MathDB
Problems
Contests
National and Regional Contests
Canada Contests
Canadian Junior Mathematical Olympiad
2024 Canadian Junior Mathematical Olympiad
2024 Canadian Junior Mathematical Olympiad
Part of
Canadian Junior Mathematical Olympiad
Subcontests
(2)
1
1
Hide problems
juniors scanning rectangles for treasure
Centuries ago, the pirate Captain Blackboard buried a vast amount of treasure in a single cell of a
2
×
4
2 \times 4
2
×
4
grid-structured island. Treasure was buried in a single cell of an
M
×
N
M\times N
M
×
N
(
2
≤
M
2\le M
2
≤
M
,
N
N
N
) grid. You and your crew have reached the island and have brought special treasure detectors to find the cell with the treasure For each detector, you can set it up to scan a specific subgrid
[
a
,
b
]
×
[
c
,
d
]
[a,b]\times[c,d]
[
a
,
b
]
×
[
c
,
d
]
with
1
≤
a
≤
b
≤
2
1\le a\le b\le 2
1
≤
a
≤
b
≤
2
and
1
≤
c
≤
d
≤
4
1\le c\le d\le 4
1
≤
c
≤
d
≤
4
. Running the detector will tell you whether the treasure is in the region or not, though it cannot say where in the region the treasure was detected. You plan on setting up
Q
Q
Q
detectors, which may only be run simultaneously after all
Q
Q
Q
detectors are ready. What is the minimum
Q
Q
Q
required to gaurantee to determine the location of the Blackboard’s legendary treasure?
2
1
Hide problems
Triple summation of fractions
Let
I
n
=
∑
i
=
1
n
∑
j
=
1
n
∑
k
=
1
n
min
(
1
i
,
1
j
,
1
k
)
I_n=\sum_{i=1}^n\sum_{j=1}^n\sum_{k=1}^n \min \left( \frac{1}{i}, \frac{1}{j}, \frac{1}{k} \right)
I
n
=
∑
i
=
1
n
∑
j
=
1
n
∑
k
=
1
n
min
(
i
1
,
j
1
,
k
1
)
and let
H
n
=
1
+
1
2
+
…
1
n
H_n=1+\frac{1}{2}+\ldots \frac{1}{n}
H
n
=
1
+
2
1
+
…
n
1
Find
I
n
−
H
n
I_n-H_n
I
n
−
H
n
in terms of
n
n
n
(Paraphrased)