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Problems
Contests
National and Regional Contests
USA Contests
USA - Middle School Tournaments
LMT
2020 LMT Spring
2020 LMT Spring
Part of
LMT
Subcontests
(30)
28
1
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Spring 2020 Team Round Problem 28
A particular country has seven distinct cities, conveniently named
C
1
,
C
2
,
…
,
C
7
.
C_1,C_2,\dots,C_7.
C
1
,
C
2
,
…
,
C
7
.
Between each pair of cities, a direction is chosen, and a one-way road is constructed in that direction connecting the two cities. After the construction is complete, it is found that any city is reachable from any other city, that is, for distinct
1
≤
i
,
j
≤
7
,
1 \leq i, j \leq 7,
1
≤
i
,
j
≤
7
,
there is a path of one-way roads leading from
C
i
C_i
C
i
to
C
j
.
C_j.
C
j
.
Compute the number of ways the roads could have been configured. Pictured on the following page are the possible configurations possible in a country with three cities, if every city is reachable from every other city. [Insert Diagram] Proposed by Ezra Erives
13
1
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Spring 2020 Team Round Problem 13
In the game of Flow, a path is drawn through a
3
×
3
3\times3
3
×
3
grid of squares obeying the following rules: i A path is continuous with no breaks (it can be drawn without lifting a pencil). ii A path that spans multiple squares can only be drawn between colored squares that share a side. iii A path cannot go through a square more than once. Compute the number of ways to color a positive number of squares on the grid such that a valid path can be drawn. An example of one such coloring and a valid path is shown below. [Insert Diagram] Proposed by Alex Li
12
1
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Spring 2020 Team Round Problem 12
In the figure above, the large triangle and all four shaded triangles are equilateral. If the areas of triangles
A
,
B
,
A, B,
A
,
B
,
and
C
C
C
are
1
,
2
,
1, 2,
1
,
2
,
and
3
,
3,
3
,
respectively, compute the smallest possible integer ratio between the area of the entire triangle to the area of triangle
D
.
D.
D
.
[Insert Diagram] Proposed by Alex Li
7
1
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Spring 2020 Team Round Problem 7
The hexagonal pattern constructed below has two smaller hexagons per side and has a total of
30
30
30
edges. A similar figure is constructed with
20
20
20
smaller hexagons per side. Compute the number of edges in this larger figure. [Insert Diagram] Proposed by Ezra Erives
2
1
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Spring 2020 Team Round Problem 2
In tetrahedron
A
B
C
D
,
ABCD,
A
BC
D
,
as shown below, compute the number of ways to start at
A
,
A,
A
,
walk along some path of edges, and arrive back at
A
A
A
without walking over the same edge twice. [Insert Diagram] Proposed by Richard Chen
30
1
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Spring 2020 Team Round Problem 30
Let
A
B
C
D
ABCD
A
BC
D
be a cyclic quadrilateral such that the ratio of its diagonals is
A
C
:
B
D
=
7
:
5.
AC:BD=7:5.
A
C
:
B
D
=
7
:
5.
Let
E
E
E
and
F
F
F
be the intersections of lines
A
B
AB
A
B
and
C
D
CD
C
D
and lines
B
C
BC
BC
and
A
D
AD
A
D
, respectively. Let
L
L
L
and
M
M
M
be the midpoints of diagonals
A
C
AC
A
C
and
B
D
BD
B
D
, respectively. Given that
E
F
=
2020
,
EF=2020,
EF
=
2020
,
the length of
L
M
LM
L
M
can be written as
p
q
\frac{p}{q}
q
p
where
p
,
q
p,q
p
,
q
are relatively prime positive integers. Compute
p
+
q
.
p+q.
p
+
q
.
29
1
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Spring 2020 Team Round Problem 29
Let
F
\mathcal{F}
F
be the set of polynomials
f
(
x
)
f(x)
f
(
x
)
with integer coefficients for which there exists an integer root of the equation
f
(
x
)
=
1
f(x)=1
f
(
x
)
=
1
. For all
k
>
1
k>1
k
>
1
, let
m
k
m_k
m
k
be the smallest integer greater than one for which there exists
f
(
x
)
∈
F
f(x)\in \mathcal{F}
f
(
x
)
∈
F
such that
f
(
x
)
=
m
k
f(x)=m_k
f
(
x
)
=
m
k
has exactly
k
k
k
distinct integer roots. If the value of
m
2021
−
m
2020
\sqrt{m_{2021}-m_{2020}}
m
2021
−
m
2020
can be written as
m
n
m\sqrt{n}
m
n
for positive integers
m
,
n
m,n
m
,
n
where
n
n
n
is squarefree, compute the largest integer value of
k
k
k
such that
2
k
2^k
2
k
divides
m
n
\frac{m}{n}
n
m
.
27
1
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Spring 2020 Team Round Problem 27
Let
S
n
=
∑
k
=
1
n
(
k
5
+
k
7
)
.
S_n=\sum_{k=1}^n (k^5+k^7).
S
n
=
∑
k
=
1
n
(
k
5
+
k
7
)
.
Let the prime factorization of
gcd
(
S
2020
,
S
6060
)
\text{gcd}(S_{2020},S_{6060})
gcd
(
S
2020
,
S
6060
)
be
p
1
k
1
⋅
p
2
k
2
⋯
p
i
k
i
p_1^{k_1}\cdot p_2^{k_2}\cdots p_i^{k_i}
p
1
k
1
⋅
p
2
k
2
⋯
p
i
k
i
. Compute
p
1
+
p
2
+
⋯
+
p
i
+
k
1
+
k
2
+
⋯
+
k
i
p_1+p_2+\cdots +p_i+k_1+k_2+\cdots + k_i
p
1
+
p
2
+
⋯
+
p
i
+
k
1
+
k
2
+
⋯
+
k
i
.
26
1
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Spring 2020 Team Round Problem 26
A magic
3
×
5
3 \times 5
3
×
5
board can toggle its cells between black and white. Define a
pattern
to be an assignment of black or white to each of the board's
15
15
15
cells (so there are
2
15
2^{15}
2
15
patterns total). Every day after Day 1, at the beginning of the day, the board gets bored with its black-white pattern and makes a new one. However, the board always wants to be unique and will die if any two of its patterns are less than
3
3
3
cells different from each other. Furthermore, the board dies if it becomes all white. If the board begins with all cells black on Day
1
1
1
, compute the maximum number of days it can stay alive.
25
1
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Spring 2020 Team Round Problem 25
Let
△
A
B
C
\triangle ABC
△
A
BC
be a triangle such that
A
B
=
5
,
A
C
=
8
,
AB=5,AC=8,
A
B
=
5
,
A
C
=
8
,
and
∠
B
A
C
=
6
0
∘
\angle BAC=60^{\circ}
∠
B
A
C
=
6
0
∘
. Let
Γ
\Gamma
Γ
denote the circumcircle of
A
B
C
ABC
A
BC
, and let
I
I
I
and
O
O
O
denote the incenter and circumcenter of
△
A
B
C
\triangle ABC
△
A
BC
, respectively. Let
P
P
P
be the intersection of ray
I
O
IO
I
O
with
Γ
\Gamma
Γ
, and let
X
X
X
be the intersection of ray
B
I
BI
B
I
with
Γ
\Gamma
Γ
. If the area of quadrilateral
X
I
C
P
XICP
X
I
CP
can be expressed as
a
b
+
c
d
e
\frac{a\sqrt{b}+c\sqrt{d}}{e}
e
a
b
+
c
d
, where
a
a
a
and
d
d
d
are squarefree positive integers and
gcd
(
a
,
c
,
e
)
=
1
\gcd(a,c,e)=1
g
cd
(
a
,
c
,
e
)
=
1
, compute
a
+
b
+
c
+
d
+
e
a+b+c+d+e
a
+
b
+
c
+
d
+
e
.
24
1
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Spring 2020 Team Round Problem 24
Let
a
a
a
,
b
b
b
, and
c
c
c
be real angles such that \newline
3
sin
a
+
4
sin
b
+
5
sin
c
=
0
3\sin a + 4\sin b + 5\sin c = 0
3
sin
a
+
4
sin
b
+
5
sin
c
=
0
3
cos
a
+
4
cos
b
+
5
cos
c
=
0.
3\cos a + 4\cos b + 5\cos c = 0.
3
cos
a
+
4
cos
b
+
5
cos
c
=
0.
\newline The maximum value of the expression
sin
b
sin
c
sin
2
a
\frac{\sin b \sin c}{\sin^2 a}
s
i
n
2
a
s
i
n
b
s
i
n
c
can be expressed as
p
q
\frac{p}{q}
q
p
for relatively prime
p
,
q
p,q
p
,
q
. Compute
p
+
q
p+q
p
+
q
.
23
1
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Spring 2020 Team Round Problem 23
Let
△
A
B
C
\triangle ABC
△
A
BC
be a triangle such that
A
B
=
A
C
=
40
AB=AC=40
A
B
=
A
C
=
40
and
B
C
=
79.
BC=79.
BC
=
79.
Let
X
X
X
and
Y
Y
Y
be the points on segments
A
B
AB
A
B
and
A
C
AC
A
C
such that
A
X
=
5
,
A
Y
=
25.
AX=5, AY=25.
A
X
=
5
,
A
Y
=
25.
Given that
P
P
P
is the intersection of lines
X
Y
XY
X
Y
and
B
C
,
BC,
BC
,
compute
P
X
⋅
P
Y
−
P
B
⋅
P
C
.
PX\cdot PY-PB\cdot PC.
PX
⋅
P
Y
−
PB
⋅
PC
.
22
1
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Spring 2020 Team Round Problem 22
The numbers one through eight are written, in that order, on a chalkboard. A mysterious higher power in possession of both an eraser and a piece of chalk chooses three distinct numbers
x
x
x
,
y
y
y
, and
z
z
z
on the board, and does the following. First,
x
x
x
is erased and replaced with
y
y
y
, after which
y
y
y
is erased and replaced with
z
z
z
, and finally
z
z
z
is erased and replaced with
x
x
x
. The higher power repeats this process some finite number of times. For example, if
(
x
,
y
,
z
)
=
(
2
,
4
,
5
)
(x,y,z)=(2,4,5)
(
x
,
y
,
z
)
=
(
2
,
4
,
5
)
is chosen, followed by
(
x
,
y
,
z
)
=
(
1
,
4
,
3
)
(x,y,z)=(1,4,3)
(
x
,
y
,
z
)
=
(
1
,
4
,
3
)
, the board would change in the following manner:
12345678
→
14352678
→
43152678
12345678 \rightarrow 14352678 \rightarrow 43152678
12345678
→
14352678
→
43152678
Compute the number of possible final orderings of the eight numbers.
21
1
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Spring 2020 Team Round Problem 21
Let
{
a
n
}
\{a_n\}
{
a
n
}
be the sequence such that
a
0
=
2019
a_0=2019
a
0
=
2019
and
a
n
=
−
2020
n
∑
k
=
0
n
−
1
a
k
.
a_n=-\frac{2020}{n}\sum_{k=0}^{n-1}a_k.
a
n
=
−
n
2020
k
=
0
∑
n
−
1
a
k
.
Compute the last three digits of
∑
n
=
1
2020
202
0
n
a
n
n
\sum_{n=1}^{2020}2020^na_nn
∑
n
=
1
2020
202
0
n
a
n
n
.
20
1
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Spring 2020 Team Round Problem 20
Let
c
1
<
c
2
<
c
3
c_1<c_2<c_3
c
1
<
c
2
<
c
3
be the three smallest positive integer values of
c
c
c
such that the distance between the parabola
y
=
x
2
+
2020
y=x^2+2020
y
=
x
2
+
2020
and the line
y
=
c
x
y=cx
y
=
c
x
is a rational multiple of
2
\sqrt{2}
2
. Compute
c
1
+
c
2
+
c
3
c_1+c_2+c_3
c
1
+
c
2
+
c
3
.
19
1
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Spring 2020 Team Round Problem 19
Let
A
B
C
ABC
A
BC
be a triangle such that such that
A
B
=
14
,
B
C
=
13
AB=14, BC=13
A
B
=
14
,
BC
=
13
, and
A
C
=
15
AC=15
A
C
=
15
. Let
X
X
X
be a point inside triangle
A
B
C
ABC
A
BC
. Compute the minimum possible value of
(
2
A
X
+
B
X
+
C
X
)
2
(\sqrt{2}AX+BX+CX)^2
(
2
A
X
+
BX
+
CX
)
2
.
18
1
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Spring 2020 Team Round Problem 18
Compute the maximum integer value of
k
k
k
such that
2
k
2^k
2
k
divides
3
2
n
+
3
+
40
n
−
27
3^{2n+3}+40n-27
3
2
n
+
3
+
40
n
−
27
for any positive integer
n
n
n
.
17
1
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Spring 2020 Team Round Problem 17
Let
A
B
C
ABC
A
BC
be a triangle such that
A
B
=
26
,
A
C
=
30
,
AB = 26, AC = 30,
A
B
=
26
,
A
C
=
30
,
and
B
C
=
28
BC = 28
BC
=
28
. Let
C
′
C'
C
′
and
B
′
B'
B
′
be the reflections of the circumcenter
O
O
O
over
A
B
AB
A
B
and
A
C
AC
A
C
, respectively. The length of the portion of line segment
B
′
C
′
B'C'
B
′
C
′
inside triangle
A
B
C
ABC
A
BC
can be written as
p
q
\frac{p}{q}
q
p
, where
p
,
q
p,q
p
,
q
are relatively prime positive integers. Compute
p
+
q
p+q
p
+
q
.
16
1
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Spring 2020 Team Round Problem 16
For non-negative integer
n
n
n
, the function
f
f
f
is given by
f
(
x
)
=
{
x
2
if
n
is even
x
−
1
if
n
is odd.
f(x)=\begin{cases} \frac{x}{2} & \text{if $n$ is even} \\ x-1 & \text{if $n$ is odd.} \end{cases}
f
(
x
)
=
{
2
x
x
−
1
if
n
is even
if
n
is odd.
Furthermore, let
h
(
n
)
h(n)
h
(
n
)
be the smallest
k
k
k
for which
f
k
(
n
)
=
0
f^k(n)=0
f
k
(
n
)
=
0
. Compute
∑
n
=
1
1024
h
(
n
)
.
\sum_{n=1}^{1024} h(n).
n
=
1
∑
1024
h
(
n
)
.
15
1
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Spring 2020 Team Round Problem 15
Let
ϕ
(
k
)
\phi(k)
ϕ
(
k
)
denote the number of positive integers less than or equal to
k
k
k
that are relatively prime to
k
k
k
. For example,
ϕ
(
2
)
=
1
\phi(2)=1
ϕ
(
2
)
=
1
and
ϕ
(
10
)
=
4
\phi(10)=4
ϕ
(
10
)
=
4
. Compute the number of positive integers
n
≤
2020
n \leq 2020
n
≤
2020
such that
ϕ
(
n
2
)
=
2
ϕ
(
n
)
2
\phi(n^2)=2\phi(n)^2
ϕ
(
n
2
)
=
2
ϕ
(
n
)
2
.
14
1
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Spring 2020 Team Round Problem 14
Let
△
A
B
C
\triangle ABC
△
A
BC
be a triangle such that
A
B
=
40
AB=40
A
B
=
40
and
A
C
=
30.
AC=30.
A
C
=
30.
Points
X
X
X
and
Y
Y
Y
are on the segment
A
B
AB
A
B
and
B
C
,
BC,
BC
,
respectively such that
A
X
:
B
X
=
3
:
2
AX:BX=3:2
A
X
:
BX
=
3
:
2
and
B
Y
:
C
Y
=
1
:
4.
BY:CY=1:4.
B
Y
:
C
Y
=
1
:
4.
Given that
X
Y
=
12
,
XY=12,
X
Y
=
12
,
the area of
△
A
B
C
\triangle ABC
△
A
BC
can be written as
a
b
a\sqrt{b}
a
b
where
a
a
a
and
b
b
b
are positive integers and
b
b
b
is squarefree. Compute
a
+
b
.
a+b.
a
+
b
.
11
1
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Spring 2020 Team Round Problem 11
Let set
S
\mathcal{S}
S
contain all positive integers less than or equal to
2020
2020
2020
that can be written in the form
n
(
n
+
1
)
n(n+1)
n
(
n
+
1
)
for some positive integer
n
n
n
. Compute the number of ordered pairs
(
a
,
b
)
(a,b)
(
a
,
b
)
such that
a
,
b
∈
S
a, b\in \mathcal{S}
a
,
b
∈
S
and
a
−
b
a-b
a
−
b
is a power of two.
10
1
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Spring 2020 Team Round Problem 10
Three mutually externally tangent circles are internally tangent to a circle with radius
1
1
1
. If two of the inner circles have radius
1
3
\frac{1}{3}
3
1
, the largest possible radius of the third inner circle can be expressed in the form
a
+
b
c
d
\frac{a+b\sqrt{c}}{d}
d
a
+
b
c
where
c
c
c
is squarefree and
gcd
(
a
,
b
,
d
)
=
1
\gcd(a,b,d)=1
g
cd
(
a
,
b
,
d
)
=
1
. Find
a
+
b
+
c
+
d
a+b+c+d
a
+
b
+
c
+
d
.
9
1
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Spring 2020 Team Round Problem 9
A function
f
(
x
)
f(x)
f
(
x
)
is such that for any integer
x
x
x
,
f
(
x
)
+
x
f
(
2
−
x
)
=
6
f(x)+xf(2-x)=6
f
(
x
)
+
x
f
(
2
−
x
)
=
6
. Compute
−
2019
f
(
2020
)
-2019f(2020)
−
2019
f
(
2020
)
.
8
1
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Spring 2020 Team Round Problem 8
Let
a
,
b
a,b
a
,
b
be real numbers satisfying
a
2
+
b
2
=
3
a
b
=
75
a^{2} + b^{2} = 3ab = 75
a
2
+
b
2
=
3
ab
=
75
and
a
>
b
a>b
a
>
b
. Compute
a
3
−
b
3
a^{3}-b^{3}
a
3
−
b
3
.
6
1
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Spring 2020 Team Round Problem 6
Let
△
A
B
C
\triangle ABC
△
A
BC
be a triangle such that
A
B
=
6
,
B
C
=
8
,
AB=6, BC=8,
A
B
=
6
,
BC
=
8
,
and
A
C
=
10
AC=10
A
C
=
10
. Let
M
M
M
be the midpoint of
B
C
BC
BC
. Circle
ω
\omega
ω
passes through
A
A
A
and is tangent to
B
C
BC
BC
at
M
M
M
. Suppose
ω
\omega
ω
intersects segments
A
B
AB
A
B
and
A
C
AC
A
C
again at points
X
X
X
and
Y
Y
Y
, respectively. If the area of
A
X
Y
AXY
A
X
Y
can be expressed as
p
q
\frac{p}{q}
q
p
where
p
,
q
p, q
p
,
q
are relatively prime integers, compute
p
+
q
p+q
p
+
q
.
5
1
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Spring 2020 Team Round Problem 5
For a positive integer
n
n
n
, let
D
(
n
)
\mathcal{D}(n)
D
(
n
)
be the value obtained by, starting from the left, alternating between adding and subtracting the digits of
n
n
n
. For example,
D
(
321
)
=
3
−
2
+
1
=
2
\mathcal{D}(321)=3-2+1=2
D
(
321
)
=
3
−
2
+
1
=
2
, while
D
(
40
)
=
4
−
0
=
4
\mathcal{D}(40)=4-0=4
D
(
40
)
=
4
−
0
=
4
. Compute the value of the sum
∑
n
=
1
100
D
(
n
)
=
D
(
1
)
+
D
(
2
)
+
⋯
+
D
(
100
)
.
\sum_{n=1}^{100}\mathcal{D}(n)=\mathcal{D}(1)+\mathcal{D}(2)+\dots+\mathcal{D}(100).
n
=
1
∑
100
D
(
n
)
=
D
(
1
)
+
D
(
2
)
+
⋯
+
D
(
100
)
.
4
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Spring 2020 Team Round Problem 4
Suppose there are
n
n
n
ordered pairs of positive integers
(
a
i
,
b
i
)
(a_i,b_i)
(
a
i
,
b
i
)
such that
a
i
+
b
i
=
2020
a_i+b_i=2020
a
i
+
b
i
=
2020
and
a
i
b
i
a_ib_i
a
i
b
i
is a multiple of
2020
2020
2020
, where
1
≤
i
≤
n
1\le i \le n
1
≤
i
≤
n
. Compute the sum
∑
i
=
1
n
a
i
+
b
i
.
\sum_{i=1}^{n} a_i+b_i.
i
=
1
∑
n
a
i
+
b
i
.
3
1
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Spring 2020 Team Round Problem 3
Let
L
M
T
LMT
L
MT
represent a 3-digit positive integer where
L
L
L
and
M
M
M
are nonzero digits. Suppose that the 2-digit number
M
T
MT
MT
divides
L
M
T
LMT
L
MT
. Compute the difference between the maximum and minimum possible values of
L
M
T
LMT
L
MT
.
1
1
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Spring 2020 Team Round Problem 1
Compute the smallest nonnegative integer that can be written as the sum of 2020 distinct integers.