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Math Open At Andover problems
2020 MOAA
2020 MOAA
Part of
Math Open At Andover problems
Subcontests
(9)
General
1
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2020 MOAA General Round - Math Open At Andover
p1. What is
20
×
20
−
19
×
19
20\times 20 - 19\times 19
20
×
20
−
19
×
19
? p2. Andover has a total of
1440
1440
1440
students and teachers as well as a
1
:
5
1 : 5
1
:
5
teacher-to-student ratio (for every teacher, there are exactly
5
5
5
students). In addition, every student is either a boarding student or a day student, and
70
%
70\%
70%
of the students are boarding students. How many day students does Andover have? p3. The time is
2
:
20
2:20
2
:
20
. If the acute angle between the hour hand and the minute hand of the clock measures
x
x
x
degrees, find
x
x
x
. https://cdn.artofproblemsolving.com/attachments/b/a/a18b089ae016b15580ec464c3e813d5cb57569.png p4. Point
P
P
P
is located on segment
A
C
AC
A
C
of square
A
B
C
D
ABCD
A
BC
D
with side length
10
10
10
such that
A
P
>
C
P
AP >CP
A
P
>
CP
. If the area of quadrilateral
A
B
P
D
ABPD
A
BP
D
is
70
70
70
, what is the area of
△
P
B
D
\vartriangle PBD
△
PB
D
? p5. Andrew always sweetens his tea with sugar, and he likes a
1
:
7
1 : 7
1
:
7
sugar-to-unsweetened tea ratio. One day, he makes a
100
100
100
ml cup of unsweetened tea but realizes that he has run out of sugar. Andrew decides to borrow his sister's jug of pre-made SUPERSWEET tea, which has a
1
:
2
1 : 2
1
:
2
sugar-to-unsweetened tea ratio. How much SUPERSWEET tea, in ml,does Andrew need to add to his unsweetened tea so that the resulting tea is his desired sweetness? p6. Jeremy the architect has built a railroad track across the equator of his spherical home planet which has a radius of exactly
2020
2020
2020
meters. He wants to raise the entire track
6
6
6
meters off the ground, everywhere around the planet. In order to do this, he must buymore track, which comes from his supplier in bundles of
2
2
2
meters. What is the minimum number of bundles he must purchase? Assume the railroad track was originally built on the ground. p7. Mr. DoBa writes the numbers
1
,
2
,
3
,
.
.
.
,
20
1, 2, 3,..., 20
1
,
2
,
3
,
...
,
20
on the board. Will then walks up to the board, chooses two of the numbers, and erases them from the board. Mr. DoBa remarks that the average of the remaining
18
18
18
numbers is exactly
11
11
11
. What is the maximum possible value of the larger of the two numbers that Will erased? p8. Nathan is thinking of a number. His number happens to be the smallest positive integer such that if Nathan doubles his number, the result is a perfect square, and if Nathan triples his number, the result is a perfect cube. What is Nathan's number? p9. Let
S
S
S
be the set of positive integers whose digits are in strictly increasing order when read from left to right. For example,
1
1
1
,
24
24
24
, and
369
369
369
are all elements of
S
S
S
, while
20
20
20
and
667
667
667
are not. If the elements of
S
S
S
are written in increasing order, what is the
100
100
100
th number written? p10. Find the largest prime factor of the expression
2
20
+
2
16
+
2
12
+
2
8
+
2
4
+
1
2^{20} + 2^{16} + 2^{12} + 2^{8} + 2^{4} + 1
2
20
+
2
16
+
2
12
+
2
8
+
2
4
+
1
. p11. Christina writes down all the numbers from
1
1
1
to
2020
2020
2020
, inclusive, on a whiteboard. What is the sum of all the digits that she wrote down? p12. Triangle
A
B
C
ABC
A
BC
has side lengths
A
B
=
A
C
=
10
AB = AC = 10
A
B
=
A
C
=
10
and
B
C
=
16
BC = 16
BC
=
16
. Let
M
M
M
and
N
N
N
be the midpoints of segments
B
C
BC
BC
and
C
A
CA
C
A
, respectively. There exists a point
P
≠
A
P \ne A
P
=
A
on segment
A
M
AM
A
M
such that
2
P
N
=
P
C
2PN = PC
2
PN
=
PC
. What is the area of
△
P
B
C
\vartriangle PBC
△
PBC
? p13. Consider the polynomial
P
(
x
)
=
x
4
+
3
x
3
+
5
x
2
+
7
x
+
9.
P(x) = x^4 + 3x^3 + 5x^2 + 7x + 9.
P
(
x
)
=
x
4
+
3
x
3
+
5
x
2
+
7
x
+
9.
Let its four roots be
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
. Evaluate the expression
(
a
+
b
+
c
)
(
a
+
b
+
d
)
(
a
+
c
+
d
)
(
b
+
c
+
d
)
.
(a + b + c)(a + b + d)(a + c + d)(b + c + d).
(
a
+
b
+
c
)
(
a
+
b
+
d
)
(
a
+
c
+
d
)
(
b
+
c
+
d
)
.
p14. Consider the system of equations
∣
y
−
1
∣
=
4
−
∣
x
−
1
∣
|y - 1| = 4 -|x - 1|
∣
y
−
1∣
=
4
−
∣
x
−
1∣
∣
y
∣
=
∣
k
−
x
∣
.
|y| =\sqrt{|k - x|}.
∣
y
∣
=
∣
k
−
x
∣
.
Find the largest
k
k
k
for which this system has a solution for real values
x
x
x
and
y
y
y
. p16. Let
T
n
=
1
+
2
+
.
.
.
+
n
T_n = 1 + 2 + ... + n
T
n
=
1
+
2
+
...
+
n
denote the
n
n
n
th triangular number. Find the number of positive integers
n
n
n
less than
100
100
100
such that
n
n
n
and
T
n
T_n
T
n
have the same number of positive integer factors. p17. Let
A
B
C
D
ABCD
A
BC
D
be a square, and let
P
P
P
be a point inside it such that
P
A
=
4
PA = 4
P
A
=
4
,
P
B
=
2
PB = 2
PB
=
2
, and
P
C
=
2
2
PC = 2\sqrt2
PC
=
2
2
. What is the area of
A
B
C
D
ABCD
A
BC
D
? p18. The Fibonacci sequence
{
F
n
}
\{F_n\}
{
F
n
}
is defined as
F
0
=
0
F_0 = 0
F
0
=
0
,
F
1
=
1
F_1 = 1
F
1
=
1
, and
F
n
+
2
=
F
n
+
1
+
F
n
F_{n+2}= F_{n+1} + F_n
F
n
+
2
=
F
n
+
1
+
F
n
for all integers
n
≥
0
n \ge 0
n
≥
0
. Let
S
=
1
F
6
+
1
F
6
+
1
F
8
+
1
F
8
+
1
F
10
+
1
F
10
+
1
F
12
+
1
F
12
+
.
.
.
S =\dfrac{1}{F_6 + \frac{1}{F_6}}+\dfrac{1}{F_8 + \frac{1}{F_8}}+\dfrac{1}{F_{10} +\frac{1}{F_{10}}}+\dfrac{1}{F_{12} + \frac{1}{F_{12}}}+ ...
S
=
F
6
+
F
6
1
1
+
F
8
+
F
8
1
1
+
F
10
+
F
10
1
1
+
F
12
+
F
12
1
1
+
...
Compute
420
S
420S
420
S
. p19. Let
A
B
C
D
ABCD
A
BC
D
be a square with side length
5
5
5
. Point
P
P
P
is located inside the square such that the distances from
P
P
P
to
A
B
AB
A
B
and
A
D
AD
A
D
are
1
1
1
and
2
2
2
respectively. A point
T
T
T
is selected uniformly at random inside
A
B
C
D
ABCD
A
BC
D
. Let
p
p
p
be the probability that quadrilaterals
A
P
C
T
APCT
A
PCT
and
B
P
D
T
BPDT
BP
D
T
are both not self-intersecting and have areas that add to no more than
10
10
10
. If
p
p
p
can be expressed in the form
m
n
\frac{m}{n}
n
m
for relatively prime positive integers
m
m
m
and
n
n
n
, find
m
+
n
m + n
m
+
n
.Note: A quadrilateral is self-intersecting if any two of its edges cross. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
Sets 6-9
1
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2020 MOAA Gunga Bowl - Math Open At Andover - last 4 sets - 12 problems
Set 6 B16. Let
ℓ
r
\ell_r
ℓ
r
denote the line
x
+
r
y
+
r
2
=
420
x + ry + r^2 = 420
x
+
ry
+
r
2
=
420
. Jeffrey draws the lines
ℓ
a
\ell_a
ℓ
a
and
ℓ
b
\ell_b
ℓ
b
and calculates their single intersection point. B17. Let set
L
L
L
consist of lines of the form
3
x
+
2
a
y
=
60
a
+
48
3x + 2ay = 60a + 48
3
x
+
2
a
y
=
60
a
+
48
across all real constants a. For every line
ℓ
\ell
ℓ
in
L
L
L
, the point on
ℓ
\ell
ℓ
closest to the origin is in set
T
T
T
. The area enclosed by the locus of all the points in
T
T
T
can be expressed in the form nπ for some positive integer
n
n
n
. Compute
n
n
n
. B18. What is remainder when the
2020
2020
2020
-digit number
202020...20
202020 ... 20
202020...20
is divided by
275
275
275
? Set 7 B19. Consider right triangle
△
A
B
C
\vartriangle ABC
△
A
BC
where
∠
A
B
C
=
9
0
o
\angle ABC = 90^o
∠
A
BC
=
9
0
o
,
∠
A
C
B
=
3
0
o
\angle ACB = 30^o
∠
A
CB
=
3
0
o
, and
A
C
=
10
AC = 10
A
C
=
10
. Suppose a beam of light is shot out from point
A
A
A
. It bounces off side
B
C
BC
BC
and then bounces off side
A
C
AC
A
C
, and then hits point
B
B
B
and stops moving. If the beam of light travelled a distance of
d
d
d
, then compute
d
2
d^2
d
2
. B20. Let
S
S
S
be the set of all three digit numbers whose digits sum to
12
12
12
. What is the sum of all the elements in
S
S
S
? B21. Consider all ordered pairs
(
m
,
n
)
(m, n)
(
m
,
n
)
where
m
m
m
is a positive integer and
n
n
n
is an integer that satisfy
m
!
=
3
n
2
+
6
n
+
15
,
m! = 3n^2 + 6n + 15,
m
!
=
3
n
2
+
6
n
+
15
,
where
m
!
=
m
×
(
m
−
1
)
×
.
.
.
×
1
m! = m \times (m - 1) \times ... \times 1
m
!
=
m
×
(
m
−
1
)
×
...
×
1
. Determine the product of all possible values of
n
n
n
. Set 8 B22. Compute the number of ordered pairs of integers
(
m
,
n
)
(m, n)
(
m
,
n
)
satisfying
1000
>
m
>
n
>
0
1000 > m > n > 0
1000
>
m
>
n
>
0
and
6
⋅
l
c
m
(
m
−
n
,
m
+
n
)
=
5
⋅
l
c
m
(
m
,
n
)
6 \cdot lcm(m - n, m + n) = 5 \cdot lcm(m, n)
6
⋅
l
c
m
(
m
−
n
,
m
+
n
)
=
5
⋅
l
c
m
(
m
,
n
)
. B23. Andrew is flipping a coin ten times. After every flip, he records the result (heads or tails). He notices that after every flip, the number of heads he had flipped was always at least the number of tails he had flipped. In how many ways could Andrew have flipped the coin? B24. Consider a triangle
A
B
C
ABC
A
BC
with
A
B
=
7
AB = 7
A
B
=
7
,
B
C
=
8
BC = 8
BC
=
8
, and
C
A
=
9
CA = 9
C
A
=
9
. Let
D
D
D
lie on
A
B
‾
\overline{AB}
A
B
and
E
E
E
lie on
A
C
‾
\overline{AC}
A
C
such that
B
C
E
D
BCED
BCE
D
is a cyclic quadrilateral and
D
,
O
,
E
D, O, E
D
,
O
,
E
are collinear, where
O
O
O
is the circumcenter of
A
B
C
ABC
A
BC
. The area of
△
A
D
E
\vartriangle ADE
△
A
D
E
can be expressed as
m
n
p
\frac{m\sqrt{n}}{p}
p
m
n
, where
m
m
m
and
p
p
p
are relatively prime positive integers, and
n
n
n
is a positive integer not divisible by the square of any prime. What is
m
+
n
+
p
m + n + p
m
+
n
+
p
? Set 9This set consists of three estimation problems, with scoring schemes described. B25. Submit one of the following ten numbers:
3
6
9
12
15
18
21
24
27
30.
3 \,\,\,\, 6\,\,\,\, 9\,\,\,\, 12\,\,\,\, 15\,\,\,\, 18\,\,\,\, 21\,\,\,\, 24\,\,\,\, 27\,\,\,\, 30.
3
6
9
12
15
18
21
24
27
30.
The number of points you will receive for this question is equal to the number you selected divided by the total number of teams that selected that number, then rounded up to the nearest integer. For example, if you and four other teams select the number
27
27
27
, you would receive
⌈
27
5
⌉
=
6
\left\lceil \frac{27}{5}\right\rceil = 6
⌈
5
27
⌉
=
6
points. B26. Submit any integer from
1
1
1
to
1
,
000
,
000
1,000,000
1
,
000
,
000
, inclusive. The standard deviation
σ
\sigma
σ
of all responses
x
i
x_i
x
i
to this question is computed by first taking the arithmetic mean
μ
\mu
μ
of all responses, then taking the square root of average of
(
x
i
−
μ
)
2
(x_i -\mu)^2
(
x
i
−
μ
)
2
over all
i
i
i
. More, precisely, if there are
N
N
N
responses, then
σ
=
1
N
∑
i
=
1
N
(
x
i
−
μ
)
2
.
\sigma =\sqrt{\frac{1}{N} \sum^N_{i=1} (x_i -\mu)^2}.
σ
=
N
1
i
=
1
∑
N
(
x
i
−
μ
)
2
.
For this problem, your goal is to estimate the standard deviation of all responses.An estimate of
e
e
e
gives
max
{
⌊
130
(
m
i
n
{
σ
e
,
e
σ
}
3
⌋
−
100
,
0
}
\max \{ \left\lfloor 130 ( min \{ \frac{\sigma }{e},\frac{e}{\sigma }\}^{3}\right\rfloor -100,0 \}
max
{
⌊
130
(
min
{
e
σ
,
σ
e
}
3
⌋
−
100
,
0
}
points. B27. For a positive integer
n
n
n
, let
f
(
n
)
f(n)
f
(
n
)
denote the number of distinct nonzero exponents in the prime factorization of
n
n
n
. For example,
f
(
36
)
=
f
(
2
2
×
3
2
)
=
1
f(36) = f(2^2 \times 3^2) = 1
f
(
36
)
=
f
(
2
2
×
3
2
)
=
1
and
f
(
72
)
=
f
(
2
3
×
3
2
)
=
2
f(72) = f(2^3 \times 3^2) = 2
f
(
72
)
=
f
(
2
3
×
3
2
)
=
2
. Estimate
N
=
f
(
2
)
+
f
(
3
)
+
.
.
+
f
(
10000
)
N = f(2) + f(3) +.. + f(10000)
N
=
f
(
2
)
+
f
(
3
)
+
..
+
f
(
10000
)
.An estimate of
e
e
e
gives
max
{
30
−
⌊
7
l
o
g
10
(
∣
N
−
e
∣
)
⌋
,
0
}
\max \{30 - \lfloor 7 log_{10}(|N - e|)\rfloor , 0\}
max
{
30
−
⌊
7
l
o
g
10
(
∣
N
−
e
∣
)⌋
,
0
}
points.PS. You had better use hide for answers. First sets have been posted [url=https://artofproblemsolving.com/community/c4h2777391p24371239]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
Sets 1-5
1
Hide problems
2020 MOAA Gunga Bowl - Math Open At Andover - first 5 sets - 15 problems
Set 1 B1. Evaluate
2
+
0
−
2
×
0
2 + 0 - 2 \times 0
2
+
0
−
2
×
0
. B2. It takes four painters four hours to paint four houses. How many hours does it take forty painters to paint forty houses? B3. Let
a
a
a
be the answer to this question. What is
1
2
−
a
\frac{1}{2-a}
2
−
a
1
? Set 2 B4. Every day at Andover is either sunny or rainy. If today is sunny, there is a
60
%
60\%
60%
chance that tomorrow is sunny and a
40
%
40\%
40%
chance that tomorrow is rainy. On the other hand, if today is rainy, there is a
60
%
60\%
60%
chance that tomorrow is rainy and a
40
%
40\%
40%
chance that tomorrow is sunny. Given that today is sunny, the probability that the day after tomorrow is sunny can be expressed as n%, where n is a positive integer. What is
n
n
n
? B5. In the diagram below, what is the value of
∠
D
D
′
Y
\angle DD'Y
∠
D
D
′
Y
in degrees? https://cdn.artofproblemsolving.com/attachments/0/8/6c966b13c840fa1885948d0e4ad598f36bee9d.png B6. Christina, Jeremy, Will, and Nathan are standing in a line. In how many ways can they be arranged such that Christina is to the left of Will and Jeremy is to the left of Nathan? Note: Christina does not have to be next to Will and Jeremy does not have to be next to Nathan. For example, arranging them as Christina, Jeremy, Will, Nathan would be valid. Set 3 B7. Let
P
P
P
be a point on side
A
B
AB
A
B
of square
A
B
C
D
ABCD
A
BC
D
with side length
8
8
8
such that
P
A
=
3
PA = 3
P
A
=
3
. Let
Q
Q
Q
be a point on side
A
D
AD
A
D
such that
P
Q
⊥
P
C
P Q \perp P C
PQ
⊥
PC
. The area of quadrilateral
P
Q
D
B
PQDB
PQ
D
B
can be expressed in the form
m
/
n
m/n
m
/
n
for relatively prime positive integers
m
m
m
and
n
n
n
. Compute
m
+
n
m + n
m
+
n
. B8. Jessica and Jeffrey each pick a number uniformly at random from the set
{
1
,
2
,
3
,
4
,
5
}
\{1, 2, 3, 4, 5\}
{
1
,
2
,
3
,
4
,
5
}
(they could pick the same number). If Jessica’s number is
x
x
x
and Jeffrey’s number is
y
y
y
, the probability that
x
y
x^y
x
y
has a units digit of
1
1
1
can be expressed as
m
/
n
m/n
m
/
n
, where
m
m
m
and
n
n
n
are relatively prime positive integers. Find
m
+
n
m + n
m
+
n
. B9. For two points
(
x
1
,
y
1
)
(x_1, y_1)
(
x
1
,
y
1
)
and
(
x
2
,
y
2
)
(x_2, y_2)
(
x
2
,
y
2
)
in the plane, we define the taxicab distance between them as
∣
x
1
−
x
2
∣
+
∣
y
1
−
y
2
∣
|x_1 - x_2| + |y_1 - y_2|
∣
x
1
−
x
2
∣
+
∣
y
1
−
y
2
∣
. For example, the taxicab distance between
(
−
1
,
2
)
(-1, 2)
(
−
1
,
2
)
and
(
3
,
2
)
(3,\sqrt2)
(
3
,
2
)
is
6
−
2
6-\sqrt2
6
−
2
. What is the largest number of points Nathan can find in the plane such that the taxicab distance between any two of the points is the same? Set 4 B10. Will wants to insert some × symbols between the following numbers:
1
2
3
4
6
1\,\,\,2\,\,\,3\,\,\,4\,\,\,6
1
2
3
4
6
to see what kinds of answers he can get. For example, here is one way he can insert
×
\times
×
symbols:
1
×
23
×
4
×
6
=
552.
1 \times 23 \times 4 \times 6 = 552.
1
×
23
×
4
×
6
=
552.
Will discovers that he can obtain the number
276
276
276
. What is the sum of the numbers that he multiplied together to get
276
276
276
? B11. Let
A
B
C
D
ABCD
A
BC
D
be a parallelogram with
A
B
=
5
AB = 5
A
B
=
5
,
B
C
=
3
BC = 3
BC
=
3
, and
∠
B
A
D
=
6
0
o
\angle BAD = 60^o
∠
B
A
D
=
6
0
o
. Let the angle bisector of
∠
A
D
C
\angle ADC
∠
A
D
C
meet
A
C
AC
A
C
at
E
E
E
and
A
B
AB
A
B
at
F
F
F
. The length
E
F
EF
EF
can be expressed as
m
/
n
m/n
m
/
n
, where
m
m
m
and
n
n
n
are relatively prime positive integers. What is
m
+
n
m + n
m
+
n
? B12. Find the sum of all positive integers
n
n
n
such that
⌊
n
2
−
2
n
+
19
⌋
=
n
\lfloor \sqrt{n^2 - 2n + 19} \rfloor = n
⌊
n
2
−
2
n
+
19
⌋
=
n
.Note:
⌊
x
⌋
\lfloor x \rfloor
⌊
x
⌋
denotes the greatest integer less than or equal to
x
x
x
. Set 5 B13. This year, February
29
29
29
fell on a Saturday. What is the next year in which February
29
29
29
will be a Saturday? B14. Let
f
(
x
)
=
1
x
−
1
f(x) = \frac{1}{x} - 1
f
(
x
)
=
x
1
−
1
. Evaluate
f
(
1
2020
)
×
f
(
2
2020
)
×
f
(
3
2020
)
×
×
.
.
.
×
f
(
2019
2020
)
.
f\left( \frac{1}{2020}\right) \times f\left( \frac{2}{2020}\right) \times f\left( \frac{3}{2020}\right) \times \times ... \times f\left( \frac{2019}{2020}\right) .
f
(
2020
1
)
×
f
(
2020
2
)
×
f
(
2020
3
)
×
×
...
×
f
(
2020
2019
)
.
B15. Square
W
X
Y
Z
WXYZ
W
X
Y
Z
is inscribed in square
A
B
C
D
ABCD
A
BC
D
with side length
1
1
1
such that
W
W
W
is on
A
B
AB
A
B
,
X
X
X
is on
B
C
BC
BC
,
Y
Y
Y
is on
C
D
CD
C
D
, and
Z
Z
Z
is on
D
A
DA
D
A
. Line
W
Y
W Y
WY
hits
A
D
AD
A
D
and
B
C
BC
BC
at points
P
P
P
and
R
R
R
respectively, and line
X
Z
XZ
XZ
hits
A
B
AB
A
B
and
C
D
CD
C
D
at points
Q
Q
Q
and
S
S
S
respectively. If the area of
W
X
Y
Z
WXYZ
W
X
Y
Z
is
13
18
\frac{13}{18}
18
13
, then the area of
P
Q
R
S
PQRS
PQRS
can be expressed as
m
/
n
m/n
m
/
n
for relatively prime positive integers
m
m
m
and
n
n
n
. What is
m
+
n
m + n
m
+
n
?PS. You had better use hide for answers. Last sets have been posted [url=https://artofproblemsolving.com/community/c4h2777424p24371574]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
1
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2020 MOAA Theme Round, Relay - Math Open At Andover
Each problem in this section will depend on the previous one! The values
A
,
B
,
C
A, B, C
A
,
B
,
C
, and
D
D
D
refer to the answers to problems
1
,
2
,
3
1, 2, 3
1
,
2
,
3
, and
4
4
4
, respectively. TR1. The number
2020
2020
2020
has three different prime factors. What is their sum? TR2. Let
A
A
A
be the answer to the previous problem. Suppose
A
B
C
ABC
A
BC
is a triangle with
A
B
=
81
AB = 81
A
B
=
81
,
B
C
=
A
BC = A
BC
=
A
, and
∠
A
B
C
=
9
0
o
\angle ABC = 90^o
∠
A
BC
=
9
0
o
. Let
D
D
D
be the midpoint of
B
C
BC
BC
. The perimeter of
△
C
A
D
\vartriangle CAD
△
C
A
D
can be written as
x
+
y
z
x + y\sqrt{z}
x
+
y
z
, where
x
,
y
x, y
x
,
y
, and
z
z
z
are positive integers and
z
z
z
is not divisible by the square of any prime. What is
x
+
y
x + y
x
+
y
? TR3. Let
B
B
B
the answer to the previous problem. What is the unique real value of
k
k
k
such that the parabola
y
=
B
x
2
+
k
y = Bx^2 + k
y
=
B
x
2
+
k
and the line
y
=
k
x
+
B
y = kx + B
y
=
k
x
+
B
are tangent? TR4. Let
C
C
C
be the answer to the previous problem. How many ordered triples of positive integers
(
a
,
b
,
c
)
(a, b, c)
(
a
,
b
,
c
)
are there such that
g
c
d
(
a
,
b
)
=
g
c
d
(
b
,
c
)
=
1
gcd(a, b) = gcd(b, c) = 1
g
c
d
(
a
,
b
)
=
g
c
d
(
b
,
c
)
=
1
and
a
b
c
=
C
abc = C
ab
c
=
C
? TR5. Let
D
D
D
be the answer to the previous problem. Let
A
B
C
D
ABCD
A
BC
D
be a square with side length
D
D
D
and circumcircle
ω
\omega
ω
. Denote points
C
′
C'
C
′
and
D
′
D'
D
′
as the reflections over line
A
B
AB
A
B
of
C
C
C
and
D
D
D
respectively. Let
P
P
P
and
Q
Q
Q
be the points on
ω
\omega
ω
, with
A
A
A
and
P
P
P
on opposite sides of line
B
C
BC
BC
and
B
B
B
and
Q
Q
Q
on opposite sides of line
A
D
AD
A
D
, such that lines
C
′
P
C'P
C
′
P
and
D
′
Q
D'Q
D
′
Q
are both tangent to
ω
\omega
ω
. If the lines
A
P
AP
A
P
and
B
Q
BQ
BQ
intersect at
T
T
T
, what is the area of
△
C
D
T
\vartriangle CDT
△
C
D
T
?PS. You had better use hide for answers.
TO5
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2020 MOAA Theme Optimization TO 5
For a real number
x
x
x
, the minimum value of the expression
2
x
2
+
x
−
3
x
2
−
2
x
+
3
\frac{2x^2 + x - 3}{x^2 - 2x + 3}
x
2
−
2
x
+
3
2
x
2
+
x
−
3
can be written in the form
a
−
b
c
\frac{a-\sqrt{b}}{c}
c
a
−
b
, where
a
,
b
a, b
a
,
b
, and
c
c
c
are positive integers such that
a
a
a
and
c
c
c
are relatively prime. Find
a
+
b
+
c
a + b + c
a
+
b
+
c
TO4
1
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2020 MOAA Theme Optimization TO 4
Over all real numbers
x
x
x
, let
k
k
k
be the minimum possible value of the expression
x
2
+
9
+
x
2
−
6
x
+
45
.
\sqrt{x^2 + 9} +\sqrt{x^2 - 6x + 45}.
x
2
+
9
+
x
2
−
6
x
+
45
.
Determine
k
2
k^2
k
2
.
TO3
1
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2020 MOAA Theme Optimization TO 3
Consider the addition \begin{tabular}{cccc} & O & N & E \\ + & T & W & O \\ \hline F & O & U & R \\ \end{tabular} where different letters represent different nonzero digits. What is the smallest possible value of the four-digit number
F
O
U
R
FOUR
FO
U
R
?
TO2
1
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2020 MOAA Theme Optimization TO 2
The Den has two deals on chicken wings. The first deal is
4
4
4
chicken wings for
3
3
3
dollars, and the second deal is
11
11
11
chicken wings for
8
8
8
dollars. If Jeremy has
18
18
18
dollars, what is the largest number of chicken wings he can buy?
TO1
1
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2020 MOAA Theme Optimization TO 1
The number
2020
2020
2020
has three different prime factors. What is their sum?