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Contests
National and Regional Contests
Saudi Arabia Contests
Saudi Arabia Pre-TST + Training Tests
2021 Saudi Arabia Training Tests
2021 Saudi Arabia Training Tests
Part of
Saudi Arabia Pre-TST + Training Tests
Subcontests
(37)
23
1
Hide problems
inradius of XYZ is R/2, symmedian point and 3 parallelograms related
Let
A
B
C
ABC
A
BC
be triangle with the symmedian point
L
L
L
and circumradius
R
R
R
. Construct parallelograms
A
D
L
E
ADLE
A
D
L
E
,
B
H
L
K
BHLK
B
H
L
K
,
C
I
L
J
CILJ
C
I
L
J
such that
D
,
H
∈
A
B
D,H \in AB
D
,
H
∈
A
B
,
K
,
I
∈
B
C
K, I \in BC
K
,
I
∈
BC
,
J
,
E
∈
C
A
J,E \in CA
J
,
E
∈
C
A
Suppose that
D
E
DE
D
E
,
H
K
HK
HK
,
I
J
IJ
I
J
pairwise intersect at
X
,
Y
,
Z
X, Y,Z
X
,
Y
,
Z
. Prove that inradius of
X
Y
Z
XYZ
X
Y
Z
is
R
2
\frac{R}{2}
2
R
.
24
1
Hide problems
circumscribed quadrilateral wanted, midpoint and 2 excenters related
Let
A
B
C
ABC
A
BC
be triangle with
M
M
M
is the midpoint of
B
C
BC
BC
and
X
,
Y
X, Y
X
,
Y
are excenters with respect to angle
B
,
C
B,C
B
,
C
. Prove that
M
X
MX
MX
,
M
Y
MY
M
Y
intersect
A
B
AB
A
B
,
A
C
AC
A
C
at four points that are vertices of circumscribed quadrilateral.
22
1
Hide problems
AT_|_MN wanted, H altitudes of ABC, T circumcenter of HBC
Let
A
B
C
ABC
A
BC
be a non-isosceles triangle with altitudes
A
D
AD
A
D
,
B
E
BE
BE
,
C
F
CF
CF
with orthocenter
H
H
H
. Suppose that
D
F
∩
H
B
=
M
DF \cap HB = M
D
F
∩
H
B
=
M
,
D
E
∩
H
C
=
N
DE \cap HC = N
D
E
∩
H
C
=
N
and
T
T
T
is the circumcenter of triangle
H
B
C
HBC
H
BC
. Prove that
A
T
⊥
M
N
AT\perp MN
A
T
⊥
MN
.
21
1
Hide problems
circle of diameter OI is tangent to 2 circles, Gauss line of cyclic ABCD
Let
A
B
C
D
ABCD
A
BC
D
be a cyclic quadrilateral with
O
O
O
is circumcenter and
A
C
AC
A
C
meets
B
D
BD
B
D
at
I
I
I
Suppose that rays
D
A
,
C
D
DA,CD
D
A
,
C
D
meet at
E
E
E
and rays
B
A
,
C
D
BA,CD
B
A
,
C
D
meet at
F
F
F
. The Gauss line of
A
B
C
D
ABCD
A
BC
D
meets
A
B
,
B
C
,
C
D
,
D
A
AB,BC,CD,DA
A
B
,
BC
,
C
D
,
D
A
at points
M
,
N
,
P
,
Q
M,N,P,Q
M
,
N
,
P
,
Q
respectively. Prove that the circle of diameter
O
I
OI
O
I
is tangent to two circles
(
E
N
Q
)
,
(
F
M
P
)
(ENQ), (FMP)
(
ENQ
)
,
(
FMP
)
20
1
Hide problems
AO bisects EF, midpoint of altitude, circumcenter related
Let
A
B
C
ABC
A
BC
be an acute, non-isosceles triangle with altitude
A
D
AD
A
D
(
D
∈
B
C
D \in BC
D
∈
BC
),
M
M
M
is the midpoint of
A
D
AD
A
D
and
O
O
O
is the circumcenter. Line
A
O
AO
A
O
meets
B
C
BC
BC
at
K
K
K
and circle of center
K
K
K
, radius
K
A
KA
K
A
cuts
A
B
,
A
C
AB,AC
A
B
,
A
C
at
E
,
F
E, F
E
,
F
respectively. Prove that
A
O
AO
A
O
bisects
E
F
EF
EF
.
19
1
Hide problems
(HRS) tangent to OD wanted, circumcircle related
Let
A
B
C
ABC
A
BC
be a triangle with
A
B
<
A
C
AB < AC
A
B
<
A
C
inscribed in
(
O
)
(O)
(
O
)
. Tangent line at
A
A
A
of
(
O
)
(O)
(
O
)
cuts
B
C
BC
BC
at
D
D
D
. Take
H
H
H
as the projection of
A
A
A
on
O
D
OD
O
D
and
E
,
F
E,F
E
,
F
as projections of
H
H
H
on
A
B
,
A
C
AB,AC
A
B
,
A
C
.Suppose that
E
F
EF
EF
cuts
(
O
)
(O)
(
O
)
at
R
,
S
R,S
R
,
S
. Prove that
(
H
R
S
)
(HRS)
(
H
RS
)
is tangent to
O
D
OD
O
D
18
1
Hide problems
midpoint wanted, CD = CK, incircle related
Let
A
B
C
ABC
A
BC
be a triangle with
A
B
<
A
C
AB < AC
A
B
<
A
C
and incircle
(
I
)
(I)
(
I
)
tangent to
B
C
BC
BC
at
D
D
D
. Take
K
K
K
on
A
D
AD
A
D
such that
C
D
=
C
K
CD = CK
C
D
=
C
K
. Suppose that
A
D
AD
A
D
cuts
(
I
)
(I)
(
I
)
at
G
G
G
and
B
G
BG
BG
cuts
C
K
CK
C
K
at
L
L
L
. Prove that K is the midpoint of
C
L
CL
C
L
.
17
1
Hide problems
(MKT) is tangent to (ABC)
Let
A
B
C
ABC
A
BC
be an acute, non-isosceles triangle with circumcenter
O
O
O
. Tangent lines to
(
O
)
(O)
(
O
)
at
B
,
C
B,C
B
,
C
meet at
T
T
T
. A line passes through
T
T
T
cuts segments
A
B
AB
A
B
at
D
D
D
and cuts ray
C
A
CA
C
A
at
E
E
E
. Take
M
M
M
as midpoint of
D
E
DE
D
E
and suppose that
M
A
MA
M
A
cuts
(
O
)
(O)
(
O
)
again at
K
K
K
. Prove that
(
M
K
T
)
(MKT)
(
M
K
T
)
is tangent to
(
O
)
(O)
(
O
)
.
39
1
Hide problems
sum a_i divides 100! + sum b_i
Determine if there exists pairwise distinct positive integers
a
1
a_1
a
1
,
a
2
a_2
a
2
,
.
.
.
...
...
,
a
101
a_{101}
a
101
,
b
1
b_1
b
1
,
b
2
b_2
b
2
,
.
.
.
...
...
,
b
101
b_{101}
b
101
satisfying the following property: for each non-empty subset
S
S
S
of
{
1
,
2
,
.
.
.
,
101
}
\{1, 2, ..., 101\}
{
1
,
2
,
...
,
101
}
the sum
∑
i
∈
S
a
i
\sum_{i \in S} a_i
∑
i
∈
S
a
i
divides
100
!
+
∑
i
∈
S
b
i
100! + \sum_{i \in S} b_i
100
!
+
∑
i
∈
S
b
i
.
40
1
Hide problems
exist distinct primes p_1, p_2, ..., p_n such that m + k is divisible by p_k
Given
m
,
n
m, n
m
,
n
such that
m
>
n
n
−
1
m > n^{n-1}
m
>
n
n
−
1
and the number
m
+
1
m+1
m
+
1
,
m
+
2
m+2
m
+
2
,
.
.
.
...
...
,
m
+
n
m+n
m
+
n
are composite. Prove that there exist distinct primes
p
1
,
p
2
,
.
.
.
,
p
n
p_1, p_2, ..., p_n
p
1
,
p
2
,
...
,
p
n
such that
m
+
k
m + k
m
+
k
is divisible by
p
k
p_k
p
k
for each
k
=
1
,
2
,
.
.
.
k = 1, 2, ...
k
=
1
,
2
,
...
38
1
Hide problems
property of set of all divisors of a positive integer not a perfect square
Prove that the set of all divisors of a positive integer which is not a perfect square can be divided into pairs so that in each pair, one number is divided by another.
37
1
Hide problems
max no of perfect cubes among their pairwise products
Given
n
≥
2
n \ge 2
n
≥
2
distinct positive integers
a
1
,
a
2
,
.
.
.
,
a
n
a_1, a_2, ..., a_n
a
1
,
a
2
,
...
,
a
n
none of which is a perfect cube. Find the maximal possible number of perfect cubes among their pairwise products.
36
1
Hide problems
330 seats in the first row of the auditorium
There are
330
330
330
seats in the first row of the auditorium. Some of these seats are occupied by
25
25
25
viewers. Prove that among the pairwise distances between these viewers, there are two equal.
35
1
Hide problems
deg P = 1 if sequence contains a b−th power of some positive integer >1
Let
P
(
x
)
P (x)
P
(
x
)
be a non constant integer polynomial and positive integer
n
n
n
. The sequence
a
0
,
a
1
,
.
.
.
a_0, a_1, ...
a
0
,
a
1
,
...
is defined by
a
0
=
n
a_0 = n
a
0
=
n
and
a
k
=
P
(
a
k
−
1
)
a_k = P (a_{k-1})
a
k
=
P
(
a
k
−
1
)
for
k
≥
1
k \ge 1
k
≥
1
. Given that for each positive integer
b
b
b
, the sequence contains a
b
b
b
-th power of some positive integer greater than
1
1
1
. Prove that deg
P
=
1
P = 1
P
=
1
34
1
Hide problems
b_1 = a_0, b_{n+1} = P (b_n) , P (x) = a_dx^d + ... + a_2x^2 + a_0
Let coefficients of the polynomial
P
(
x
)
=
a
d
x
d
+
.
.
.
+
a
2
x
2
+
a
0
P (x) = a_dx^d + ... + a_2x^2 + a_0
P
(
x
)
=
a
d
x
d
+
...
+
a
2
x
2
+
a
0
where
d
≥
2
d \ge 2
d
≥
2
, are positive integers. The sequences
(
b
n
)
(b_n)
(
b
n
)
is defined by
b
1
=
a
0
b_1 = a_0
b
1
=
a
0
and
b
n
+
1
=
P
(
b
n
)
b_{n+1} = P (b_n)
b
n
+
1
=
P
(
b
n
)
for
n
≥
1
n \ge 1
n
≥
1
. Prove that for any
n
≥
2
n \ge 2
n
≥
2
, there exists a prime number
p
p
p
such that
p
∣
b
n
p|b_n
p
∣
b
n
but it does not divide
b
1
,
b
2
,
.
.
.
,
b
n
−
1
b_1, b_2, ..., b_{n-1}
b
1
,
b
2
,
...
,
b
n
−
1
.
33
1
Hide problems
each integer k satisfies both |x - k^2| > 10^6 and |x - k^3| > 10^6
Call a positive integer
x
x
x
to be remote from squares and cubes if each integer
k
k
k
satisfies both
∣
x
−
k
2
∣
>
1
0
6
|x - k^2| > 10^6
∣
x
−
k
2
∣
>
1
0
6
and
∣
x
−
k
3
∣
>
1
0
6
|x - k^3| > 10^6
∣
x
−
k
3
∣
>
1
0
6
. Prove that there exist infinitely many positive integer
n
n
n
such that
2
n
2^n
2
n
is remote from squares and cubes.
32
1
Hide problems
exist M for which a_n = a_M for n >= M, remainder a_k is divided by 2^n
Let
N
N
N
be a positive integer. Consider the sequence
a
1
,
a
2
,
.
.
.
,
a
N
a_1, a_2, ..., a_N
a
1
,
a
2
,
...
,
a
N
of positive integers, none of which is a multiple of
2
N
+
1
2^{N+1}
2
N
+
1
. For
n
≥
N
+
1
n \ge N +1
n
≥
N
+
1
, the number
a
n
a_n
a
n
is defined as follows: choose
k
k
k
to be the number among
1
,
2
,
.
.
.
,
n
−
1
1, 2, ..., n - 1
1
,
2
,
...
,
n
−
1
for which the remainder obtained when
a
k
a_k
a
k
is divided by
2
n
2^n
2
n
is the smallest, and define
a
n
=
2
a
k
a_n = 2a_k
a
n
=
2
a
k
(if there are more than one such
k
k
k
, choose the largest such
k
k
k
). Prove that there exist
M
M
M
for which
a
n
=
a
M
a_n = a_M
a
n
=
a
M
holds for every
n
≥
M
n \ge M
n
≥
M
.
31
1
Hide problems
partition set M = {n, n + 1, ..., m}
Let
n
n
n
be a positive integer. What is the smallest value of
m
m
m
with
m
>
n
m > n
m
>
n
such that the set
M
=
{
n
,
n
+
1
,
.
.
.
,
m
}
M = \{n, n + 1, ..., m\}
M
=
{
n
,
n
+
1
,
...
,
m
}
can be partitioned into subsets so that in each subset, there is a number which equals to the sum of all other numbers of this subset?
30
1
Hide problems
f(k) is number of m such that remainder of km modulo 2019^3 > m
For a positive integer
k
k
k
, denote by
f
(
k
)
f(k)
f
(
k
)
the number of positive integer
m
m
m
such that the remainder of
k
m
km
km
modulo
201
9
3
2019^3
201
9
3
is greater than
m
m
m
. Find the amount of different numbers among
f
(
1
)
,
f
(
2
)
,
.
.
.
,
f
(
201
9
3
)
f(1), f(2), ..., f(2019^3)
f
(
1
)
,
f
(
2
)
,
...
,
f
(
201
9
3
)
.
29
1
Hide problems
fill cells of an 8x 8 table with numbers from 1 to 64
Prove that it is impossible to fill the cells of an
8
×
8
8 \times 8
8
×
8
table with the numbers from
1
1
1
to
64
64
64
(each number must be used once) so that for each
2
×
2
2\times 2
2
×
2
square, the difference between products of the numbers on it’s diagonals will be equal to
1
1
1
.
28
1
Hide problems
mark the vertices of a regular n- gon with numbers from 1 to n
Find all positive integer
n
≥
3
n\ge 3
n
≥
3
such that it is possible to mark the vertices of a regular
n
n
n
- gon with the number from 1 to n so that for any three vertices
A
,
B
A, B
A
,
B
and
C
C
C
with
A
B
=
A
C
AB = AC
A
B
=
A
C
, the number in
A
A
A
is greater or smaller than both numbers in
B
,
C
B, C
B
,
C
.
27
1
Hide problems
N people have chosen 5 elements from a 23-element set
Each of
N
N
N
people have chosen some
5
5
5
elements from a
23
23
23
-element set so that any two people share at most
3
3
3
chosen elements. Does this mean that
N
≤
2020
N \le 2020
N
≤
2020
? Answer the same question with
25
25
25
instead of
23
23
23
.
26
1
Hide problems
periodic sequence if a_k = a_{k+t} = a_{k+2t} = .... ?
Given an infinite sequence of numbers
a
1
,
a
2
,
a
3
,
.
.
.
a_1, a_2, a_3, ...
a
1
,
a
2
,
a
3
,
...
such that for each positive integer
k
k
k
, there exists positive integer
t
t
t
for which
a
k
=
a
k
+
t
=
a
k
+
2
t
=
.
.
.
.
a_k = a_{k+t} = a_{k+2t} = ....
a
k
=
a
k
+
t
=
a
k
+
2
t
=
....
Does this sequences must be periodic?
25
1
Hide problems
Magician and Assistant trick with sequence of N digits. on board
The Magician and his Assistant show trick. The Viewer writes on the board the sequence of
N
N
N
digits. Then the Assistant covers some pair of adjacent digits so that they become invisible. Finally, the Magician enters the show, looks at the board and guesses the covered digits and their order. Find the minimal
N
N
N
such that the Magician and his Assistant can agree in advance so that the Magician always guesses right
16
1
Hide problems
(PQD) bisects segment BC
Let
A
B
C
ABC
A
BC
be an acute, non-isosceles triangle with circumcenter
O
O
O
, incenter
I
I
I
and
(
I
)
(I)
(
I
)
tangent to
B
C
BC
BC
,
C
A
CA
C
A
,
A
B
AB
A
B
at
D
,
E
,
F
D, E, F
D
,
E
,
F
respectively. Suppose that
E
F
EF
EF
cuts
(
O
)
(O)
(
O
)
at
P
,
Q
P, Q
P
,
Q
. Prove that
(
P
Q
D
)
(PQD)
(
PQ
D
)
bisects segment
B
C
BC
BC
.
15
1
Hide problems
<XAD+<XCD=<XBC+<XDC if XA x XC^2 = XB x XD^2, <AXD+<BXC=<CXD
Let
A
B
C
ABC
A
BC
be convex quadrilateral and
X
X
X
lying inside it such that
X
A
⋅
X
C
2
=
X
B
⋅
X
D
2
XA \cdot XC^2 = XB \cdot XD^2
X
A
⋅
X
C
2
=
XB
⋅
X
D
2
and
∠
A
X
D
+
∠
B
X
C
=
∠
C
X
D
\angle AXD + \angle BXC = \angle CXD
∠
A
X
D
+
∠
BXC
=
∠
CX
D
. Prove that
∠
X
A
D
+
∠
X
C
D
=
∠
X
B
C
+
∠
X
D
C
\angle XAD + \angle XCD = \angle XBC + \angle XDC
∠
X
A
D
+
∠
XC
D
=
∠
XBC
+
∠
X
D
C
.
12
1
Hide problems
AT passes through an intersection of (J) and (DEF ), incenter, excenter
Let
A
B
C
ABC
A
BC
be a triangle with circumcenter
O
O
O
and incenter
I
I
I
, ex-center in angle
A
A
A
is
J
J
J
. Denote
D
D
D
as the tangent point of
(
I
)
(I)
(
I
)
on
B
C
BC
BC
and the angle bisector of angle
A
A
A
cuts
B
C
BC
BC
,
(
O
)
(O)
(
O
)
respectively at
E
,
F
E, F
E
,
F
. The circle
(
D
E
F
)
(DEF )
(
D
EF
)
meets
(
O
)
(O)
(
O
)
again at
T
T
T
. Prove that
A
T
AT
A
T
passes through an intersection of
(
J
)
(J)
(
J
)
and
(
D
E
F
)
(DEF )
(
D
EF
)
.
13
1
Hide problems
<ADC = 2<ABE. if <A =<B = 90^o, AB = AD., CD = BC + AD, AD > BC
Let
A
B
C
D
ABCD
A
BC
D
be a quadrilateral with
∠
A
=
∠
B
=
9
0
o
\angle A = \angle B = 90^o
∠
A
=
∠
B
=
9
0
o
,
A
B
=
A
D
AB = AD
A
B
=
A
D
. Denote
E
E
E
as the midpoint of
A
D
AD
A
D
, suppose that
C
D
=
B
C
+
A
D
CD = BC + AD
C
D
=
BC
+
A
D
,
A
D
>
B
C
AD > BC
A
D
>
BC
. Prove that
∠
A
D
C
=
2
∠
A
B
E
\angle ADC = 2\angle ABE
∠
A
D
C
=
2∠
A
BE
.
10
1
Hide problems
AB = 2CD wanted, circle tangent to chord and another circle
Let
A
B
AB
A
B
be a chord of the circle
(
O
)
(O)
(
O
)
. Denote M as the midpoint of the minor arc
A
B
AB
A
B
. A circle
(
O
′
)
(O')
(
O
′
)
tangent to segment
A
B
AB
A
B
and internally tangent to
(
O
)
(O)
(
O
)
. A line passes through
M
M
M
, perpendicular to
O
′
A
O'A
O
′
A
,
O
′
B
O'B
O
′
B
and cuts
A
B
AB
A
B
respectively at
C
,
D
C, D
C
,
D
. Prove that
A
B
=
2
C
D
AB = 2CD
A
B
=
2
C
D
.
6
1
Hide problems
XY _|_ AM if RX, SY _|_ DE, MD = ME, AB, AC tangents of (O)
Let
A
A
A
be a point lies outside circle
(
O
)
(O)
(
O
)
and tangent lines
A
B
AB
A
B
,
A
C
AC
A
C
of
(
O
)
(O)
(
O
)
. Consider points
D
,
E
,
M
D, E, M
D
,
E
,
M
on
(
O
)
(O)
(
O
)
such that
M
D
=
M
E
MD = ME
M
D
=
ME
. The line
D
E
DE
D
E
cuts
M
B
MB
MB
,
M
C
MC
MC
at
R
,
S
R, S
R
,
S
. Take
X
∈
O
B
X \in OB
X
∈
OB
,
Y
∈
O
C
Y \in OC
Y
∈
OC
such that
R
X
,
S
Y
⊥
D
E
RX, SY \perp DE
RX
,
S
Y
⊥
D
E
. Prove that
X
Y
⊥
A
M
XY \perp AM
X
Y
⊥
A
M
.
5
1
Hide problems
BM = DN wanted , ABCD rectangle, QB = QC, RC = RD, 2 circles
Let
A
B
C
D
ABCD
A
BC
D
be a rectangle with
P
P
P
lies on the segment
A
C
AC
A
C
. Denote
Q
Q
Q
as a point on minor arc
P
B
PB
PB
of
(
P
A
B
)
(PAB)
(
P
A
B
)
such that
Q
B
=
Q
C
QB = QC
QB
=
QC
. Denote
R
R
R
as a point on minor arc
P
D
PD
P
D
of
(
P
A
D
)
(PAD)
(
P
A
D
)
such that
R
C
=
R
D
RC = RD
RC
=
R
D
. The lines
C
B
CB
CB
,
C
D
CD
C
D
meet
(
C
Q
R
)
(CQR)
(
CQR
)
again at
M
,
N
M, N
M
,
N
respectively. Prove that
B
M
=
D
N
BM = DN
BM
=
D
N
.by Tran Quang Hung
4
1
Hide problems
<BAC <= 60^o if AX=IK, touchpoints of incircle, CM//AB, AP//BC, BN//AC, AQ//BC
Let
A
B
C
ABC
A
BC
be a triangle with incircle
(
I
)
(I)
(
I
)
, tangent to
B
C
BC
BC
,
C
A
CA
C
A
,
A
B
AB
A
B
at
D
,
E
,
F
D, E, F
D
,
E
,
F
respectively. On the line
D
F
DF
D
F
, take points
M
,
P
M, P
M
,
P
such that
C
M
∥
A
B
CM \parallel AB
CM
∥
A
B
,
A
P
∥
B
C
AP \parallel BC
A
P
∥
BC
. On the line
D
E
DE
D
E
, take points
N
N
N
,
Q
Q
Q
such that
B
N
∥
A
C
BN \parallel AC
BN
∥
A
C
,
A
Q
∥
B
C
AQ \parallel BC
A
Q
∥
BC
. Denote
X
X
X
as intersection of
P
E
PE
PE
,
Q
F
QF
QF
and
K
K
K
as the midpoint of
B
C
BC
BC
. Prove that if
A
X
=
I
K
AX = IK
A
X
=
I
K
then
∠
B
A
C
≤
6
0
o
\angle BAC \le 60^o
∠
B
A
C
≤
6
0
o
.
3
1
Hide problems
perpendicular bisectors of PQ bisects two segments AO, BC, altitudes
Let
A
B
C
ABC
A
BC
be an acute, non-isosceles triangle inscribed in (O) and
B
B
′
BB'
B
B
′
,
C
C
′
CC'
C
C
′
are altitudes. Denote
E
,
F
E, F
E
,
F
as the intersections of
B
B
′
BB'
B
B
′
,
C
C
′
CC'
C
C
′
with
(
O
)
(O)
(
O
)
and
D
,
P
,
Q
D, P, Q
D
,
P
,
Q
are projections of
A
A
A
on
B
C
BC
BC
,
C
E
CE
CE
,
B
F
BF
BF
. Prove that the perpendicular bisectors of
P
Q
PQ
PQ
bisects two segments
A
O
AO
A
O
,
B
C
BC
BC
.
1
1
Hide problems
tangent of circumcircle of ABC is also tangent to D-excircle of DPQ, altitudes
Let
A
B
C
ABC
A
BC
be an acute, non-isosceles triangle with
A
D
AD
A
D
,
B
E
BE
BE
,
C
F
CF
CF
are altitudes and
d
d
d
is the tangent line of the circumcircle of triangle
A
B
C
ABC
A
BC
at
A
A
A
. The line through
H
H
H
and parallel to
E
F
EF
EF
cuts
D
E
DE
D
E
,
D
F
DF
D
F
at
Q
,
P
Q, P
Q
,
P
respectively. Prove that
d
d
d
is tangent to the ex-circle respect to vertex
D
D
D
of triangle
D
P
Q
DPQ
D
PQ
.
9
1
Hide problems
ARBP is a harmonic quadrilateral
Let
A
B
C
ABC
A
BC
be a triangle inscribed in circle
(
O
)
(O)
(
O
)
with diamter
K
L
KL
K
L
passes through the midpoint
M
M
M
of
A
B
AB
A
B
such that
L
,
C
L, C
L
,
C
lie on the different sides respect to
A
B
AB
A
B
. A circle passes through
M
,
K
M, K
M
,
K
cuts
L
C
LC
L
C
at
P
,
Q
P, Q
P
,
Q
(point
P
P
P
lies between
Q
,
C
Q, C
Q
,
C
). The line
K
Q
KQ
K
Q
cuts
(
L
M
Q
)
(LMQ)
(
L
MQ
)
at
R
R
R
. Prove that
A
R
B
P
ARBP
A
RBP
is cyclic and
A
B
AB
A
B
is the symmedian of triangle
A
P
R
APR
A
PR
.Please help :)
8
1
Hide problems
Median IPQ bisects arc BAC
Let
A
B
C
ABC
A
BC
be an non-isosceles triangle with incenter
I
I
I
, circumcenter
O
O
O
and a point
D
D
D
on segment
B
C
BC
BC
such that
(
B
I
D
)
(BID)
(
B
I
D
)
cut segments
A
B
AB
A
B
at
E
E
E
and
(
C
I
D
)
(CID)
(
C
I
D
)
cuts segment
A
C
AC
A
C
at
F
F
F
Circle
(
D
E
F
)
(DEF)
(
D
EF
)
cuts segments
A
B
AB
A
B
,
A
C
AC
A
C
again at
M
,
N
M,N
M
,
N
. Let
P
P
P
The intersection of
I
B
IB
I
B
and
D
E
DE
D
E
,
Q
Q
Q
The intersection of
I
C
IC
I
C
and
D
F
DF
D
F
. Prove that
E
N
,
F
M
,
P
Q
EN,FM,PQ
EN
,
FM
,
PQ
are parallel and the median of vertex
I
I
I
in triangle
I
P
Q
IPQ
I
PQ
bisects the arc
B
A
C
BAC
B
A
C
of
(
O
)
(O)
(
O
)
.
2
1
Hide problems
HJ || BC
Let
A
B
C
ABC
A
BC
be an acute, non isosceles triangle with the orthocenter
H
H
H
, circumcenter
O
O
O
and
A
D
AD
A
D
is the diameter of
(
O
)
(O)
(
O
)
. Suppose that the circle
(
A
H
D
)
(AHD)
(
A
HD
)
meets the lines
A
B
,
A
C
AB, AC
A
B
,
A
C
at
F
F
F
, respectively. Denote
J
,
K
J, K
J
,
K
as orthocenter and nine- point center of
A
E
F
AEF
A
EF
. Prove that
H
J
∥
B
C
HJ \parallel BC
H
J
∥
BC
and
K
O
=
K
H
KO = KH
K
O
=
KH
.