MathDB
Problems
Contests
National and Regional Contests
USA Contests
USA - College-Hosted Events
Stanford Mathematics Tournament
2023 Stanford Mathematics Tournament
2023 Stanford Mathematics Tournament
Part of
Stanford Mathematics Tournament
Subcontests
(20)
R9
1
Hide problems
2023 SMT Guts Round 9 p25-27 - Stanford Math Tournament
p25. You are given that
1000
!
1000!
1000
!
has
2568
2568
2568
decimal digits. Call a permutation
π
\pi
π
of length
1000
1000
1000
good if
π
(
2
i
)
>
π
(
2
i
−
1
)
\pi(2i) > \pi (2i - 1)
π
(
2
i
)
>
π
(
2
i
−
1
)
for all
1
≤
i
≤
500
1 \le i \le 500
1
≤
i
≤
500
and
π
(
2
i
)
>
π
(
2
i
+
1
)
\pi (2i) > \pi (2i + 1)
π
(
2
i
)
>
π
(
2
i
+
1
)
for all
1
≤
i
≤
499
1 \le i \le 499
1
≤
i
≤
499
. Let
N
N
N
be the number of good permutations. Estimate
D
D
D
, the number of decimal digits in
N
N
N
. You will get
max
(
0
,
25
−
⌈
∣
D
−
X
∣
10
⌉
)
\max \left( 0, 25 - \left\lceil \frac{|D-X|}{10} \right\rceil \right)
max
(
0
,
25
−
⌈
10
∣
D
−
X
∣
⌉
)
points, where
X
X
X
is the true answer. p26. A year is said to be interesting if it is the product of
3
3
3
, not necessarily distinct, primes (for example
2
2
⋅
5
2^2 \cdot 5
2
2
⋅
5
is interesting, but
2
2
⋅
3
⋅
5
2^2 \cdot 3 \cdot 5
2
2
⋅
3
⋅
5
is not). How many interesting years are there between
5000
5000
5000
and
10000
10000
10000
, inclusive? For an estimate of
E
E
E
, you will get
max
(
0
,
25
−
⌈
∣
E
−
X
∣
10
⌉
)
\max \left( 0, 25 - \left\lceil \frac{|E-X|}{10} \right\rceil \right)
max
(
0
,
25
−
⌈
10
∣
E
−
X
∣
⌉
)
points, where
X
X
X
is the true answer. p27. Sam chooses
1000
1000
1000
random lattice points
(
x
,
y
)
(x, y)
(
x
,
y
)
with
1
≤
x
,
y
≤
1000
1 \le x, y \le 1000
1
≤
x
,
y
≤
1000
such that all pairs
(
x
,
y
)
(x, y)
(
x
,
y
)
are distinct. Let
N
N
N
be the expected size of the maximum collinear set among them. Estimate
⌊
100
N
⌋
\lfloor 100N \rfloor
⌊
100
N
⌋
. Let
S
S
S
be the answer you provide and
X
X
X
be the true value of
⌊
100
N
⌋
\lfloor 100N \rfloor
⌊
100
N
⌋
. You will get
max
(
0
,
25
−
⌈
∣
S
−
X
∣
10
⌉
)
\max \left( 0, 25 - \left\lceil \frac{|S-X|}{10} \right\rceil \right)
max
(
0
,
25
−
⌈
10
∣
S
−
X
∣
⌉
)
points for your estimate. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
R8
1
Hide problems
2023 SMT Guts Round 8 p22-24 - Stanford Math Tournament
p22. Consider the series
{
A
n
}
n
=
0
∞
\{A_n\}^{\infty}_{n=0}
{
A
n
}
n
=
0
∞
, where
A
0
=
1
A_0 = 1
A
0
=
1
and for every
n
>
0
n > 0
n
>
0
,
A
n
=
A
[
n
2023
]
+
A
[
n
202
3
2
]
+
A
[
n
202
3
3
]
,
A_n = A_{\left[ \frac{n}{2023}\right]} + A_{\left[ \frac{n}{2023^2}\right]}+A_{\left[ \frac{n}{2023^3}\right]},
A
n
=
A
[
2023
n
]
+
A
[
202
3
2
n
]
+
A
[
202
3
3
n
]
,
where
[
x
]
[x]
[
x
]
denotes the largest integer value smaller than or equal to
x
x
x
. Find the
(
202
3
3
2
+
20
)
(2023^{3^2}+20)
(
202
3
3
2
+
20
)
-th element of the series. p23. The side lengths of triangle
△
A
B
C
\vartriangle ABC
△
A
BC
are
5
5
5
,
7
7
7
and
8
8
8
. Construct equilateral triangles
△
A
1
B
C
\vartriangle A_1BC
△
A
1
BC
,
△
B
1
C
A
\vartriangle B_1CA
△
B
1
C
A
, and
△
C
1
A
B
\vartriangle C_1AB
△
C
1
A
B
such that
A
1
A_1
A
1
,
B
1
B_1
B
1
,
C
1
C_1
C
1
lie outside of
△
A
B
C
\vartriangle ABC
△
A
BC
. Let
A
2
A_2
A
2
,
B
2
B_2
B
2
, and
C
2
C_2
C
2
be the centers of
△
A
1
B
C
\vartriangle A_1BC
△
A
1
BC
,
△
B
1
C
A
\vartriangle B_1CA
△
B
1
C
A
, and
△
C
1
A
B
\vartriangle C_1AB
△
C
1
A
B
, respectively. What is the area of
△
A
2
B
2
C
2
\vartriangle A_2B_2C_2
△
A
2
B
2
C
2
? p24. There are
20
20
20
people participating in a random tag game around an
20
20
20
-gon. Whenever two people end up at the same vertex, if one of them is a tagger then the other also becomes a tagger. A round consists of everyone moving to a random vertex on the
20
20
20
-gon (no matter where they were at the beginning). If there are currently
10
10
10
taggers, let
E
E
E
be the expected number of untagged people at the end of the next round. If
E
E
E
can be written as
a
b
\frac{a}{b}
b
a
for
a
,
b
a, b
a
,
b
relatively prime positive integers, compute
a
+
b
a + b
a
+
b
. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
R7
1
Hide problems
2023 SMT Guts Round 7 p19-21 - Stanford Math Tournament
p19.
A
1
A
2
.
.
.
A
12
A_1A_2...A_{12}
A
1
A
2
...
A
12
is a regular dodecagon with side length
1
1
1
and center at point
O
O
O
. What is the area of the region covered by circles
(
A
1
A
2
O
)
(A_1A_2O)
(
A
1
A
2
O
)
,
(
A
3
A
4
O
)
(A_3A_4O)
(
A
3
A
4
O
)
,
(
A
5
A
6
O
)
(A_5A_6O)
(
A
5
A
6
O
)
,
(
A
7
A
8
O
)
(A_7A_8O)
(
A
7
A
8
O
)
,
(
A
9
A
10
O
)
(A_9A_{10}O)
(
A
9
A
10
O
)
, and
(
A
11
A
12
O
)
(A_{11}A_{12}O)
(
A
11
A
12
O
)
?
(
A
B
C
)
(ABC)
(
A
BC
)
denotes the circle passing through points
A
,
B
A,B
A
,
B
, and
C
C
C
. p20. Let
N
=
2000...0
x
0...00023
N = 2000... 0x0 ... 00023
N
=
2000...0
x
0...00023
be a
2023
2023
2023
-digit number where the
x
x
x
is the
23
23
23
rd digit from the right. If
N
N
N
is divisible by
13
13
13
, compute
x
x
x
. p21. Alice and Bob each visit the dining hall to get a grilled cheese at a uniformly random time between
12
12
12
PM and
1
1
1
PM (their arrival times are independent) and, after arrival, will wait there for a uniformly random amount of time between
0
0
0
and
30
30
30
minutes. What is the probability that they will meet? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
R6
1
Hide problems
2023 SMT Guts Round 6 p16-18 - Stanford Math Tournament
p16. When not writing power rounds, Eric likes to climb trees. The strength in his arms as a function of time is
s
(
t
)
=
t
3
−
3
t
2
s(t) = t^3 - 3t^2
s
(
t
)
=
t
3
−
3
t
2
. His climbing velocity as a function of the strength in his arms is
v
(
s
)
=
s
5
+
9
s
4
+
19
s
3
−
9
s
2
−
20
s
v(s) = s^5 + 9s^4 + 19s^3 - 9s^2 - 20s
v
(
s
)
=
s
5
+
9
s
4
+
19
s
3
−
9
s
2
−
20
s
. At how many (possibly negative) points in time is Eric stationary? p17. Consider a triangle
△
A
B
C
\vartriangle ABC
△
A
BC
with angles
∠
A
C
B
=
6
0
o
\angle ACB = 60^o
∠
A
CB
=
6
0
o
,
∠
A
B
C
=
4
5
o
\angle ABC = 45^o
∠
A
BC
=
4
5
o
. The circumcircle around
△
A
B
H
\vartriangle ABH
△
A
B
H
, where
H
H
H
is the orthocenter of
△
A
B
C
\vartriangle ABC
△
A
BC
, intersects
B
C
BC
BC
for a second time in point
P
P
P
, and the center of that circumcircle is
O
c
O_c
O
c
. The line
P
H
PH
P
H
intersects
A
C
AC
A
C
in point
Q
Q
Q
, and
N
N
N
is center of the circumcircle around
△
A
Q
P
\vartriangle AQP
△
A
QP
. Find
∠
N
O
c
P
\angle NO_cP
∠
N
O
c
P
. p18. If
x
,
y
x, y
x
,
y
are positive real numbers and
x
y
3
=
16
9
xy^3 = \frac{16}{9}
x
y
3
=
9
16
, what is the minimum possible value of
3
x
+
y
3x + y
3
x
+
y
? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
R5
1
Hide problems
2023 SMT Guts Round 5 p13-15 - Stanford Math Tournament
p13. Let
△
A
B
C
\vartriangle ABC
△
A
BC
be an equilateral triangle with side length
1
1
1
. Let the unit circles centered at
A
A
A
,
B
B
B
, and
C
C
C
be
Ω
A
\Omega_A
Ω
A
,
Ω
B
\Omega_B
Ω
B
, and
Ω
C
\Omega_C
Ω
C
, respectively. Then, let
Ω
A
\Omega_A
Ω
A
and
Ω
C
\Omega_C
Ω
C
intersect again at point
D
D
D
, and
Ω
B
\Omega_B
Ω
B
and
Ω
C
\Omega_C
Ω
C
intersect again at point
E
E
E
. Line
B
D
BD
B
D
intersects
Ω
B
\Omega_B
Ω
B
at point
F
F
F
where
F
F
F
lies between
B
B
B
and
D
D
D
, and line
A
E
AE
A
E
intersects
Ω
A
\Omega_A
Ω
A
at
G
G
G
where
G
G
G
lies between
A
A
A
and
E
E
E
.
B
D
BD
B
D
and
A
E
AE
A
E
intersect at
H
H
H
. Finally, let
C
H
CH
C
H
and
F
G
FG
FG
intersect at
I
I
I
. Compute
I
H
IH
I
H
. p14. Suppose Bob randomly fills in a
45
×
45
45 \times 45
45
×
45
grid with the numbers from
1
1
1
to
2025
2025
2025
, using each number exactly once. For each of the
45
45
45
rows, he writes down the largest number in the row. Of these
45
45
45
numbers, he writes down the second largest number. The probability that this final number is equal to
2023
2023
2023
can be expressed as
p
q
\frac{p}{q}
q
p
where
p
p
p
and
q
q
q
are relatively prime positive integers. Compute the value of
p
p
p
. p15.
f
f
f
is a bijective function from the set
{
0
,
1
,
2
,
.
.
.
,
11
}
\{0, 1, 2, ..., 11\}
{
0
,
1
,
2
,
...
,
11
}
to
{
0
,
1
,
2
,
.
.
.
,
11
}
\{0, 1, 2, ... , 11\}
{
0
,
1
,
2
,
...
,
11
}
, with the property that whenever
a
a
a
divides
b
b
b
,
f
(
a
)
f(a)
f
(
a
)
divides
f
(
b
)
f(b)
f
(
b
)
. How many such
f
f
f
are there? A bijective function maps each element in its domain to a distinct element in its range. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
R4
1
Hide problems
2023 SMT Guts Round 4 p10-12 - Stanford Math Tournament
p10. Three rectangles of dimension
X
×
2
X \times 2
X
×
2
and four rectangles of dimension
Y
×
1
Y \times 1
Y
×
1
are the pieces that form a rectangle of area
3
X
Y
3XY
3
X
Y
where
X
X
X
and
Y
Y
Y
are positive, integer values. What is the sum of all possible values of
X
X
X
? p11. Suppose we have a polynomial
p
(
x
)
=
x
2
+
a
x
+
b
p(x) = x^2 + ax + b
p
(
x
)
=
x
2
+
a
x
+
b
with real coefficients
a
+
b
=
1000
a + b = 1000
a
+
b
=
1000
and
b
>
0
b > 0
b
>
0
. Find the smallest possible value of
b
b
b
such that
p
(
x
)
p(x)
p
(
x
)
has two integer roots. p12. Ten square slips of paper of the same size, numbered
0
,
1
,
2
,
.
.
.
,
9
0, 1, 2, ..., 9
0
,
1
,
2
,
...
,
9
, are placed into a bag. Four of these squares are then randomly chosen and placed into a two-by-two grid of squares. What is the probability that the numbers in every pair of blocks sharing a side have an absolute difference no greater than two? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
R3
1
Hide problems
2023 SMT Guts Round 3 p7-9 - Stanford Math Tournament
p7. An ant starts at the point
(
0
,
0
)
(0, 0)
(
0
,
0
)
. It travels along the integer lattice, at each lattice point choosing the positive
x
x
x
or
y
y
y
direction with equal probability. If the ant reaches
(
20
,
23
)
(20, 23)
(
20
,
23
)
, what is the probability it did not pass through
(
20
,
20
)
(20, 20)
(
20
,
20
)
? p8. Let
a
0
=
2023
a_0 = 2023
a
0
=
2023
and
a
n
a_n
a
n
be the sum of all divisors of
a
n
−
1
a_{n-1}
a
n
−
1
for all
n
≥
1
n \ge 1
n
≥
1
. Compute the sum of the prime numbers that divide
a
3
a_3
a
3
. p9. Five circles of radius one are stored in a box of base length five as in the following diagram. How far above the base of the box are the upper circles touching the sides of the box? https://cdn.artofproblemsolving.com/attachments/7/c/c20b5fa21fbd8ce791358fd888ed78fcdb7646.png PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
R2
1
Hide problems
2023 SMT Guts Round 2 p4-6 - Stanford Math Tournament
p4. For how many three-digit multiples of
11
11
11
in the form
a
b
c
‾
\underline{abc}
ab
c
does the quadratic
a
x
2
+
b
x
+
c
ax^2 + bx + c
a
x
2
+
b
x
+
c
have real roots? p5. William draws a triangle
△
A
B
C
\vartriangle ABC
△
A
BC
with
A
B
=
3
AB =\sqrt3
A
B
=
3
,
B
C
=
1
BC = 1
BC
=
1
, and
A
C
=
2
AC = 2
A
C
=
2
on a piece of paper and cuts out
△
A
B
C
\vartriangle ABC
△
A
BC
. Let the angle bisector of
∠
A
B
C
\angle ABC
∠
A
BC
meet
A
C
AC
A
C
at point
D
D
D
. He folds
△
A
B
D
\vartriangle ABD
△
A
B
D
over
B
D
BD
B
D
. Denote the new location of point
A
A
A
as
A
′
A'
A
′
. After William folds
△
A
′
C
D
\vartriangle A'CD
△
A
′
C
D
over
C
D
CD
C
D
, what area of the resulting figure is covered by three layers of paper? p6. Compute
(
1
)
(
2
)
(
3
)
+
(
2
)
(
3
)
(
4
)
+
.
.
.
+
(
18
)
(
19
)
(
20
)
(1)(2)(3) + (2)(3)(4) + ... + (18)(19)(20)
(
1
)
(
2
)
(
3
)
+
(
2
)
(
3
)
(
4
)
+
...
+
(
18
)
(
19
)
(
20
)
. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
R1
1
Hide problems
2023 SMT Guts Round 1 p1-3 - Stanford Math Tournament
p1. To convert between Fahrenheit,
F
F
F
, and Celsius,
C
C
C
, the formula is
F
=
9
5
C
+
32
F = \frac95 C + 32
F
=
5
9
C
+
32
. Jennifer, having no time to be this precise, instead approximates the temperature of Fahrenheit,
F
^
\widehat F
F
, as
F
^
=
2
C
+
30
\widehat F = 2C + 30
F
=
2
C
+
30
. There is a range of temperatures
C
1
≤
C
≤
C
2
C_1 \le C \le C_2
C
1
≤
C
≤
C
2
such that for any
C
C
C
in this range,
∣
F
^
−
F
∣
≤
5
| \widehat F - F| \le 5
∣
F
−
F
∣
≤
5
. Compute the ordered pair
(
C
1
,
C
2
)
(C_1,C_2)
(
C
1
,
C
2
)
. p2. Compute integer
x
x
x
such that
x
23
=
27368747340080916343
x^{23} = 27368747340080916343
x
23
=
27368747340080916343
. p3. The number of ways to flip
n
n
n
fair coins such that there are no three heads in a row can be expressed with the recurrence relation
S
(
n
+
1
)
=
a
0
S
(
n
)
+
a
1
S
(
n
−
1
)
+
.
.
.
+
a
k
S
(
n
−
k
)
S(n + 1) = a_0 S(n) + a_1 S(n - 1) + ... + a_k S(n - k)
S
(
n
+
1
)
=
a
0
S
(
n
)
+
a
1
S
(
n
−
1
)
+
...
+
a
k
S
(
n
−
k
)
for sufficiently large
n
n
n
and
k
k
k
where
S
(
n
)
S(n)
S
(
n
)
is the number of valid sequences of length
n
n
n
. What is
∑
n
=
0
k
∣
a
n
∣
\sum^k_{n=0}|a_n|
∑
n
=
0
k
∣
a
n
∣
? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
1
Hide problems
SMT 2023 Team
In the spirit of parmenides51, I guess.p1. We call a time on a
12
12
12
hour digital clock nice if the sum of the minutes digits is equal to the hour. For example,
10
:
55
10:55
10
:
55
,
3
:
12
3:12
3
:
12
and
5
:
05
5:05
5
:
05
are nice times. How many nice times occur during the course of one day? (We do not consider times of the form
00
:
XX
00:\text{XX}
00
:
XX
.)p2. Along Stanford’s University Avenue are
2023
2023
2023
palm trees which are either red, green, or blue. Let the positive integers
R
R
R
,
G
G
G
,
B
B
B
be the number of red, green, and blue palm trees respectively. Given that
R
3
+
2
B
+
G
=
12345
,
R^3+2B+G=12345,
R
3
+
2
B
+
G
=
12345
,
compute
R
R
R
.p3.
5
5
5
integers are each selected uniformly at random from the range
1
1
1
to
5
5
5
inclusive and put into a set
S
S
S
. Each integer is selected independently of the others. What is the expected value of the minimum element of
S
S
S
?p4. Cornelius chooses three complex numbers
a
,
b
,
c
a,b,c
a
,
b
,
c
uniformly at random from the complex unit circle. Given that real parts of
a
⋅
c
‾
a\cdot\overline{c}
a
⋅
c
and
b
⋅
c
‾
b\cdot\overline{c}
b
⋅
c
are
1
10
\tfrac{1}{10}
10
1
, compute the expected value of the real part of
a
⋅
b
‾
a\cdot\overline{b}
a
⋅
b
.p5. A computer virus starts off infecting a single device. Every second an infected computer has a
7
/
30
7/30
7/30
chance to stay infected and not do anything else, a
7
/
15
7/15
7/15
chance to infect a new computer, and a
1
/
6
1/6
1/6
chance to infect two new computers. Otherwise (a
2
/
15
2/15
2/15
chance), the virus gets exterminated, but other copies of it on other computers are unaffected. Compute the probability that a single infected computer produces an infinite chain of infections.p6. In the language of Blah, there is a unique word for every integer between
0
0
0
and
98
98
98
inclusive. A team of students has an unordered list of these
99
99
99
words, but do not know what integer each word corresponds to. However, the team is given access to a machine that, given two, not necessarily distinct, words in Blah, outputs the word in Blah corresponding to the sum modulo
99
99
99
of their corresponding integers. What is the minimum
N
N
N
such that the team can narrow down the possible translations of "
1
1
1
" to a list of
N
N
N
Blah words, using the machine as many times as they want?p7. Compute
6
∑
t
=
1
∞
(
1
+
∑
k
=
1
∞
(
∑
j
=
1
∞
(
1
+
k
)
−
j
)
2
)
−
t
.
\sqrt{6\sum_{t=1}^\infty\left(1+\sum_{k=1}^\infty\left(\sum_{j=1}^\infty(1+k)^{-j}\right)^2\right)^{-t}}.
6
t
=
1
∑
∞
1
+
k
=
1
∑
∞
(
j
=
1
∑
∞
(
1
+
k
)
−
j
)
2
−
t
.
p8. What is the area that is swept out by a regular hexagon of side length
1
1
1
as it rotates
3
0
∘
30^\circ
3
0
∘
about its center?p9. Let
A
A
A
be the the area enclosed by the relation
x
2
+
y
2
≤
2023.
x^2+y^2\le2023.
x
2
+
y
2
≤
2023.
Let
B
B
B
be the area enclosed by the relation
x
2
n
+
y
2
n
≤
(
A
⋅
7
16
π
)
n
/
2
.
x^{2n}+y^{2n}\le\left(A\cdot\frac{7}{16\pi}\right)^{n/2}.
x
2
n
+
y
2
n
≤
(
A
⋅
16
π
7
)
n
/2
.
Compute the limit of
B
B
B
as
n
→
∞
n\rightarrow\infty
n
→
∞
for
n
∈
N
n\in\mathbb{N}
n
∈
N
.p10. Let
S
=
{
1
,
6
,
10
,
…
}
\mathcal{S}=\{1,6,10,\dots\}
S
=
{
1
,
6
,
10
,
…
}
be the set of positive integers which are the product of an even number of distinct primes, including
1
1
1
. Let
T
=
{
2
,
3
,
…
,
}
\mathcal{T}=\{2,3,\dots,\}
T
=
{
2
,
3
,
…
,
}
be the set of positive integers which are the product of an odd number of distinct primes. Compute
∑
n
∈
S
⌊
2023
n
⌋
−
∑
n
∈
T
⌊
2023
n
⌋
.
\sum_{n\in\mathcal{S}}\left\lfloor\frac{2023}{n}\right\rfloor-\sum_{n\in\mathcal{T}}\left\lfloor\frac{2023}{n}\right\rfloor.
n
∈
S
∑
⌊
n
2023
⌋
−
n
∈
T
∑
⌊
n
2023
⌋
.
p11. Define the Fibonacci sequence by
F
0
=
0
F_0=0
F
0
=
0
,
F
1
=
1
F_1=1
F
1
=
1
, and
F
i
=
F
i
−
1
+
F
i
−
2
F_i=F_{i-1}+F_{i-2}
F
i
=
F
i
−
1
+
F
i
−
2
for
i
≥
2
i\ge2
i
≥
2
. Compute
lim
n
→
∞
F
F
n
+
1
+
1
F
F
n
⋅
F
F
n
−
1
−
1
.
\lim_{n\rightarrow\infty}\frac{F_{F_{n+1}+1}}{F_{F_n}\cdot F_{F_{n-1}-1}}.
n
→
∞
lim
F
F
n
⋅
F
F
n
−
1
−
1
F
F
n
+
1
+
1
.
p12. Let
A
A
A
,
B
B
B
,
C
C
C
, and
D
D
D
be points in the plane with integer coordinates such that no three of them are collinear, and where the distances
A
B
AB
A
B
,
A
C
AC
A
C
,
A
D
AD
A
D
,
B
C
BC
BC
,
B
D
BD
B
D
, and
C
D
CD
C
D
are all integers. Compute the smallest possible length of a side of a convex quadrilateral formed by such points.p13. Suppose the real roots of
p
(
x
)
=
x
9
+
16
x
8
+
60
x
7
+
1920
x
2
+
2048
x
+
512
p(x)=x^9+16x^8+60x^7+1920x^2+2048x+512
p
(
x
)
=
x
9
+
16
x
8
+
60
x
7
+
1920
x
2
+
2048
x
+
512
are
r
1
,
r
2
,
…
,
r
k
r_1,r_2,\dots,r_k
r
1
,
r
2
,
…
,
r
k
(roots may be repeated). Compute
∑
i
=
1
k
1
2
−
r
i
.
\sum_{i=1}^k\frac{1}{2-r_i}.
i
=
1
∑
k
2
−
r
i
1
.
p14. A teacher stands at
(
0
,
10
)
(0,10)
(
0
,
10
)
and has some students, where there is exactly one student at each integer position in the following triangle: https://cdn.artofproblemsolving.com/attachments/2/2/0cddcedf318d7b53bd33bd14353ece9614ec44.png Here, the circle denotes the teacher at
(
0
,
10
)
(0,10)
(
0
,
10
)
and the triangle extends until and includes the column
(
21
,
y
)
(21,y)
(
21
,
y
)
. A teacher can see a student
(
i
,
j
)
(i,j)
(
i
,
j
)
if there is no student in the direct line of sight between the teacher and the position
(
i
,
j
)
(i,j)
(
i
,
j
)
. Compute the number of students the teacher can see (assume that each student has no width—that is, each student is a point).p15. Suppose we have a right triangle
△
A
B
C
\triangle ABC
△
A
BC
where
A
A
A
is the right angle and lengths
A
B
=
A
C
=
2
AB=AC=2
A
B
=
A
C
=
2
. Suppose we have points
D
D
D
,
E
E
E
, and
F
F
F
on
A
B
AB
A
B
,
A
C
AC
A
C
, and
B
C
BC
BC
respectively with
D
E
⊥
E
F
DE\perp EF
D
E
⊥
EF
. What is the minimum possible length of
D
F
DF
D
F
?
10
3
Hide problems
SMT 2023 Algebra #10
Suppose that
p
(
x
)
,
q
(
x
)
p(x),q(x)
p
(
x
)
,
q
(
x
)
are monic polynomials with nonnegative integer coefficients such that
1
5
x
≥
1
q
(
x
)
−
1
p
(
x
)
≥
1
3
x
2
\frac{1}{5x}\ge\frac{1}{q(x)}-\frac{1}{p(x)}\ge\frac{1}{3x^2}
5
x
1
≥
q
(
x
)
1
−
p
(
x
)
1
≥
3
x
2
1
for all integers
x
≥
2
x\ge2
x
≥
2
. Compute the minimum possible value of
p
(
1
)
⋅
q
(
1
)
p(1)\cdot q(1)
p
(
1
)
⋅
q
(
1
)
.
SMT 2023 Discrete #10
Colin has a peculiar
12
12
12
-sided dice: it is made up of two regular hexagonal pyramids. Colin wants to paint each face one of three colors so that no two adjacent faces on the same pyramid have the same color. How many ways can he do this? Two paintings are considered identical if there is a way to rotate or flip the dice to go from one to the other. Faces are adjacent if they share an edge. https://cdn.artofproblemsolving.com/attachments/b/2/074e9a4bc404d45546661a5ae269248d20ed5a.png
SMT 2023 Geometry #10
Let
△
A
B
C
\vartriangle ABC
△
A
BC
be a triangle with side lengths
A
B
=
13
AB = 13
A
B
=
13
,
B
C
=
14
BC = 14
BC
=
14
, and
C
A
=
15
CA = 15
C
A
=
15
. The angle bisector of
∠
B
A
C
\angle BAC
∠
B
A
C
, the angle bisector of
∠
A
B
C
\angle ABC
∠
A
BC
, and the angle bisector of
∠
A
C
B
\angle ACB
∠
A
CB
intersect the circumcircle of
△
A
B
C
\vartriangle ABC
△
A
BC
again at points
D
D
D
,
E
E
E
and
F
F
F
, respectively. Compute the area of hexagon
A
F
B
D
C
E
AF BDCE
A
FB
D
CE
.
9
3
Hide problems
SMT 2023 Algebra #9
Let
x
,
y
,
z
x,y,z
x
,
y
,
z
be nonzero numbers, not necessarily real, such that
(
x
−
y
)
2
+
(
y
−
z
)
2
+
(
z
−
x
)
2
=
24
y
z
(x-y)^2+(y-z)^2+(z-x)^2=24yz
(
x
−
y
)
2
+
(
y
−
z
)
2
+
(
z
−
x
)
2
=
24
yz
and
x
2
y
z
+
y
2
z
x
+
z
2
x
y
=
3.
\tfrac{x^2}{yz}+\tfrac{y^2}{zx}+\tfrac{z^2}{xy}=3.
yz
x
2
+
z
x
y
2
+
x
y
z
2
=
3.
Compute
x
2
y
z
\tfrac{x^2}{yz}
yz
x
2
.
SMT 2023 Discrete #9
Suppose
a
a
a
and
b
b
b
are positive integers with a curious property:
(
a
3
−
3
a
b
+
1
2
)
n
+
(
b
3
+
1
2
)
n
(a^3 - 3ab +\tfrac{1}{2})^n + (b^3 +\tfrac{1}{2})^n
(
a
3
−
3
ab
+
2
1
)
n
+
(
b
3
+
2
1
)
n
is an integer for at least
3
3
3
, but at most finitely many different choices of positive integers
n
n
n
. What is the least possible value of
a
+
b
a+b
a
+
b
?
SMT 2023 Geometry #9
Triangle
△
A
B
C
\vartriangle ABC
△
A
BC
is isosceles with
A
C
=
A
B
AC = AB
A
C
=
A
B
,
B
C
=
1
BC = 1
BC
=
1
, and
∠
B
A
C
=
3
6
o
\angle BAC = 36^o
∠
B
A
C
=
3
6
o
. Let
ω
\omega
ω
be a circle with center B and radius
r
ω
=
P
A
B
C
4
r_{\omega}= \frac{P_{ABC}}{4}
r
ω
=
4
P
A
BC
, where
P
A
B
C
P_{ABC}
P
A
BC
denotes the perimeter of
△
A
B
C
\vartriangle ABC
△
A
BC
. Let
ω
\omega
ω
intersect line
A
B
AB
A
B
at
P
P
P
and line
B
C
BC
BC
at
Q
Q
Q
. Let
I
B
I_B
I
B
be the center of the excircle with of
△
A
B
C
\vartriangle ABC
△
A
BC
with respect to point
B
B
B
, and let
B
I
B
BI_B
B
I
B
intersect
P
Q
P Q
PQ
at
S
S
S
. We draw a tangent line from
S
S
S
to
⊙
I
B
\odot I_B
⊙
I
B
that intersects
⊙
I
B
\odot I_B
⊙
I
B
at point
T
T
T
. Compute the length of ST.
7
3
Hide problems
SMT 2023 Algebra #7
Consider a sequence
F
0
=
2
F_0=2
F
0
=
2
,
F
1
=
3
F_1=3
F
1
=
3
that has the property
F
n
+
1
F
n
−
1
−
F
n
2
=
(
−
1
)
n
⋅
2
F_{n+1}F_{n-1}-F_n^2=(-1)^n\cdot2
F
n
+
1
F
n
−
1
−
F
n
2
=
(
−
1
)
n
⋅
2
. If each term of the sequence can be written in the form
a
⋅
r
1
n
+
b
⋅
r
2
n
a\cdot r_1^n+b\cdot r_2^n
a
⋅
r
1
n
+
b
⋅
r
2
n
, what is the positive difference between
r
1
r_1
r
1
and
r
2
r_2
r
2
?
SMT 2023 Discrete #7
Let
S
S
S
be the number of bijective functions
f
:
{
0
,
1
,
…
,
288
}
→
{
0
,
1
,
…
,
288
}
f:\{0,1,\dots,288\}\rightarrow\{0,1,\dots,288\}
f
:
{
0
,
1
,
…
,
288
}
→
{
0
,
1
,
…
,
288
}
such that
f
(
(
m
+
n
)
(
m
o
d
17
)
)
f((m+n)\pmod{17})
f
((
m
+
n
)
(
mod
17
))
is divisible by
17
17
17
if and only if
f
(
m
)
+
f
(
n
)
f(m)+f(n)
f
(
m
)
+
f
(
n
)
is divisible by
17
17
17
. Compute the largest positive integer
n
n
n
such that
2
n
2^n
2
n
divides
S
S
S
.
SMT 2023 Geometry #7
Triangle
A
B
C
ABC
A
BC
has
A
C
=
5
AC = 5
A
C
=
5
.
D
D
D
and
E
E
E
are on side
B
C
BC
BC
such that
A
D
AD
A
D
and
A
E
AE
A
E
trisect
∠
B
A
C
\angle BAC
∠
B
A
C
, with
D
D
D
closer to
B
B
B
and
D
E
=
3
2
DE =\frac32
D
E
=
2
3
,
E
C
=
5
2
EC =\frac52
EC
=
2
5
. From
B
B
B
and
E
E
E
, altitudes
B
F
BF
BF
and
E
G
EG
EG
are drawn onto side
A
C
AC
A
C
. Compute
C
F
C
G
−
A
F
A
G
\frac{CF}{CG}-\frac{AF}{AG}
CG
CF
−
A
G
A
F
.
6
3
Hide problems
SMT 2023 Geometry #6
Let ABC be a triangle and
ω
1
\omega_1
ω
1
its incircle. Let points
D
D
D
and
E
E
E
be on segments
A
B
AB
A
B
,
A
C
AC
A
C
respectively such that
D
E
DE
D
E
is parallel to
B
C
BC
BC
and tangent to
ω
1
\omega_1
ω
1
. Now let
ω
2
\omega_2
ω
2
be the incircle of
△
A
D
E
\vartriangle ADE
△
A
D
E
and let points
F
F
F
and
G
G
G
be on segments
A
D
,
AD,
A
D
,
A
E
AE
A
E
respectively such that F G is parallel to
D
E
DE
D
E
and tangent to
ω
2
\omega_2
ω
2
. Given that
ω
2
\omega_2
ω
2
is tangent to line
A
F
AF
A
F
at point X and line
A
G
AG
A
G
at point
Y
Y
Y
, the radius of
ω
1
\omega_1
ω
1
is
60
60
60
, and
4
(
A
X
)
=
5
(
F
G
)
=
4
(
A
Y
)
,
4(AX) = 5(F G) = 4(AY),
4
(
A
X
)
=
5
(
FG
)
=
4
(
A
Y
)
,
compute the radius of
ω
2
\omega_2
ω
2
.
SMT 2023 Algebra #6
What is the area of the figure in the complex plane enclosed by the origin and the set of all points
1
z
\tfrac{1}{z}
z
1
such that
(
1
−
2
i
)
z
+
(
−
2
i
−
1
)
z
‾
=
6
i
(1-2i)z+(-2i-1)\overline{z}=6i
(
1
−
2
i
)
z
+
(
−
2
i
−
1
)
z
=
6
i
?
SMT 2023 Discrete #6
We say that an integer
x
∈
{
1
,
…
,
102
}
x\in\{1,\dots,102\}
x
∈
{
1
,
…
,
102
}
is
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
s
q
u
a
r
e
−
i
s
h
<
/
s
p
a
n
>
<span class='latex-italic'>square-ish</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
s
q
u
a
re
−
i
s
h
<
/
s
p
an
>
if there exists some integer
n
n
n
such that
x
≡
n
2
+
n
(
m
o
d
103
)
x\equiv n^2+n\pmod{103}
x
≡
n
2
+
n
(
mod
103
)
. Compute the product of all
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
s
q
u
a
r
e
−
i
s
h
<
/
s
p
a
n
>
<span class='latex-italic'>square-ish</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
s
q
u
a
re
−
i
s
h
<
/
s
p
an
>
integers modulo
103
103
103
.
5
3
Hide problems
SMT 2023 Algebra #5
Suppose
α
,
β
,
γ
∈
{
−
2
,
3
}
\alpha,\beta,\gamma\in\{-2,3\}
α
,
β
,
γ
∈
{
−
2
,
3
}
are chosen such that
M
=
max
x
∈
R
min
y
∈
R
≥
0
α
x
+
β
y
+
γ
x
y
M=\max_{x\in\mathbb{R}}\min_{y\in\mathbb{R}_{\ge0}}\alpha x+\beta y+\gamma xy
M
=
x
∈
R
max
y
∈
R
≥
0
min
αx
+
β
y
+
γ
x
y
is finite and positive (note:
R
≥
0
\mathbb{R}_{\ge0}
R
≥
0
is the set of nonnegative real numbers). What is the sum of the possible values of
M
M
M
?
SMT 2023 Discrete #5
Ryan chooses five subsets
S
1
,
S
2
,
S
3
,
S
4
,
S
5
S_1,S_2,S_3,S_4,S_5
S
1
,
S
2
,
S
3
,
S
4
,
S
5
of
{
1
,
2
,
3
,
4
,
5
,
6
,
7
}
\{1, 2, 3, 4, 5, 6, 7\}
{
1
,
2
,
3
,
4
,
5
,
6
,
7
}
such that
∣
S
1
∣
=
1
|S_1| = 1
∣
S
1
∣
=
1
,
∣
S
2
∣
=
2
|S_2| = 2
∣
S
2
∣
=
2
,
∣
S
3
∣
=
3
|S_3| = 3
∣
S
3
∣
=
3
,
∣
S
4
∣
=
4
|S_4| = 4
∣
S
4
∣
=
4
, and
∣
S
5
∣
=
5
|S_5| = 5
∣
S
5
∣
=
5
. Moreover, for all
1
≤
i
<
j
≤
5
1 \le i < j \le 5
1
≤
i
<
j
≤
5
, either
S
i
∩
S
j
=
S
i
S_i \cap S_j = S_i
S
i
∩
S
j
=
S
i
or
S
i
∩
S
j
=
∅
S_i \cap S_j = \emptyset
S
i
∩
S
j
=
∅
(in other words, the intersection of
S
i
S_i
S
i
and
S
j
S_j
S
j
is either
S
i
S_i
S
i
or the empty set). In how many ways can Ryan select the sets?
SMT 2023 Geometry #5
Equilateral triangle
△
A
B
C
\vartriangle ABC
△
A
BC
has side length
12
12
12
and equilateral triangles of side lengths
a
,
b
,
c
<
6
a, b, c < 6
a
,
b
,
c
<
6
are each cut from a vertex of
△
A
B
C
\vartriangle ABC
△
A
BC
, leaving behind an equiangular hexagon
A
1
A
2
B
1
B
2
C
1
C
2
A_1A_2B_1B_2C_1C_2
A
1
A
2
B
1
B
2
C
1
C
2
, where
A
1
A_1
A
1
lies on
A
C
AC
A
C
,
A
2
A_2
A
2
lies on
A
B
AB
A
B
, and the rest of the vertices are similarly defined. Let
A
3
A_3
A
3
be the midpoint of
A
1
A
2
A_1A_2
A
1
A
2
and define
B
3
B_3
B
3
,
C
3
C_3
C
3
similarly. Let the center of
△
A
B
C
\vartriangle ABC
△
A
BC
be
O
O
O
. Note that
O
A
3
OA_3
O
A
3
,
O
B
3
OB_3
O
B
3
,
O
C
3
OC_3
O
C
3
split the hexagon into three pentagons. If the sum of the areas of the equilateral triangles cut out is
18
3
18\sqrt3
18
3
and the ratio of the areas of the pentagons is
5
:
6
:
7
5 : 6 : 7
5
:
6
:
7
, what is the value of
a
b
c
abc
ab
c
?
4
3
Hide problems
SMT 2023 Algebra #4
If the sum of the real roots
x
x
x
to each of the equations
2
2
x
−
2
x
+
1
+
1
−
1
k
2
=
0
2^{2x}-2^{x+1}+1-\frac{1}{k^2}=0
2
2
x
−
2
x
+
1
+
1
−
k
2
1
=
0
for
k
=
2
,
3
,
…
,
2023
k=2,3,\dots,2023
k
=
2
,
3
,
…
,
2023
is
N
N
N
, what is
2
N
2^N
2
N
?
SMT 2023 Geometry #4
Equilateral triangle
△
A
B
C
\vartriangle ABC
△
A
BC
is inscribed in circle
Ω
\Omega
Ω
, which has a radius of
1
1
1
. Let the midpoint of
B
C
BC
BC
be
M
M
M
. Line
A
M
AM
A
M
intersects
Ω
\Omega
Ω
again at point
D
D
D
. Let
ω
\omega
ω
be the circle with diameter
M
D
MD
M
D
. Compute the radius of the circle that is tangent to BC on the same side of
B
C
BC
BC
as
ω
\omega
ω
, internally tangent to
Ω
\Omega
Ω
, and externally tangent to
ω
\omega
ω
.
SMT 2023 Discrete #4
Michelle is drawing segments in the plane. She begins from the origin facing up the
y
y
y
-axis and draws a segment of length
1
1
1
. Now, she rotates her direction by
12
0
∘
120^\circ
12
0
∘
, with equal probability clockwise or counterclockwise, and draws another segment of length
1
1
1
beginning from the end of the previous segment. She then continues this until she hits an already drawn segment. What is the expected number of segments she has drawn when this happens?
3
6
Show problems
2
6
Show problems
1
6
Show problems
8
3
Hide problems
SMT 2023 Algebra #8
If
x
x
x
and
y
y
y
are real numbers, compute the minimum possible value of
4
x
y
(
3
x
2
+
10
x
y
+
6
y
2
)
x
4
+
4
y
4
.
\frac{4xy(3x^2+10xy+6y^2)}{x^4+4y^4}.
x
4
+
4
y
4
4
x
y
(
3
x
2
+
10
x
y
+
6
y
2
)
.
SMT 2023 Discrete #8
Define the Fibonacci numbers via
F
0
=
0
F_0=0
F
0
=
0
,
f
1
=
1
f_1=1
f
1
=
1
, and
F
n
−
1
+
F
n
−
2
F_{n-1}+F_{n-2}
F
n
−
1
+
F
n
−
2
.Olivia flips two fair coins at the same time, repeatedly, until she has flipped a tails on both, not necessarily on the same throw. She records the number of pairs of flips
c
c
c
until this happens (not including the last pair, so if on the last flip both coins turned up tails
c
c
c
would be
0
0
0
). What is the expected value
F
c
F_c
F
c
?
SMT 2023 Geometry #8
In acute triangle
△
A
B
C
\triangle ABC
△
A
BC
, point
R
R
R
lies on the perpendicular bisector of
A
C
AC
A
C
such that
C
A
‾
\overline{CA}
C
A
bisects
∠
B
A
R
\angle BAR
∠
B
A
R
. Let
Q
Q
Q
be the intersection of lines
A
C
AC
A
C
and
B
R
BR
BR
. The circumcircle of
△
A
R
C
\triangle ARC
△
A
RC
intersects segment
A
B
‾
\overline{AB}
A
B
at
P
≠
A
P\neq A
P
=
A
, with
A
P
=
1
AP=1
A
P
=
1
,
P
B
=
5
PB=5
PB
=
5
, and
A
Q
=
2
AQ=2
A
Q
=
2
. Compute
A
R
AR
A
R
.