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Problems
Contests
International Contests
Tuymaada Olympiad
2014 Tuymaada Olympiad
2014 Tuymaada Olympiad
Part of
Tuymaada Olympiad
Subcontests
(8)
7
1
Hide problems
A line, a median and a bisector have a common point
A parallelogram
A
B
C
D
ABCD
A
BC
D
is given. The excircle of triangle
△
A
B
C
\triangle{ABC}
△
A
BC
touches the sides
A
B
AB
A
B
at
L
L
L
and the extension of
B
C
BC
BC
at
K
K
K
. The line
D
K
DK
DK
meets the diagonal
A
C
AC
A
C
at point
X
X
X
; the line
B
X
BX
BX
meets the median
C
C
1
CC_1
C
C
1
of trianlge
△
A
B
C
\triangle{ABC}
△
A
BC
at
Y
{Y}
Y
. Prove that the line
Y
L
YL
Y
L
, median
B
B
1
BB_1
B
B
1
of triangle
△
A
B
C
\triangle{ABC}
△
A
BC
and its bisector
C
C
′
CC^\prime
C
C
′
have a common point.(A. Golovanov)
8
2
Hide problems
g(a,b,c)\ge \sqrt{2abc}
Let positive integers
a
,
b
,
c
a,\ b,\ c
a
,
b
,
c
be pairwise coprime. Denote by
g
(
a
,
b
,
c
)
g(a, b, c)
g
(
a
,
b
,
c
)
the maximum integer not representable in the form
x
a
+
y
b
+
z
c
xa+yb+zc
x
a
+
y
b
+
zc
with positive integral
x
,
y
,
z
x,\ y,\ z
x
,
y
,
z
. Prove that
g
(
a
,
b
,
c
)
≥
2
a
b
c
g(a, b, c)\ge \sqrt{2abc}
g
(
a
,
b
,
c
)
≥
2
ab
c
(M. Ivanov)[hide="Remarks (containing spoilers!)"] 1. It can be proven that
g
(
a
,
b
,
c
)
≥
3
a
b
c
g(a,b,c)\ge \sqrt{3abc}
g
(
a
,
b
,
c
)
≥
3
ab
c
. 2. The constant
3
3
3
is the best possible, as proved by the equation
g
(
3
,
3
k
+
1
,
3
k
+
2
)
=
9
k
+
5
g(3,3k+1,3k+2)=9k+5
g
(
3
,
3
k
+
1
,
3
k
+
2
)
=
9
k
+
5
.
Islands and numbered feriboats
There are
m
m
m
villages on the left bank of the Lena,
n
n
n
villages on the right bank and one village on an island. It is known that
(
m
+
1
,
n
+
1
)
>
1
(m+1,n+1)>1
(
m
+
1
,
n
+
1
)
>
1
. Every two villages separated by water are connected by ferry with positive integral number. The inhabitants of each village say that all the ferries operating in their village have different numbers and these numbers form a segment of the series of the integers. Prove that at least some of them are wrong.(K. Kokhas)
6
2
Hide problems
From 2n squares, n of them have a common point
Each of
n
n
n
black squares and
n
n
n
white squares can be obtained by a translation from each other. Every two squares of different colours have a common point. Prove that ther is a point belonging at least to
n
n
n
squares.(V. Dolnikov)
If two circles are tangent, then all three are
Radius of the circle
ω
A
\omega_A
ω
A
with centre at vertex
A
A
A
of a triangle
△
A
B
C
\triangle{ABC}
△
A
BC
is equal to the radius of the excircle tangent to
B
C
BC
BC
. The circles
ω
B
\omega_B
ω
B
and
ω
C
\omega_C
ω
C
are defined similarly. Prove that if two of these circles are tangent then every two of them are tangent to each other.(L. Emelyanov)
5
2
Hide problems
Pairing cards such that numbers differ by 1
There is an even number of cards on a table; a positive integer is written on each card. Let
a
k
a_k
a
k
be the number of cards having
k
k
k
written on them. It is known that
a
n
−
a
n
−
1
+
a
n
−
2
−
⋯
≥
0
a_n-a_{n-1}+a_{n-2}- \cdots \ge 0
a
n
−
a
n
−
1
+
a
n
−
2
−
⋯
≥
0
for each positive integer
n
n
n
. Prove that the cards can be partitioned into pairs so that the numbers in each pair differ by
1
1
1
.(A. Golovanov)
Quadratic trinomials and linear function
For two quadratic trinomials
P
(
x
)
P(x)
P
(
x
)
and
Q
(
x
)
Q(x)
Q
(
x
)
there is a linear function
ℓ
(
x
)
\ell(x)
ℓ
(
x
)
such that
P
(
x
)
=
Q
(
ℓ
(
x
)
)
P(x)=Q(\ell(x))
P
(
x
)
=
Q
(
ℓ
(
x
))
for all real
x
x
x
. How many such linear functions
ℓ
(
x
)
\ell(x)
ℓ
(
x
)
can exist?(A. Golovanov)
3
1
Hide problems
Sum of 1/\sqrt{x^3+1} when sum of 1/x=3
Positive numbers
a
,
b
,
c
a,\ b,\ c
a
,
b
,
c
satisfy
1
a
+
1
b
+
1
c
=
3
\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}=3
a
1
+
b
1
+
c
1
=
3
. Prove the inequality
1
a
3
+
1
+
1
b
3
+
1
+
1
c
3
+
1
≤
3
2
.
\dfrac{1}{\sqrt{a^3+1}}+\dfrac{1}{\sqrt{b^3+1}}+\dfrac{1}{\sqrt{c^3+1}}\le \dfrac{3}{\sqrt{2}}.
a
3
+
1
1
+
b
3
+
1
1
+
c
3
+
1
1
≤
2
3
.
(N. Alexandrov)
2
1
Hide problems
Midpoint of altitude, of a segment and the circumc. are coll
The points
K
K
K
and
L
L
L
on the side
B
C
BC
BC
of a triangle
△
A
B
C
\triangle{ABC}
△
A
BC
are such that
B
A
K
^
=
C
A
L
^
=
9
0
∘
\widehat{BAK}=\widehat{CAL}=90^\circ
B
A
K
=
C
A
L
=
9
0
∘
. Prove that the midpoint of the altitude drawn from
A
A
A
, the midpoint of
K
L
KL
K
L
and the circumcentre of
△
A
B
C
\triangle{ABC}
△
A
BC
are collinear.(A. Akopyan, S. Boev, P. Kozhevnikov)
4
1
Hide problems
Vertical sides in hexagonal parallelogram
A
k
×
ℓ
k\times \ell
k
×
ℓ
'parallelogram' is drawn on a paper with hexagonal cells (it consists of
k
k
k
horizontal rows of
ℓ
\ell
ℓ
cells each). In this parallelogram a set of non-intersecting sides of hexagons is chosen; it divides all the vertices into pairs. Juniors) How many vertical sides can there be in this set? Seniors) How many ways are there to do that? [asy] size(120); defaultpen(linewidth(0.8)); path hex = dir(30)--dir(90)--dir(150)--dir(210)--dir(270)--dir(330)--cycle; for(int i=0;i<=3;i=i+1) { for(int j=0;j<=2;j=j+1) { real shiftx=j*sqrt(3)/2+i*sqrt(3),shifty=j*3/2; draw(shift(shiftx,shifty)*hex); } } [/asy] (T. Doslic)
1
2
Hide problems
Minimum number of remainders
Four consecutive three-digit numbers are divided respectively by four consecutive two-digit numbers. What minimum number of different remainders can be obtained? (A. Golovanov)
Maximum number of primes who divide their sum
Given are three different primes. What maximum number of these primes can divide their sum?(A. Golovanov)