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National and Regional Contests
Azerbaijan Contests
Azerbaijan IZHO TST
2018 Azerbaijan IZhO TST
2018 Azerbaijan IZhO TST
Part of
Azerbaijan IZHO TST
Subcontests
(5)
4
1
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3 countries with cities connected with road
There are
10
10
10
cities in each of the three countries. Each road connects two cities from two different countries (there is at most one road between any two cities.) There are more than
200
200
200
roads between these three countries. Prove that three cities, one city from each country, can be chosen such that there is a road between any two of these cities.
5
1
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Angle chasing I believe...
Let
ω
\omega
ω
be the incircle of
△
A
B
C
\triangle ABC
△
A
BC
and
D
,
E
,
F
D,E,F
D
,
E
,
F
be the tangency points on
B
C
,
C
A
,
A
B
BC ,CA, AB
BC
,
C
A
,
A
B
. In
△
D
E
F
\triangle DEF
△
D
EF
let the altitudes from
E
,
F
E,F
E
,
F
to
F
D
,
D
E
FD,DE
F
D
,
D
E
intersect
A
B
,
A
C
AB, AC
A
B
,
A
C
at
X
,
Y
X ,Y
X
,
Y
. Prove that the second intersection of
(
A
E
X
)
(AEX)
(
A
EX
)
and
(
A
F
Y
)
(AFY)
(
A
F
Y
)
lies on
ω
\omega
ω
3
1
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TST TAJIKISTAN Day 2 Problem 2
Problem 5. Consider the integer number n>2. Let a_1,a_2,…,a_n and b_1,b_2,…,b_n be two permutations of 0,1,2,…,n-1. Prove that there exist some i≠j such that: n|a_i b_i-a_j b_j[color=#00f]Moved to HSO. ~ oVlad
2
1
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TST TAJIKISTAN 2018 Day 2 Problem 1
Problem 4. Let a,b be positive real numbers and let x,y be positive real numbers less than 1, such that: a/(1-x)+b/(1-y)=1 Prove that: ∛ay+∛bx≤1.
1
1
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TST Tajikistan 2018 Day 1 Problem 3
Problem 3. Suppose that the equation x^3-ax^2+bx-a=0 has three positive real roots (b>0). Find the minimum value of the expression: (b-a)(b^3+3a^3)