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National and Regional Contests
Mexico Contests
OMMock - Mexico National Olympiad Mock Exam
2020 OMMock - Mexico National Olympiad Mock Exam
2020 OMMock - Mexico National Olympiad Mock Exam
Part of
OMMock - Mexico National Olympiad Mock Exam
Subcontests
(6)
6
1
Hide problems
f(f(x) - y) = f(xy) + f(x)f(-y)
Find all functions
f
:
R
→
R
f \colon \mathbb{R} \to \mathbb{R}
f
:
R
→
R
such that
f
(
f
(
x
)
−
y
)
=
f
(
x
y
)
+
f
(
x
)
f
(
−
y
)
f(f(x) - y) = f(xy) + f(x)f(-y)
f
(
f
(
x
)
−
y
)
=
f
(
x
y
)
+
f
(
x
)
f
(
−
y
)
for any two real numbers
x
,
y
x, y
x
,
y
.Proposed by Pablo Valeriano
5
1
Hide problems
Ladder game, victory if you get the zero ladder
A ladder is a non-decreasing sequence
a
1
,
a
2
,
…
,
a
2020
a_1, a_2, \dots, a_{2020}
a
1
,
a
2
,
…
,
a
2020
of non-negative integers. Diego and Pablo play by turns with the ladder
1
,
2
,
…
,
2020
1, 2, \dots, 2020
1
,
2
,
…
,
2020
, starting with Diego. In each turn, the player replaces an entry
a
i
a_i
a
i
by
a
i
′
<
a
i
a_i'<a_i
a
i
′
<
a
i
, with the condition that the sequence remains a ladder. The player who gets
(
0
,
0
,
…
,
0
)
(0, 0, \dots, 0)
(
0
,
0
,
…
,
0
)
wins. Who has a winning strategy?Proposed by Violeta Hernández
4
1
Hide problems
Proving angle equality in a simple configuration
Let
A
B
C
ABC
A
BC
be a triangle. Suppose that the perpendicular bisector of
B
C
BC
BC
meets the circle of diameter
A
B
AB
A
B
at a point
D
D
D
at the opposite side of
B
C
BC
BC
with respect to
A
A
A
, and meets the circle through
A
,
C
,
D
A, C, D
A
,
C
,
D
again at
E
E
E
. Prove that
∠
A
C
E
=
∠
B
C
D
\angle ACE=\angle BCD
∠
A
CE
=
∠
BC
D
.Proposed by José Manuel Guerra and Victor Domínguez
3
1
Hide problems
Certain numbers appear in process with n double-sided cards
Let
n
n
n
be a fixed positive integer. Oriol has
n
n
n
cards, each of them with a
0
0
0
written on one side and
1
1
1
on the other. We place these cards in line, some face up and some face down (possibly all on the same side). We begin the following process consisting of
n
n
n
steps:1) At the first step, Oriol flips the first card 2) At the second step, Oriol flips the first card and second card . . . n) At the last step Oriol flips all the cardsLet
s
0
,
s
1
,
s
2
,
…
,
s
n
s_0, s_1, s_2, \dots, s_n
s
0
,
s
1
,
s
2
,
…
,
s
n
be the sum of the numbers seen in the cards at the beggining, after the first step, after the second step,
…
\dots
…
after the last step, respectively. a) Find the greatest integer
k
k
k
such that, no matter the initial card configuration, there exists at least
k
k
k
distinct numbers between
s
0
,
s
1
,
…
,
s
n
s_0, s_1, \dots, s_n
s
0
,
s
1
,
…
,
s
n
. b) Find all positive integers
m
m
m
such that, for each initial card configuration, there exists an index
r
r
r
such that
s
r
=
m
s_r = m
s
r
=
m
.Proposed by Dorlir Ahmeti
2
1
Hide problems
At least 2020 permutations giving distinct remainder for some n
We say that a permutation
(
a
1
,
…
,
a
n
)
(a_1, \dots, a_n)
(
a
1
,
…
,
a
n
)
of
(
1
,
2
,
…
,
n
)
(1, 2, \dots, n)
(
1
,
2
,
…
,
n
)
is good if the sums
a
1
+
a
2
+
⋯
+
a
i
a_1 + a_2 + \dots + a_i
a
1
+
a
2
+
⋯
+
a
i
are all distinct modulo
n
n
n
. Prove that there exists a positive integer
n
n
n
such that there are at least
2020
2020
2020
good permutations of
(
1
,
2
,
…
,
n
)
(1, 2, \dots, n)
(
1
,
2
,
…
,
n
)
.Proposed by Ariel García
1
1
Hide problems
Pair of similar inequalities with square root implies sum greater than 1
Let
a
a
a
,
b
b
b
,
c
c
c
and
d
d
d
positive real numbers with
a
>
c
a > c
a
>
c
and
b
<
d
b < d
b
<
d
. Assume that
a
+
b
≥
c
+
d
and
a
+
b
≤
c
+
d
a + \sqrt{b} \ge c + \sqrt{d} \qquad \text{and} \qquad \sqrt{a} + b \le \sqrt{c} + d
a
+
b
≥
c
+
d
and
a
+
b
≤
c
+
d
Prove that
a
+
b
+
c
+
d
>
1
a + b + c + d > 1
a
+
b
+
c
+
d
>
1
.Proposed by Victor Domínguez