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Contests
National and Regional Contests
Argentina Contests
Argentina National Olympiad
2019 Argentina National Olympiad
2019 Argentina National Olympiad
Part of
Argentina National Olympiad
Subcontests
(6)
2
1
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min sum a_i^2/(a_i+b_i) when sum a_i=1, sum b_i=1
Let
n
≥
1
n\geq1
n
≥
1
be an integer. We have two sequences, each of
n
n
n
positive real numbers
a
1
,
a
2
,
…
,
a
n
a_1,a_2,\ldots ,a_n
a
1
,
a
2
,
…
,
a
n
and
b
1
,
b
2
,
…
,
b
n
b_1,b_2,\ldots ,b_n
b
1
,
b
2
,
…
,
b
n
such that
a
1
+
a
2
+
…
+
a
n
=
1
a_1+a_2+\ldots +a_n=1
a
1
+
a
2
+
…
+
a
n
=
1
and
b
1
+
b
2
+
…
+
b
n
=
1
b_1+b_2+\ldots +b_n=1
b
1
+
b
2
+
…
+
b
n
=
1
. Find the smallest possible value that the sum can take
a
1
2
a
1
+
b
1
+
a
2
2
a
2
+
b
2
+
…
+
a
n
2
a
n
+
b
n
.
\frac{a_1^2}{a_1+b_1}+\frac{a_2^2}{a_2+b_2}+\ldots +\frac{a_n^2}{a_n +b_n}.
a
1
+
b
1
a
1
2
+
a
2
+
b
2
a
2
2
+
…
+
a
n
+
b
n
a
n
2
.
4
1
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a\sqrt2 is rational if a^2+b \sqrt2 is a rational number
If we have a set
M
M
M
of
2019
2019
2019
real numbers such that for every even
a
a
a
,
b
b
b
of numbers of
M
M
M
it is verified that
a
2
+
b
2
a^2+b \sqrt2
a
2
+
b
2
is a rational number. Show that for all
a
a
a
of
M
M
M
,
a
2
a\sqrt2
a
2
is a rational number.
5
1
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arithmetic progression of 7 different prime terms
There is an arithmetic progression of
7
7
7
terms in which all the terms are different prime numbers. Determine the smallest possible value of the last term of such a progression.Clarification: In an arithmetic progression of difference
d
d
d
each term is equal to the previous one plus
d
d
d
.
6
1
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numbers 1 - 300 around a circle
The natural numbers from
1
1
1
up to
300
300
300
are evenly located around a circle. We say that such an ordering is alternate [/i ]if each number is less than its two neighbors or is greater than its two neighbors. We will call a pair of neighboring numbers a good pair if, by removing that pair from the circumference, the remaining numbers form an alternate ordering. Determine the least possible number of good pairs in which there can be an alternate ordering of the numbers from
1
1
1
at
300
300
300
inclusive.
1
1
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singular set A of distinct positive integers with n items
A set of distinct positive integers is called singular if, for each of its elements, after crossing out that element, the remaining ones can be grouped into two sets with no common elements such that the sum of the elements in the two groups is the same. Find the smallest positive integer
n
>
1
n>1
n
>
1
such that there exists a singular set
A
A
A
with
n
n
n
items.
3
1
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<BAC = 3 <BAP wanted, <ACB = 2<ABC, AP = AC, PB = PC given
In triangle
A
B
C
ABC
A
BC
it is known that
∠
A
C
B
=
2
∠
A
B
C
\angle ACB = 2\angle ABC
∠
A
CB
=
2∠
A
BC
. Furthermore
P
P
P
is an interior point of the triangle
A
B
C
ABC
A
BC
such that
A
P
=
A
C
AP = AC
A
P
=
A
C
and
P
B
=
P
C
PB = PC
PB
=
PC
. Prove that
∠
B
A
C
=
3
∠
B
A
P
\angle BAC = 3 \angle BAP
∠
B
A
C
=
3∠
B
A
P
.