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Undergraduate contests
Brazil Undergrad MO
2018 Brazil Undergrad MO
2018 Brazil Undergrad MO
Part of
Brazil Undergrad MO
Subcontests
(23)
16
1
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Ascending numbers
A positive integer of at least two digits written in the base
10
10
10
is called 'ascending' if the digits increase in value from left to right. For example,
123
123
123
is 'ascending', but
132
132
132
and
122
122
122
is not. How many 'ascending' numbers are there?
25
1
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Additive group
Consider the
Z
/
(
10
)
\mathbb {Z} / (10)
Z
/
(
10
)
additive group automorphism group of integers module
10
10
10
, that is,
A
=
{
ϕ
:
Z
/
(
10
)
→
Z
/
(
10
)
∣
ϕ
−
a
u
t
o
m
o
r
p
h
i
s
m
}
A = \left \{\phi: \mathbb {Z} / (10) \to \mathbb {Z} / (10) | \phi-automorphism \right \}
A
=
{
ϕ
:
Z
/
(
10
)
→
Z
/
(
10
)
∣
ϕ
−
a
u
t
o
m
or
p
hi
s
m
}
24
1
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Calculate the value of the series
What is the value of the series
∑
1
≤
l
<
m
<
n
1
5
l
3
m
2
n
\sum_{1 \leq l <m<n} \frac{1}{5^l3^m2^n}
∑
1
≤
l
<
m
<
n
5
l
3
m
2
n
1
23
1
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Perfect squares
How many prime numbers
p
p
p
the number
p
3
−
4
p
+
9
p ^ 3-4 p + 9
p
3
−
4
p
+
9
is a perfect square
22
1
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What is the value of the improper integral
What is the value of the improper integral
∫
0
π
log
(
sin
(
x
)
)
d
x
\int_0 ^ {\pi} \log (\sin (x)) dx
∫
0
π
lo
g
(
sin
(
x
))
d
x
?
21
1
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Grade n polynomials
Consider
p
(
x
)
=
x
n
+
a
n
−
1
x
n
−
1
+
.
.
.
+
a
1
x
+
1
p (x) = x ^ n + a_ {n-1} x ^ {n-1} + ... + a_ {1} x + 1
p
(
x
)
=
x
n
+
a
n
−
1
x
n
−
1
+
...
+
a
1
x
+
1
a polynomial of positive real coefficients, degree
n
≥
2
n \geq 2
n
≥
2
e with
n
n
n
real roots. Which of the following statements is always true?a)
p
(
2
)
<
2
(
2
n
−
1
+
1
)
p (2) <2 (2 ^ {n-1} +1)
p
(
2
)
<
2
(
2
n
−
1
+
1
)
(b)
p
(
1
)
<
3
p (1) <3
p
(
1
)
<
3
c)
p
(
1
)
>
2
n
p (1)> 2 ^ n
p
(
1
)
>
2
n
d)
p
(
3
)
<
3
(
2
n
−
1
−
2
)
p (3 ) <3 (2 ^ {n-1} -2)
p
(
3
)
<
3
(
2
n
−
1
−
2
)
20
1
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Point Numbers in plan
What is the largest number of points that can exist on a plane so that each distance between any two of them is an odd integer?
19
1
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Number of Complex Solutions
What is the largest amount of complex
z
z
z
solutions a system can have?
∣
z
−
1
∣
∣
z
+
1
∣
=
1
| z-1 || z + 1 | = 1
∣
z
−
1∣∣
z
+
1∣
=
1
I
m
(
z
)
=
b
?
Im (z) = b?
I
m
(
z
)
=
b
?
(where
b
b
b
is a real constant)
15
1
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probability that all roots
A real number
t
o
to
t
o
is randomly and uniformly chosen from the
[
−
3
,
4
]
[- 3,4]
[
−
3
,
4
]
interval. What is the probability that all roots of the polynomial
x
3
+
a
x
2
+
a
x
+
1
x ^ 3 + ax ^ 2 + ax + 1
x
3
+
a
x
2
+
a
x
+
1
are real?
14
1
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Arithmetic mean of all values
What is the arithmetic mean of all values of the expression
∣
a
1
−
a
2
∣
+
∣
a
3
−
a
4
∣
| a_1-a_2 | + | a_3-a_4 |
∣
a
1
−
a
2
∣
+
∣
a
3
−
a
4
∣
Where
a
1
,
a
2
,
a
3
,
a
4
a_1, a_2, a_3, a_4
a
1
,
a
2
,
a
3
,
a
4
is a permutation of the elements of the set {
1
,
2
,
3
,
4
1,2,3,4
1
,
2
,
3
,
4
}?
13
1
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Calculate the value of f (2018)
A continuous function
f
:
R
→
R
f: \mathbb {R} \to \mathbb {R}
f
:
R
→
R
satisfies
f
(
x
)
f
(
f
(
x
)
)
=
1
f (x) f (f (x)) = 1
f
(
x
)
f
(
f
(
x
))
=
1
for every real
x
x
x
and
f
(
2020
)
=
2019
f (2020) = 2019
f
(
2020
)
=
2019
. What is the value of
f
(
2018
)
f (2018)
f
(
2018
)
?
12
1
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Probability of the measures
Let
A
B
C
ABC
A
BC
be an equilateral triangle.
A
A
A
point
P
P
P
is chosen at random within this triangle. What is the probability that the sum of the distances from point
P
P
P
to the sides of triangle
A
B
C
ABC
A
BC
are measures of the sides of a triangle?
17
1
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Calculate the value of AN
In the figure, a semicircle is folded along the
A
N
AN
A
N
string and intersects the
M
N
MN
MN
diameter in
B
B
B
.
M
B
:
B
N
=
2
:
3
MB: BN = 2: 3
MB
:
BN
=
2
:
3
and
M
N
=
10
MN = 10
MN
=
10
are known to be. If
A
N
=
x
AN = x
A
N
=
x
, what is the value of
x
2
x ^ 2
x
2
?
7
1
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isomorphisms
Unless of isomorphisms, how many simple four-vertex graphs are there?
10
1
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Satisfaction for ordered pairs
How many ordered pairs of real numbers
(
a
,
b
)
(a, b)
(
a
,
b
)
satisfy equality
lim
x
→
0
sin
2
x
e
a
x
−
2
b
x
−
1
=
1
2
\lim_{x \to 0} \frac{\sin^2x}{e^{ax}-2bx-1}= \frac{1}{2}
lim
x
→
0
e
a
x
−
2
b
x
−
1
s
i
n
2
x
=
2
1
?
9
1
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How many functions satisfy for every $ x $?
How many functions
f
:
{
1
,
2
,
3
}
→
{
1
,
2
,
3
}
f: \left\{1,2,3\right\} \to \left\{1,2,3 \right\}
f
:
{
1
,
2
,
3
}
→
{
1
,
2
,
3
}
satisfy
f
(
f
(
x
)
)
=
f
(
f
(
f
(
x
)
)
)
f(f(x))=f(f(f(x)))
f
(
f
(
x
))
=
f
(
f
(
f
(
x
)))
for every
x
x
x
?
8
1
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Examination probability
A student will take an exam in which they have to solve three chosen problems by chance of a list of
10
10
10
possible problems. It will be approved if it correctly resolves two problems. Considering that the student can solve five of the problems on the list and not know how to solve others, how likely is he to pass the exam?
6
1
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Isosceles or non-degenerate triangles
Given an equilateral triangle
A
B
C
ABC
A
BC
in the plane, how many points
P
P
P
in the plane are such that the three triangles
A
P
B
,
B
P
C
AP B, BP C
A
PB
,
BPC
and
C
P
A
CP A
CP
A
are isosceles and not degenerate?
5
1
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Smallest number that belongs
Consider the set
A
=
{
j
4
+
100
j
∣
j
=
1
,
2
,
3
,
.
.
}
A = \left\{\frac{j}{4}+\frac{100}{j}|j=1,2,3,..\right\}
A
=
{
4
j
+
j
100
∣
j
=
1
,
2
,
3
,
..
}
What is the smallest number that belongs to the
A
A
A
set?
4
1
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Element of a group G
Consider the property that each a element of a group
G
G
G
satisfies
a
2
=
e
a ^ 2 = e
a
2
=
e
, where e is the identity element of the group. Which of the following statements is not always valid for a group
G
G
G
with this property? (a)
G
G
G
is commutative (b)
G
G
G
has infinite or even order (c)
G
G
G
is Noetherian (d)
G
G
G
is vector space over
Z
2
\mathbb{Z}_2
Z
2
3
1
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Permutations sequence
How many permutations
a
1
,
a
2
,
a
3
,
a
4
a_1, a_2, a_3, a_4
a
1
,
a
2
,
a
3
,
a
4
of
1
,
2
,
3
,
4
1, 2, 3, 4
1
,
2
,
3
,
4
satisfy the condition that for
k
=
1
,
2
,
3
,
k = 1, 2, 3,
k
=
1
,
2
,
3
,
the list
a
1
,
.
.
.
,
a
k
a_1,. . . , a_k
a
1
,
...
,
a
k
contains a number greater than
k
k
k
?
2
1
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What is the value of $ g (x + f (y) $?
Let
f
,
g
:
R
→
R
f, g: \mathbb {R} \to \mathbb {R}
f
,
g
:
R
→
R
function such that
f
(
x
+
g
(
y
)
)
=
−
x
+
y
+
1
f (x + g (y)) = - x + y + 1
f
(
x
+
g
(
y
))
=
−
x
+
y
+
1
for each pair of real numbers
x
x
x
e
y
y
y
. What is the value of
g
(
x
+
f
(
y
)
g (x + f (y)
g
(
x
+
f
(
y
)
?
1
1
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Cuts in a triangle
An equilateral triangle is cut as shown in figure 1 and the parts are used to form figure 2. What is the shape of figure 2?