MathDB
Problems
Contests
National and Regional Contests
Canada Contests
Canadian Junior Mathematical Olympiad
2021 Canadian Junior Mathematical Olympiad
2021 Canadian Junior Mathematical Olympiad
Part of
Canadian Junior Mathematical Olympiad
Subcontests
(2)
2
1
Hide problems
first k elements of permutation of first n have distinct remainder mod k
How many ways are there to permute the first
n
n
n
positive integers such that in the permutation, for each value of
k
≤
n
k \le n
k
≤
n
, the first
k
k
k
elements of the permutation have distinct remainder mod
k
k
k
?
1
1
Hide problems
P_1X bisects <Q_1P_1Q_2, start with concentric circles
Let
C
1
C_1
C
1
and
C
2
C_2
C
2
be two concentric circles with
C
1
C_1
C
1
inside
C
2
C_2
C
2
. Let
P
1
P_1
P
1
and
P
2
P_2
P
2
be two points on
C
1
C_1
C
1
that are not diametrically opposite. Extend the segment
P
1
P
2
P_1P_2
P
1
P
2
past
P
2
P_2
P
2
until it meets the circle
C
2
C_2
C
2
in
Q
2
Q_2
Q
2
. The tangent to
C
2
C_2
C
2
at
Q
2
Q_2
Q
2
and the tangent to
C
1
C_1
C
1
at
P
1
P_1
P
1
meet in a point
X
X
X
. Draw from X the second tangent to
C
2
C_2
C
2
which meets
C
2
C_2
C
2
at the point
Q
1
Q_1
Q
1
. Show that
P
1
X
P_1X
P
1
X
bisects angle
Q
1
P
1
Q
2
Q_1P_1Q_2
Q
1
P
1
Q
2
.