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Girls in Math at Yale
2022 Girls in Math at Yale
R2
R2
Part of
2022 Girls in Math at Yale
Problems
(1)
Girls in Math at Yale 2022 Mathathon Round 2
Source:
3/7/2022
p4 Define the sequence
a
n
{a_n}
a
n
as follows: 1)
a
1
=
−
1
a_1 = -1
a
1
=
−
1
, and 2) for all
n
≥
2
n \ge 2
n
≥
2
,
a
n
=
1
+
2
+
.
.
.
+
n
−
(
n
+
1
)
a_n = 1 + 2 + . . . + n - (n + 1)
a
n
=
1
+
2
+
...
+
n
−
(
n
+
1
)
. For example,
a
3
=
1
+
2
+
3
−
4
=
2
a_3 = 1+2+3-4 = 2
a
3
=
1
+
2
+
3
−
4
=
2
. Find the largest possible value of
k
k
k
such that
a
k
+
a
k
+
1
=
a
k
+
2
a_k+a_{k+1} = a_{k+2}
a
k
+
a
k
+
1
=
a
k
+
2
. p5 The taxicab distance between two points
(
a
,
b
)
(a, b)
(
a
,
b
)
and
(
c
,
d
)
(c, d)
(
c
,
d
)
on the coordinate plane is
∣
c
−
a
∣
+
∣
d
−
b
∣
|c-a|+|d-b|
∣
c
−
a
∣
+
∣
d
−
b
∣
. Given that the taxicab distance between points
A
A
A
and
B
B
B
is
8
8
8
and that the length of
A
B
AB
A
B
is
k
k
k
, find the minimum possible value of
k
2
k^2
k
2
. p6 For any two-digit positive integer
A
B
‾
\overline{AB}
A
B
, let
f
(
A
B
‾
)
=
A
B
‾
−
A
⋅
B
f(\overline{AB}) = \overline{AB}-A\cdot B
f
(
A
B
)
=
A
B
−
A
⋅
B
, or in other words, the result of subtracting the product of its digits from the integer itself. For example,
f
(
72
‾
)
=
72
−
7
⋅
2
=
58
f(\overline{72}) = 72-7\cdot 2 = 58
f
(
72
)
=
72
−
7
⋅
2
=
58
. Find the maximum possible
n
n
n
such that there exist distinct two-digit integers
X
Y
‾
\overline{XY}
X
Y
and
W
Z
‾
\overline{WZ}
W
Z
such that
f
(
X
Y
‾
)
=
f
(
W
Z
‾
)
=
n
f(\overline{XY} ) = f(\overline{WZ}) = n
f
(
X
Y
)
=
f
(
W
Z
)
=
n
.
algebra
number theory
Yale