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Contests
International Contests
IMO Longlists
1976 IMO Longlists
1976 IMO Longlists
Part of
IMO Longlists
Subcontests
(39)
34
1
Hide problems
Prove that a unique constant satifies inequality.
Let
{
a
n
}
0
∞
\{a_n\}^{\infty}_0
{
a
n
}
0
∞
and
{
b
n
}
0
∞
\{b_n\}^{\infty}_0
{
b
n
}
0
∞
be two sequences determined by the recursion formulas
a
n
+
1
=
a
n
+
b
n
,
a_{n+1} = a_n + b_n,
a
n
+
1
=
a
n
+
b
n
,
b
n
+
1
=
3
a
n
+
b
n
,
n
=
0
,
1
,
2
,
⋯
,
b_{n+1} = 3a_n + b_n, n= 0, 1, 2, \cdots,
b
n
+
1
=
3
a
n
+
b
n
,
n
=
0
,
1
,
2
,
⋯
,
and the initial values
a
0
=
b
0
=
1
a_0 = b_0 = 1
a
0
=
b
0
=
1
. Prove that there exists a uniquely determined constant
c
c
c
such that
n
∣
c
a
n
−
b
n
∣
<
2
n|ca_n-b_n| < 2
n
∣
c
a
n
−
b
n
∣
<
2
for all nonnegative integers
n
n
n
.
36
1
Hide problems
Area of region enclosed by tangents of 3 concentric circles
Three concentric circles with common center
O
O
O
are cut by a common chord in successive points
A
,
B
,
C
A, B, C
A
,
B
,
C
. Tangents drawn to the circles at the points
A
,
B
,
C
A, B, C
A
,
B
,
C
enclose a triangular region. If the distance from point
O
O
O
to the common chord is equal to
p
p
p
, prove that the area of the region enclosed by the tangents is equal to
A
B
⋅
B
C
⋅
C
A
2
p
\frac{AB \cdot BC \cdot CA}{2p}
2
p
A
B
⋅
BC
⋅
C
A
43
1
Hide problems
Existence of a polynomial given another special polynomial
Prove that if for a polynomial
P
(
x
,
y
)
P(x, y)
P
(
x
,
y
)
, we have
P
(
x
−
1
,
y
−
2
x
+
1
)
=
P
(
x
,
y
)
,
P(x - 1, y - 2x + 1) = P(x, y),
P
(
x
−
1
,
y
−
2
x
+
1
)
=
P
(
x
,
y
)
,
then there exists a polynomial
Φ
(
x
)
\Phi(x)
Φ
(
x
)
with
P
(
x
,
y
)
=
Φ
(
y
−
x
2
)
.
P(x, y) = \Phi(y - x^2).
P
(
x
,
y
)
=
Φ
(
y
−
x
2
)
.
49
1
Hide problems
1976 non similar triangles each with specific angles
Determine whether there exist
1976
1976
1976
nonsimilar triangles with angles
α
,
β
,
γ
,
\alpha, \beta, \gamma,
α
,
β
,
γ
,
each of them satisfying the relations
sin
α
+
sin
β
+
sin
γ
cos
α
+
cos
β
+
cos
γ
=
12
7
and
sin
α
sin
β
sin
γ
=
12
25
\frac{\sin \alpha + \sin\beta + \sin\gamma}{\cos \alpha + \cos \beta + \cos \gamma}=\frac{12}{7}\text{ and }\sin \alpha \sin \beta \sin \gamma =\frac{12}{25}
cos
α
+
cos
β
+
cos
γ
sin
α
+
sin
β
+
sin
γ
=
7
12
and
sin
α
sin
β
sin
γ
=
25
12
38
1
Hide problems
Prove that the fractional part of x exceeds 10^{-19.76}.
Let
x
=
a
+
b
x =\sqrt{a}+\sqrt{b}
x
=
a
+
b
, where
a
a
a
and
b
b
b
are natural numbers,
x
x
x
is not an integer, and
x
<
1976
x < 1976
x
<
1976
. Prove that the fractional part of
x
x
x
exceeds
1
0
−
19.76
10^{-19.76}
1
0
−
19.76
.
40
1
Hide problems
Prove that f(x) is not divisible by x^2 - x + 1
Let
g
(
x
)
g(x)
g
(
x
)
be a fixed polynomial with real coefficients and define
f
(
x
)
f(x)
f
(
x
)
by
f
(
x
)
=
x
2
+
x
g
(
x
3
)
f(x) =x^2 + xg(x^3)
f
(
x
)
=
x
2
+
xg
(
x
3
)
. Show that
f
(
x
)
f(x)
f
(
x
)
is not divisible by
x
2
−
x
+
1
x^2 - x + 1
x
2
−
x
+
1
.
42
1
Hide problems
Ratio of lengths being integers.
For a point
O
O
O
inside a triangle
A
B
C
ABC
A
BC
, denote by
A
1
,
B
1
,
C
1
,
A_1,B_1, C_1,
A
1
,
B
1
,
C
1
,
the respective intersection points of
A
O
,
B
O
,
C
O
AO, BO, CO
A
O
,
BO
,
CO
with the corresponding sides. Let
n
1
=
A
O
A
1
O
,
n
2
=
B
O
B
1
O
,
n
3
=
C
O
C
1
O
.
n_1 =\frac{AO}{A_1O}, n_2 = \frac{BO}{B_1O}, n_3 = \frac{CO}{C_1O}.
n
1
=
A
1
O
A
O
,
n
2
=
B
1
O
BO
,
n
3
=
C
1
O
CO
.
What possible values of
n
1
,
n
2
,
n
3
n_1, n_2, n_3
n
1
,
n
2
,
n
3
can all be positive integers?
50
1
Hide problems
f(x+ 2) - f(x) = x^2 + 2x + 4
Find a function
f
(
x
)
f(x)
f
(
x
)
defined for all real values of
x
x
x
such that for all
x
x
x
,
f
(
x
+
2
)
−
f
(
x
)
=
x
2
+
2
x
+
4
,
f(x+ 2) - f(x) = x^2 + 2x + 4,
f
(
x
+
2
)
−
f
(
x
)
=
x
2
+
2
x
+
4
,
and if
x
∈
[
0
,
2
)
x \in [0, 2)
x
∈
[
0
,
2
)
, then
f
(
x
)
=
x
2
.
f(x) = x^2.
f
(
x
)
=
x
2
.
33
1
Hide problems
Maybe well known as Sylvester Gallai result
A finite set of points
P
P
P
in the plane has the following property: Every line through two points in
P
P
P
contains at least one more point belonging to
P
P
P
. Prove that all points in
P
P
P
lie on a straight line.[hide="Remark."]This may be a well known theorem called "Sylvester Gallai", but I didn't find this problem (I mean, exactly this one) using search function. So please discuss about the problem here, in this topic. Thanks :)
51
1
Hide problems
The swallows can catch the fly
Four swallows are catching a fly. At first, the swallows are at the four vertices of a tetrahedron, and the fly is in its interior. Their maximal speeds are equal. Prove that the swallows can catch the fly.
28
1
Hide problems
There exists an integer n for every disk D ⊂ Q
Let
Q
Q
Q
be a unit square in the plane:
Q
=
[
0
,
1
]
×
[
0
,
1
]
Q = [0, 1] \times [0, 1]
Q
=
[
0
,
1
]
×
[
0
,
1
]
. Let
T
:
Q
⟶
Q
T :Q \longrightarrow Q
T
:
Q
⟶
Q
be defined as follows: T(x, y) =\begin{cases} (2x, \frac{y}{2}) &\mbox{ if } 0 \le x \le \frac{1}{2};\$$2x - 1, \frac{y}{2}+ \frac{1}{2})&\mbox{ if } \frac{1}{2} < x \le 1.\end{cases} Show that for every disk
D
⊂
Q
D \subset Q
D
⊂
Q
there exists an integer
n
>
0
n > 0
n
>
0
such that
T
n
(
D
)
∩
D
≠
∅
.
T^n(D) \cap D \neq \emptyset.
T
n
(
D
)
∩
D
=
∅.
31
1
Hide problems
Inscribed circles in quadrangular pyramid...
Into every lateral face of a quadrangular pyramid a circle is inscribed. The circles inscribed into adjacent faces are tangent (have one point in common). Prove that the points of contact of the circles with the base of the pyramid lie on a circle.
32
1
Hide problems
Infinite chessboard with each entry average of four entries.
We consider the infinite chessboard covering the whole plane. In every field of the chessboard there is a nonnegative real number. Every number is the arithmetic mean of the numbers in the four adjacent fields of the chessboard. Prove that the numbers occurring in the fields of the chessboard are all equal.
37
1
Hide problems
Covering a 11x11 board with central square missing...
From a square board
11
11
11
squares long and
11
11
11
squares wide, the central square is removed. Prove that the remaining
120
120
120
squares cannot be covered by
15
15
15
strips each
8
8
8
units long and one unit wide.
44
1
Hide problems
Circle rolling around another n times.
A circle of radius
1
1
1
rolls around a circle of radius
2
\sqrt{2}
2
. Initially, the tangent point is colored red. Afterwards, the red points map from one circle to another by contact. How many red points will be on the bigger circle when the center of the smaller one has made
n
n
n
circuits around the bigger one?
45
1
Hide problems
Every three circles have a common point...
We are given
n
(
n
≥
5
)
n (n \ge 5)
n
(
n
≥
5
)
circles in a plane. Suppose that every three of them have a common point. Prove that all
n
n
n
circles have a common point.
30
1
Hide problems
Show that(x-a)^kQ(x) has at least k+1 non zero coefficients
Prove that if
P
(
x
)
=
(
x
−
a
)
k
Q
(
x
)
P(x) = (x-a)^kQ(x)
P
(
x
)
=
(
x
−
a
)
k
Q
(
x
)
, where
k
k
k
is a positive integer,
a
a
a
is a nonzero real number,
Q
(
x
)
Q(x)
Q
(
x
)
is a nonzero polynomial, then
P
(
x
)
P(x)
P
(
x
)
has at least
k
+
1
k + 1
k
+
1
nonzero coefficients.
27
1
Hide problems
Constructing a triangle passing through given points
In a plane three points
P
,
Q
,
R
,
P,Q,R,
P
,
Q
,
R
,
not on a line, are given. Let
k
,
l
,
m
k, l, m
k
,
l
,
m
be positive numbers. Construct a triangle
A
B
C
ABC
A
BC
whose sides pass through
P
,
Q
,
P, Q,
P
,
Q
,
and
R
R
R
such that
P
P
P
divides the segment
A
B
AB
A
B
in the ratio
1
:
k
1 : k
1
:
k
,
Q
Q
Q
divides the segment
B
C
BC
BC
in the ratio
1
:
l
1 : l
1
:
l
, and
R
R
R
divides the segment
C
A
CA
C
A
in the ratio
1
:
m
.
1 : m.
1
:
m
.
24
1
Hide problems
There exists interval whose elements satisfy inequality.
Let
0
≤
x
1
≤
x
2
≤
⋯
≤
x
n
≤
1
0 \le x_1 \le x_2\le\cdots\le x_n \le 1
0
≤
x
1
≤
x
2
≤
⋯
≤
x
n
≤
1
. Prove that for all
A
≥
1
A \ge 1
A
≥
1
, there exists an interval
I
I
I
of length
2
A
n
2\sqrt[n]{A}
2
n
A
such that for all
x
∈
I
x \in I
x
∈
I
,
∣
(
x
−
x
1
)
(
x
−
x
2
)
⋯
(
x
−
x
n
)
∣
≤
A
.
|(x - x_1)(x - x_2) \cdots (x -x_n)| \le A.
∣
(
x
−
x
1
)
(
x
−
x
2
)
⋯
(
x
−
x
n
)
∣
≤
A
.
23
1
Hide problems
Concentric circles-inscribed triangle-one irrational side
Prove that in a Euclidean plane there are infinitely many concentric circles
C
C
C
such that all triangles inscribed in
C
C
C
have at least one irrational side.
22
1
Hide problems
Regular pentagon and spheres at each vertex
A regular pentagon
A
1
A
2
A
3
A
4
A
5
A_1A_2A_3A_4A_5
A
1
A
2
A
3
A
4
A
5
with side length
s
s
s
is given. At each point
A
i
A_i
A
i
, a sphere
K
i
K_i
K
i
of radius
s
2
\frac{s}{2}
2
s
is constructed. There are two spheres
K
1
K_1
K
1
and
K
2
K_2
K
2
each of radius
s
2
\frac{s}{2}
2
s
touching all the five spheres
K
i
.
K_i.
K
i
.
Decide whether
K
1
K_1
K
1
and
K
2
K_2
K
2
intersect each other, touch each other, or have no common points.
21
1
Hide problems
Find largest p for which inequalityin n variables hold.
Find the largest positive real number
p
p
p
(if it exists) such that the inequality
x
1
2
+
x
2
2
+
⋯
+
x
n
2
≥
p
(
x
1
x
2
+
x
2
x
3
+
⋯
+
x
n
−
1
x
n
)
x^2_1+ x_2^2+ \cdots + x^2_n\ge p(x_1x_2 + x_2x_3 + \cdots + x_{n-1}x_n)
x
1
2
+
x
2
2
+
⋯
+
x
n
2
≥
p
(
x
1
x
2
+
x
2
x
3
+
⋯
+
x
n
−
1
x
n
)
is satisfied for all real numbers
x
i
x_i
x
i
, and
(
a
)
n
=
2
;
(
b
)
n
=
5.
(a) n = 2; (b) n = 5.
(
a
)
n
=
2
;
(
b
)
n
=
5.
Find the largest positive real number
p
p
p
(if it exists) such that the inequality holds for all real numbers
x
i
x_i
x
i
and all natural numbers
n
,
n
≥
2.
n, n \ge 2.
n
,
n
≥
2.
20
1
Hide problems
Existence of positive number that satisfies inequality
Let
(
a
n
)
,
n
=
0
,
1
,
.
.
.
,
(a_n), n = 0, 1, . . .,
(
a
n
)
,
n
=
0
,
1
,
...
,
be a sequence of real numbers such that
a
0
=
0
a_0 = 0
a
0
=
0
and
a
n
+
1
3
=
1
2
a
n
2
−
1
,
n
=
0
,
1
,
⋯
a^3_{n+1} = \frac{1}{2} a^2_n -1, n= 0, 1,\cdots
a
n
+
1
3
=
2
1
a
n
2
−
1
,
n
=
0
,
1
,
⋯
Prove that there exists a positive number
q
,
q
<
1
q, q < 1
q
,
q
<
1
, such that for all
n
=
1
,
2
,
…
,
n = 1, 2, \ldots ,
n
=
1
,
2
,
…
,
∣
a
n
+
1
−
a
n
∣
≤
q
∣
a
n
−
a
n
−
1
∣
,
|a_{n+1} - a_n| \leq q|a_n - a_{n-1}|,
∣
a
n
+
1
−
a
n
∣
≤
q
∣
a
n
−
a
n
−
1
∣
,
and give one such
q
q
q
explicitly.
19
1
Hide problems
Proving that a fraction is a national number
For a positive integer
n
n
n
, let
6
(
n
)
6^{(n)}
6
(
n
)
be the natural number whose decimal representation consists of
n
n
n
digits
6
6
6
. Let us define, for all natural numbers
m
m
m
,
k
k
k
with
1
≤
k
≤
m
1 \leq k \leq m
1
≤
k
≤
m
[
m
k
]
=
6
(
m
)
6
(
m
−
1
)
⋯
6
(
m
−
k
+
1
)
6
(
1
)
6
(
2
)
⋯
6
(
k
)
.
\left[\begin{array}{ccc}m\\ k\end{array}\right] =\frac{ 6^{(m)} 6^{(m-1)}\cdots 6^{(m-k+1)}}{6^{(1)} 6^{(2)}\cdots 6^{(k)}} .
[
m
k
]
=
6
(
1
)
6
(
2
)
⋯
6
(
k
)
6
(
m
)
6
(
m
−
1
)
⋯
6
(
m
−
k
+
1
)
.
Prove that for all
m
,
k
m, k
m
,
k
,
[
m
k
]
\left[\begin{array}{ccc}m\\ k\end{array}\right]
[
m
k
]
is a natural number whose decimal representation consists of exactly
k
(
m
+
k
−
1
)
−
1
k(m + k - 1) - 1
k
(
m
+
k
−
1
)
−
1
digits.
18
1
Hide problems
Proving that a number is divisible by 4th Fermat prime F_4
Prove that the number
1
9
1976
+
7
6
1976
19^{1976} + 76^{1976}
1
9
1976
+
7
6
1976
:
(
a
)
(a)
(
a
)
is divisible by the (Fermat) prime number
F
4
=
2
2
4
+
1
F_4 = 2^{2^4} + 1
F
4
=
2
2
4
+
1
;
(
b
)
(b)
(
b
)
is divisible by at least four distinct primes other than
F
4
F_4
F
4
.
17
1
Hide problems
Existence of convex polyhedron with vertices on sphere
Show that there exists a convex polyhedron with all its vertices on the surface of a sphere and with all its faces congruent isosceles triangles whose ratio of sides are
3
:
3
:
2
\sqrt{3} :\sqrt{3} :2
3
:
3
:
2
.
16
1
Hide problems
7^n contains block of m consecutive zeroes for any m
Prove that there is a positive integer
n
n
n
such that the decimal representation of
7
n
7^n
7
n
contains a block of at least
m
m
m
consecutive zeros, where
m
m
m
is any given positive integer.
15
1
Hide problems
Concurrency on coplanar triangles
Let
A
B
C
ABC
A
BC
and
A
′
B
′
C
′
A'B'C'
A
′
B
′
C
′
be any two coplanar triangles. Let
L
L
L
be a point such that
A
L
∣
∣
B
C
,
A
′
L
∣
∣
B
′
C
′
AL || BC, A'L || B'C'
A
L
∣∣
BC
,
A
′
L
∣∣
B
′
C
′
, and
M
,
N
M,N
M
,
N
similarly defined. The line
B
C
BC
BC
meets
B
′
C
′
B'C'
B
′
C
′
at
P
P
P
, and similarly defined are
Q
Q
Q
and
R
R
R
. Prove that
P
L
,
Q
M
,
R
N
PL, QM, RN
P
L
,
QM
,
RN
are concurrent.
14
1
Hide problems
Each term of sequence differs by 2 from an integral square
A sequence
{
u
n
}
\{ u_n \}
{
u
n
}
of integers is defined by
u
1
=
2
,
u
2
=
u
3
=
7
,
u_1 = 2, u_2 = u_3 = 7,
u
1
=
2
,
u
2
=
u
3
=
7
,
u
n
+
1
=
u
n
u
n
−
1
−
u
n
−
2
,
for
n
≥
3
u_{n+1} = u_nu_{n-1} - u_{n-2}, \text{ for }n \geq 3
u
n
+
1
=
u
n
u
n
−
1
−
u
n
−
2
,
for
n
≥
3
Prove that for each
n
≥
1
n \geq 1
n
≥
1
,
u
n
u_n
u
n
differs by
2
2
2
from an integral square.
12
1
Hide problems
Maximum of smallest distance between two points
Five points lie on the surface of a ball of unit radius. Find the maximum of the smallest distance between any two of them.
10
1
Hide problems
Writing reciprocal of 2(m^2+m+1) as sum of consecutive terms
Show that the reciprocal of any number of the form
2
(
m
2
+
m
+
1
)
2(m^2+m+1)
2
(
m
2
+
m
+
1
)
, where
m
m
m
is a positive integer, can be represented as a sum of consecutive terms in the sequence
(
a
j
)
j
=
1
∞
(a_j)_{j=1}^{\infty}
(
a
j
)
j
=
1
∞
a
j
=
1
j
(
j
+
1
)
(
j
+
2
)
a_j = \frac{1}{j(j + 1)(j + 2)}
a
j
=
j
(
j
+
1
)
(
j
+
2
)
1
9
1
Hide problems
Solve in eight variables
Find all (real) solutions of the system
3
x
1
−
x
2
−
x
3
−
x
5
=
0
,
3x_1-x_2-x_3-x_5 = 0,
3
x
1
−
x
2
−
x
3
−
x
5
=
0
,
−
x
1
+
3
x
2
−
x
4
−
x
6
=
0
,
-x_1+3x_2-x_4-x_6= 0,
−
x
1
+
3
x
2
−
x
4
−
x
6
=
0
,
−
x
1
+
3
x
3
−
x
4
−
x
7
=
0
,
-x_1 + 3x_3 - x_4 - x_7 = 0,
−
x
1
+
3
x
3
−
x
4
−
x
7
=
0
,
−
x
2
−
x
3
+
3
x
4
−
x
8
=
0
,
-x_2 - x_3 + 3x_4 - x_8 = 0,
−
x
2
−
x
3
+
3
x
4
−
x
8
=
0
,
−
x
1
+
3
x
5
−
x
6
−
x
7
=
0
,
-x_1 + 3x_5 - x_6 - x_7 = 0,
−
x
1
+
3
x
5
−
x
6
−
x
7
=
0
,
−
x
2
−
x
5
+
3
x
6
−
x
8
=
0
,
-x_2 - x_5 + 3x_6 - x_8 = 0,
−
x
2
−
x
5
+
3
x
6
−
x
8
=
0
,
−
x
3
−
x
5
+
3
x
7
−
x
8
=
0
,
-x_3 - x_5 + 3x_7 - x_8 = 0,
−
x
3
−
x
5
+
3
x
7
−
x
8
=
0
,
−
x
4
−
x
6
−
x
7
+
3
x
8
=
0.
-x_4 - x_6 - x_7 + 3x_8 = 0.
−
x
4
−
x
6
−
x
7
+
3
x
8
=
0.
7
1
Hide problems
Translating a triangle by a given vector
Let
P
P
P
be a fixed point and
T
T
T
a given triangle that contains the point
P
P
P
. Translate the triangle
T
T
T
by a given vector
v
\bold{v}
v
and denote by
T
′
T'
T
′
this new triangle. Let
r
,
R
r, R
r
,
R
, respectively, be the radii of the smallest disks centered at
P
P
P
that contain the triangles
T
,
T
′
T , T'
T
,
T
′
, respectively. Prove that
r
+
∣
v
∣
≤
3
R
r + |\bold{v}| \leq 3R
r
+
∣
v
∣
≤
3
R
and find an example to show that equality can occur.
6
1
Hide problems
Sum of distances of point from all faces of polytope
For each point
X
X
X
of a given polytope, denote by
f
(
X
)
f(X)
f
(
X
)
the sum of the distances of the point
X
X
X
from all the planes of the faces of the polytope. Prove that if
f
f
f
attains its maximum at an interior point of the polytope, then
f
f
f
is constant.
5
1
Hide problems
Intersection of plane with pyramid being a parallelogram
Let
A
B
C
D
S
ABCDS
A
BC
D
S
be a pyramid with four faces and with
A
B
C
D
ABCD
A
BC
D
as a base, and let a plane
α
\alpha
α
through the vertex
A
A
A
meet its edges
S
B
SB
SB
and
S
D
SD
S
D
at points
M
M
M
and
N
N
N
, respectively. Prove that if the intersection of the plane
α
\alpha
α
with the pyramid
A
B
C
D
S
ABCDS
A
BC
D
S
is a parallelogram, then
S
M
⋅
S
N
>
B
M
⋅
D
N
SM \cdot SN > BM \cdot DN
SM
⋅
SN
>
BM
⋅
D
N
.
4
1
Hide problems
Find natural m, n for which 2^m3^n +1 is a perfect square
Find all pairs of natural numbers
(
m
,
n
)
(m, n)
(
m
,
n
)
for which
2
m
3
n
+
1
2^m3^n +1
2
m
3
n
+
1
is the square of some integer.
2
1
Hide problems
Inequality on number of points and planes
Let
P
P
P
be a set of
n
n
n
points and
S
S
S
a set of
l
l
l
segments. It is known that:
(
i
)
(i)
(
i
)
No four points of
P
P
P
are coplanar.
(
i
i
)
(ii)
(
ii
)
Any segment from
S
S
S
has its endpoints at
P
P
P
.
(
i
i
i
)
(iii)
(
iii
)
There is a point, say
g
g
g
, in
P
P
P
that is the endpoint of a maximal number of segments from
S
S
S
and that is not a vertex of a tetrahedron having all its edges in
S
S
S
. Prove that
l
≤
n
2
3
l \leq \frac{n^2}{3}
l
≤
3
n
2
46
1
Hide problems
5 lines for Turkevici's Inequality
Let
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
be nonnegative real numbers. Prove that a^4\plus{}b^4\plus{}c^4\plus{}d^4\plus{}2abcd \ge a^2b^2\plus{}a^2c^2\plus{}a^2d^2\plus{}b^2c^2\plus{}b^2d^2\plus{}c^2d^2.
39
1
Hide problems
may be ez
In
A
B
C
ABC
A
BC
, the inscribed circle is tangent to side
B
C
BC
BC
at
X
X
X
. Segment
A
X
AX
A
X
is drawn. Prove that the line joining the midpoint of
A
X
AX
A
X
to the midpoint of side
B
C
BC
BC
passes through center
I
I
I
of the inscribed circle.