MathDB
Problems
Contests
National and Regional Contests
Chile Contests
Chile Junior Math Olympiad
2022 Chile Junior Math Olympiad
2022 Chile Junior Math Olympiad
Part of
Chile Junior Math Olympiad
Subcontests
(3)
4
1
Hide problems
47 divides sum ab, if a,b \in 1-46 , 2022 Chile NMO L1 P4 - Juniors
Let
S
S
S
be the sum of all products
a
b
ab
ab
where
a
a
a
and
b
b
b
are distinct elements of the set
{
1
,
2
,
.
.
.
,
46
}
\{1,2,...,46\}
{
1
,
2
,
...
,
46
}
. Prove that
47
47
47
divides
S
S
S
.
3
1
Hide problems
number of m, so that 23 = remainder 2023:m 2022 Chile NMO L1 P3 - Juniors
By dividing
2023
2023
2023
by a natural number
m
m
m
, the remainder is
23
23
23
. How many numbers
m
m
m
are there with this property?
2
1
Hide problems
ratio of areas (APM)/(ABCD), trapezoid ABCD, AB/CD=3/2, BN = 3NC, AM = 2MB
In a trapezoid
A
B
C
D
ABCD
A
BC
D
whose parallel sides
A
B
AB
A
B
and
C
D
CD
C
D
are in ratio
A
B
C
D
=
3
2
\frac{AB}{CD}=\frac32
C
D
A
B
=
2
3
, the points
N
N
N
and
M
M
M
are marked on the sides
B
C
BC
BC
and
A
B
AB
A
B
respectively, in such a way that
B
N
=
3
N
C
BN = 3NC
BN
=
3
NC
and
A
M
=
2
M
B
AM = 2MB
A
M
=
2
MB
and segments
A
N
AN
A
N
and
D
M
DM
D
M
are drawn that intersect at point
P
P
P
, find the ratio between the areas of triangle
A
P
M
APM
A
PM
and trapezoid
A
B
C
D
ABCD
A
BC
D
. https://cdn.artofproblemsolving.com/attachments/7/8/21d59ca995d638dfcb76f9508e439fd93a5468.png