiperimeter of M and for n ≥ 4
the area of M
1
is not less than a half area of M.
9.100. In a unit circle a polygon the lengths of whose sides are confined between 1 and
√
2 is inscribed. Find how many sides does the polygon have.
Supplement. Certain inequalities
1. The inequality between the mean ari thmeti c and the mean geometric of two numbers
√
ab ≤
1
2
(a + b), where a and b are positive numbers, is often encountered. This inequality
follows from the fact that a − 2
√
ab + b = (
√
a −
√
b)
2
≥ 0, where the equality takes place
only if a = b.
This inequality implies several useful inequalities, for example:
x(a − x) ≤

x+a−x
2

2
=
a
2
4
;
a +
1
a
≥ 2

a ·
1
a
= 2 fora > 0.
2. The inequality between the mean arithmetic and the mean geometric of n positive
numbers (a
1
a
2
. . . a
n
)
1
n
≤
a
1
+···+a
n
n
is sometimes used. In this inequality the equality takes
place only if a
1
= ··· = a
n
.
First, let us prove this inequality for the numbers of the form n = 2
m
by induction on
m. For m = 1 the equality was proved above.
Suppose that it is proved for m and let us prove it for m + 1. Clearly, a
k
a
k+2
m
≤

a
k
+a
k+2
m
2

2
. Therefore,
(a
1
a
2
. . . a
2
m+1
)
1
2
m+1
≤ (b
1
b
2
. . . b
2
m
)
1
2
m
,
where b
k
=
1
2
(a
k
+ a
k+2
m
) and by the inductive hypothesis
(b
1
. . . b
2
m
)
1
2
m
≤
1
2
m
(b
1
+ ··· + b
2
m
) =
1
2
m+1
(a
1
+ ··· + a
2
m+1
).
Now, let n be an arbitrary number. Then n < 2
m
for some m. Suppose a
n+1
= ··· = a
2
m
=
a
1
+···+a
n
n
= A. Clearly,
(a
1
+ ··· + a
n
) + (a
n+1
+ ··· + a
2
m
) = nA + (2
m
− n)A = 2
m
A

Xem chi tiết: Tài liệu Collection of all geometric world champion P3 pptx