MathDB
Problems
Contests
National and Regional Contests
USA Contests
USA - College-Hosted Events
UMD Math Competition
2023 UMD Math Competition Part I
2023 UMD Math Competition Part I
Part of
UMD Math Competition
Subcontests
(25)
#8
1
Hide problems
Five positive divisors
How many positive integers less than
1
1
1
million have exactly
5
5
5
positive divisors?
a
.
1
b
.
5
c
.
11
d
.
23
e
.
24
\mathrm a. ~ 1\qquad \mathrm b.~5\qquad \mathrm c. ~11 \qquad \mathrm d. ~23 \qquad \mathrm e. ~24
a
.
1
b
.
5
c
.
11
d
.
23
e
.
24
#6
1
Hide problems
Let and what
Let
A
=
log
(
1
)
+
log
2
+
log
(
3
)
+
⋯
+
log
(
2023
)
A = \log (1) + \log 2 + \log(3) + \cdots + \log(2023)
A
=
lo
g
(
1
)
+
lo
g
2
+
lo
g
(
3
)
+
⋯
+
lo
g
(
2023
)
and
B
=
log
(
1
/
1
)
+
log
(
1
/
2
)
+
log
(
1
/
3
)
+
⋯
+
log
(
1
/
2023
)
.
B = \log(1/1) + \log(1/2) + \log(1/3) + \cdots + \log(1/2023).
B
=
lo
g
(
1/1
)
+
lo
g
(
1/2
)
+
lo
g
(
1/3
)
+
⋯
+
lo
g
(
1/2023
)
.
What is the value of
A
+
B
?
A + B\ ?
A
+
B
?
(
(
(
logs are logs base
10
)
10)
10
)
a
.
0
b
.
1
c
.
−
log
(
2023
!
)
d
.
log
(
2023
!
)
e
.
−
2023
\mathrm a. ~ 0\qquad \mathrm b.~1\qquad \mathrm c. ~{-\log(2023!)} \qquad \mathrm d. ~\log(2023!) \qquad \mathrm e. ~{-2023}
a
.
0
b
.
1
c
.
−
lo
g
(
2023
!)
d
.
lo
g
(
2023
!)
e
.
−
2023
#1
1
Hide problems
An ant walks
An ant walks a distance
A
=
1
0
9
A = 10^9
A
=
1
0
9
millimeters. A bear walks
B
=
1
0
6
B = 10^6
B
=
1
0
6
feet. A chicken walks
C
=
1
0
8
C = 10^8
C
=
1
0
8
inches. What is the correct ordering of
A
,
B
,
C
?
A, B, C?
A
,
B
,
C
?
(Note there are
25.4
25.4
25.4
millimeters in an inch, and there are
12
12
12
inches in a foot.)
a
.
A
<
B
<
C
b
.
A
<
C
<
B
c
.
C
<
B
<
A
d
.
B
<
A
<
C
e
.
B
<
C
<
A
\mathrm a. ~ A<B<C\qquad \mathrm b.~A<C<B\qquad \mathrm c. ~C<B<A \qquad \mathrm d. ~B<A<C \qquad \mathrm e. ~B<C<A
a
.
A
<
B
<
C
b
.
A
<
C
<
B
c
.
C
<
B
<
A
d
.
B
<
A
<
C
e
.
B
<
C
<
A
#9
1
Hide problems
The Amazing Prime
The Amazing Prime company ships its products in boxes whose length, width, and height (in inches) are prime numbers. If the volume of one of their boxes is
105
105
105
cubic inches, what is its surface area (that is, the sum of the areas of the 6 sides of the box) in square inches?
a
.
21
b
.
71
c
.
77
d
.
05
e
.
142
\mathrm a. ~ 21\qquad \mathrm b.~71\qquad \mathrm c. ~77 \qquad \mathrm d. ~05 \qquad \mathrm e. ~142
a
.
21
b
.
71
c
.
77
d
.
05
e
.
142
#7
1
Hide problems
Set with four
Suppose
S
=
{
1
,
2
,
3
,
x
}
S = \{1, 2, 3, x\}
S
=
{
1
,
2
,
3
,
x
}
is a set with four distinct real numbers for which the difference between the largest and smallest values of
S
S
S
is equal to the sum of elements of
S
.
S.
S
.
What is the value of
x
?
x?
x
?
a
.
−
1
b
.
−
3
/
2
c
.
−
2
d
.
−
2
/
3
e
.
−
3
\mathrm a. ~ {-1}\qquad \mathrm b.~{-3/2}\qquad \mathrm c. ~{-2} \qquad \mathrm d. ~{-2/3} \qquad \mathrm e. ~{-3}
a
.
−
1
b
.
−
3/2
c
.
−
2
d
.
−
2/3
e
.
−
3
#4
1
Hide problems
Euler is selling
Euler is selling Mathematician cards to Gauss. Three Fermat cards plus
5
5
5
Newton cards costs
95
95
95
Euros, while
5
5
5
Fermat cards plus
2
2
2
Newton cards also costs
95
95
95
Euros. How many Euroes does one Fermat card cost?
a
.
10
b
.
15
c
.
20
d
.
30
e
.
35
\mathrm a. ~ 10\qquad \mathrm b.~15\qquad \mathrm c. ~20 \qquad \mathrm d. ~30 \qquad \mathrm e. ~35
a
.
10
b
.
15
c
.
20
d
.
30
e
.
35
#2
1
Hide problems
Peter Rabbit
Peter Rabbit is hopping along the number line, always jumping in the positive
x
x
x
direction. For his first jump, he starts at
0
0
0
and jumps
1
1
1
unit to get to the number
1.
1.
1.
For his second jump, he jumps
4
4
4
units to get to the number
5.
5.
5.
He continues jumping by jumping
1
1
1
unit whenever he is on a multiple of
3
3
3
and by jumping
4
4
4
units whenever he is on a number that is not a multiple of
3.
3.
3.
What number does he land on at the end of his
100
100
100
th jump?
a
.
297
b
.
298
c
.
299
d
.
300
e
.
301
\mathrm a. ~ 297\qquad \mathrm b.~298\qquad \mathrm c. ~299 \qquad \mathrm d. ~300 \qquad \mathrm e. ~301
a
.
297
b
.
298
c
.
299
d
.
300
e
.
301
#5
1
Hide problems
Shoot an arrow
You shoot an arrow in the air. It falls to earth, you know not where. But you do know that the arrow’s height in feet after
t
{t}
t
seconds is
−
16
t
2
+
80
t
+
96.
-16t^2 + 80t + 96.
−
16
t
2
+
80
t
+
96.
After how many seconds does the arrow hit the ground? (the ground has height 0)
a
.
2
b
.
3
c
.
4
d
.
5
e
.
6
\mathrm a. ~ 2\qquad \mathrm b.~3\qquad \mathrm c. ~4 \qquad \mathrm d. ~5 \qquad \mathrm e. ~6
a
.
2
b
.
3
c
.
4
d
.
5
e
.
6
#12
1
Hide problems
Real numbers
Suppose for real numbers
a
,
b
,
c
a, b, c
a
,
b
,
c
we know
a
+
1
b
=
3
,
a + \dfrac 1b = 3,
a
+
b
1
=
3
,
and
b
+
3
c
=
1
3
.
b + \dfrac 3c = \dfrac 13.
b
+
c
3
=
3
1
.
What is the value of
c
+
27
a
?
c + \dfrac{27}a?
c
+
a
27
?
a
.
1
b
.
3
c
.
8
d
.
9
e
.
21
\mathrm a. ~ 1\qquad \mathrm b.~3\qquad \mathrm c. ~8 \qquad \mathrm d. ~9 \qquad \mathrm e. ~21
a
.
1
b
.
3
c
.
8
d
.
9
e
.
21
#3
1
Hide problems
Adam is walking
Adam is walking in the city. In order to get around a large building, he walks
12
12
12
miles east and then
5
5
5
miles north, then stop. His friend Neutrino, who can go through buildings, starts in the same place as Adam but walks in a straight line to where Adam stops. How much farther than Neutrino does Adam walk?
a
.
1
m
i
l
e
b
.
2
m
i
l
e
s
c
.
3
m
i
l
e
s
d
.
4
m
i
l
e
s
e
.
5
m
i
l
e
s
\mathrm a. ~ 1~\mathrm{mile}\qquad \mathrm b.~2 ~\mathrm{miles}\qquad \mathrm c. ~3~\mathrm{miles} \qquad \mathrm d. ~4~\mathrm{miles} \qquad \mathrm e. ~5~\mathrm{miles}
a
.
1
mile
b
.
2
miles
c
.
3
miles
d
.
4
miles
e
.
5
miles
#10
1
Hide problems
Optimists
There are
100
100
100
people in a room. Some are wise and some are optimists. \bullet~ A wise person can look at someone and know if they are wise or if they are an optimist. \bullet~ An optimist thinks everyone is wise (including themselves).Everyone in the room writes down what they think is the number of wise people in the room. What is the smallest possible value for the average?
a
.
10
b
.
25
c
.
50
d
.
75
e
.
100
\mathrm a. ~ 10\qquad \mathrm b.~25\qquad \mathrm c. ~50 \qquad \mathrm d. ~75 \qquad \mathrm e. ~100
a
.
10
b
.
25
c
.
50
d
.
75
e
.
100
#11
1
Hide problems
Circle inscribed
Let
S
1
S_1
S
1
be a square with side
s
s
s
and
C
1
C_1
C
1
be the circle inscribed in it. Let
C
2
C_2
C
2
be a circle with radius
r
r
r
and
S
2
S_2
S
2
be a square inscribed in it. We are told that the area of
S
1
−
C
1
S_1 - C_1
S
1
−
C
1
is the same as the area of
C
2
−
S
2
.
C_2 - S_2.
C
2
−
S
2
.
Which of the following numbers is closest to
s
/
r
?
s/r?
s
/
r
?
a
.
1
b
.
2
c
.
3
d
.
4
e
.
5
\mathrm a. ~ 1\qquad \mathrm b.~2\qquad \mathrm c. ~3 \qquad \mathrm d. ~4 \qquad \mathrm e. ~5
a
.
1
b
.
2
c
.
3
d
.
4
e
.
5
#13
1
Hide problems
The orthocenter
The orthocenter of triangle
A
B
C
ABC
A
BC
lies on its circumcircle. One of the angles of
A
B
C
ABC
A
BC
must equal: (The orthocenter of a triangle is the point where all three altitudes intersect.)
a
.
3
0
∘
b
.
6
0
∘
c
.
9
0
∘
d
.
12
0
∘
e
.
It cannot be deduced from the given information.
\mathrm a. ~ 30^\circ\qquad \mathrm b.~60^\circ\qquad \mathrm c. ~90^\circ \qquad \mathrm d. ~120^\circ \qquad \mathrm e. ~\text{It cannot be deduced from the given information.}
a
.
3
0
∘
b
.
6
0
∘
c
.
9
0
∘
d
.
12
0
∘
e
.
It cannot be deduced from the given information.
#14
1
Hide problems
The quadratic equation
Let
m
≠
−
1
m \neq -1
m
=
−
1
be a real number. Consider the quadratic equation
(
m
+
1
)
x
2
+
4
m
x
+
m
−
3
=
0.
(m + 1)x^2 + 4mx + m - 3 =0.
(
m
+
1
)
x
2
+
4
m
x
+
m
−
3
=
0.
Which of the following must be true? \rm(I) Both roots of this equation must be real. \rm(II) If both roots are real, then one of the roots must be less than
−
1.
-1.
−
1.
\rm(III) If both roots are real, then one of the roots must be larger than
1.
1.
1.
a
.
Only
(
I
)
b
.
(
I
)
a
n
d
(
I
I
)
c
.
O
n
l
y
(
I
I
I
)
d
.
B
o
t
h
(
I
)
a
n
d
(
I
I
I
)
e
.
(
I
)
,
(
I
I
)
,
a
n
d
(
I
I
I
)
\mathrm a. ~ \text{Only} ~(\mathrm I)\rm \qquad \mathrm b. ~(I)~and~(II)\qquad \mathrm c. ~Only~(III) \qquad \mathrm d. ~Both~(I)~and~(III) \qquad \mathrm e. ~(I), (II),~and~(III)
a
.
Only
(
I
)
b.
(
I
)
and
(
II
)
c.
Only
(
III
)
d.
Both
(
I
)
and
(
III
)
e.
(
I
)
,
(
II
)
,
and
(
III
)
#15
1
Hide problems
Two of them
What is the least positive integer
m
m
m
such that the following is true?Given
m
\it m
m
integers between
1
\it1
1
and
2023
,
\it{2023},
2023
,
inclusive, there must exist two of them
a
,
b
\it a, b
a
,
b
such that
1
<
a
b
≤
2.
1 < \frac ab \le 2.
1
<
b
a
≤
2.
a
.
10
b
.
11
c
.
12
d
.
13
e
.
1415
\mathrm a. ~ 10\qquad \mathrm b.~11\qquad \mathrm c. ~12 \qquad \mathrm d. ~13 \qquad \mathrm e. ~1415
a
.
10
b
.
11
c
.
12
d
.
13
e
.
1415
#16
1
Hide problems
Two identical digits
How many integers between
123
123
123
and
789
789
789
have at least two identical digits, when written in base
10
?
10?
10
?
a
.
180
b
.
184
c
.
186
d
.
189
e
.
191
\mathrm a. ~ 180\qquad \mathrm b.~184\qquad \mathrm c. ~186 \qquad \mathrm d. ~189 \qquad \mathrm e. ~191
a
.
180
b
.
184
c
.
186
d
.
189
e
.
191
#17
1
Hide problems
Three medians
The lengths of the sides of triangle
A
′
B
′
C
′
A'B'C'
A
′
B
′
C
′
are equal to the lengths of the three medians of triangle
A
B
C
.
ABC.
A
BC
.
Then the ratio
A
r
e
a
(
A
′
B
′
C
′
)
/
A
r
e
a
(
A
B
C
)
\mathrm{Area} (A'B'C') / \mathrm{Area} (ABC)
Area
(
A
′
B
′
C
′
)
/
Area
(
A
BC
)
equals
a
.
1
2
b
.
2
3
c
.
3
4
d
.
5
6
e
.
Cannot be determined from the information given.
\mathrm a. ~ \frac 12\qquad \mathrm b.~\frac 23\qquad \mathrm c. ~\frac34 \qquad \mathrm d. ~\frac56 \qquad \mathrm e. ~\text{Cannot be determined from the information given.}
a
.
2
1
b
.
3
2
c
.
4
3
d
.
6
5
e
.
Cannot be determined from the information given.
#19
1
Hide problems
Three positive
Three positive real numbers
a
,
b
,
c
a, b, c
a
,
b
,
c
satisfy
a
b
=
343
,
b
c
=
10
,
a
c
=
7.
a^b = 343, b^c = 10, a^c = 7.
a
b
=
343
,
b
c
=
10
,
a
c
=
7.
Find
b
b
.
b^b.
b
b
.
a
.
1000
b
.
900
c
.
1200
d
.
4000
e
.
100
\mathrm a. ~ 1000\qquad \mathrm b.~900\qquad \mathrm c. ~1200 \qquad \mathrm d. ~4000 \qquad \mathrm e. ~100
a
.
1000
b
.
900
c
.
1200
d
.
4000
e
.
100
#18
1
Hide problems
Ordered triples
How many ordered triples of integers
(
a
,
b
,
c
)
(a, b, c)
(
a
,
b
,
c
)
satisfy the following system?
{
a
b
+
c
=
17
a
+
b
c
=
19
\begin{cases} ab + c &= 17 \\ a + bc &= 19 \end{cases}
{
ab
+
c
a
+
b
c
=
17
=
19
a
.
2
b
.
3
c
.
4
d
.
5
e
.
6
\mathrm a. ~ 2\qquad \mathrm b.~3\qquad \mathrm c. ~4 \qquad \mathrm d. ~5 \qquad \mathrm e. ~6
a
.
2
b
.
3
c
.
4
d
.
5
e
.
6
#22
1
Hide problems
Sequence
A sequence
a
1
,
a
2
,
…
a_1, a_2, \ldots
a
1
,
a
2
,
…
satisfies
a
1
=
5
2
a_1 = \dfrac 52
a
1
=
2
5
and
a
n
+
1
=
a
n
2
−
2
a_{n + 1} = {a_n}^2 - 2
a
n
+
1
=
a
n
2
−
2
for all
n
≥
1.
n \ge 1.
n
≥
1.
Let
M
M
M
be the integer which is closest to
a
2023
.
a_{2023}.
a
2023
.
The last digit of
M
M
M
equals
a
.
0
b
.
2
c
.
4
d
.
6
e
.
8
\mathrm a. ~ 0\qquad \mathrm b.~2\qquad \mathrm c. ~4 \qquad \mathrm d. ~6 \qquad \mathrm e. ~8
a
.
0
b
.
2
c
.
4
d
.
6
e
.
8
#21
1
Hide problems
Least possible value
Let
a
,
b
,
c
,
d
,
e
a, b, c, d, e
a
,
b
,
c
,
d
,
e
be real numbers such that
a
<
b
<
c
<
d
<
e
.
a<b<c<d<e.
a
<
b
<
c
<
d
<
e
.
The least possible value of the function
f
:
R
→
R
f: \mathbb R \to \mathbb R
f
:
R
→
R
with
f
(
x
)
=
∣
x
−
a
∣
+
∣
x
−
b
∣
+
∣
x
−
c
∣
+
∣
x
−
d
∣
+
∣
x
−
e
∣
f(x) = |x-a| + |x - b|+ |x - c| + |x - d|+ |x - e|
f
(
x
)
=
∣
x
−
a
∣
+
∣
x
−
b
∣
+
∣
x
−
c
∣
+
∣
x
−
d
∣
+
∣
x
−
e
∣
is
a
.
e
+
d
+
c
+
b
+
a
b
.
e
+
d
+
c
−
b
−
a
c
.
e
+
d
+
∣
c
∣
−
b
−
a
d
.
e
+
d
+
b
−
a
e
.
e
+
d
−
b
−
a
\mathrm a. ~ e+d+c+b+a\qquad \mathrm b.~e+d+c-b-a\qquad \mathrm c. ~e+d+|c|-b-a \qquad \mathrm d. ~e+d+b-a \qquad \mathrm e. ~e+d-b-a
a
.
e
+
d
+
c
+
b
+
a
b
.
e
+
d
+
c
−
b
−
a
c
.
e
+
d
+
∣
c
∣
−
b
−
a
d
.
e
+
d
+
b
−
a
e
.
e
+
d
−
b
−
a
#25
1
Hide problems
Series of real numbers
Suppose that
S
S
S
is a series of real numbers between
2
2
2
and
8
8
8
inclusive, and that for any two elements
y
>
x
y > x
y
>
x
in
S
,
S,
S
,
98
y
−
102
x
−
x
y
≥
4.
98y - 102x - xy \ge 4.
98
y
−
102
x
−
x
y
≥
4.
What is the maximum possible size for the set
S
?
S?
S
?
a
.
12
b
.
14
c
.
16
d
.
18
e
.
20
\mathrm a. ~ 12\qquad \mathrm b.~14\qquad \mathrm c. ~16 \qquad \mathrm d. ~18 \qquad \mathrm e. 20
a
.
12
b
.
14
c
.
16
d
.
18
e
.20
#20
1
Hide problems
A strip is
A strip is defined as the region between two parallel lines; the width of the strip is the distance between the two lines. Two strips of width
1
1
1
intersect in a parallelogram whose area is
2.
2.
2.
What is the angle between the strips?
a
.
1
5
∘
b
.
3
0
∘
c
.
4
5
∘
d
.
6
0
∘
e
.
9
0
∘
\mathrm a. ~ 15^\circ\qquad \mathrm b.~30^\circ \qquad \mathrm c. ~45^\circ \qquad \mathrm d. ~60^\circ \qquad \mathrm e. ~90^\circ
a
.
1
5
∘
b
.
3
0
∘
c
.
4
5
∘
d
.
6
0
∘
e
.
9
0
∘
#23
1
Hide problems
Assume a triangle
Assume a triangle
A
B
C
ABC
A
BC
satisfies
∣
A
B
∣
=
1
,
∣
A
C
∣
=
2
|AB| = 1, |AC| = 2
∣
A
B
∣
=
1
,
∣
A
C
∣
=
2
and
∠
A
B
C
=
∠
A
C
B
+
9
0
∘
.
\angle ABC = \angle ACB + 90^\circ.
∠
A
BC
=
∠
A
CB
+
9
0
∘
.
What is the area of
A
B
C
?
ABC?
A
BC
?
a
.
6
/
7
b
.
5
/
7
c
.
1
/
2
d
.
4
/
5
e
.
3
/
5
\mathrm a. ~ 6/7\qquad \mathrm b.~5/7\qquad \mathrm c. ~1/2 \qquad \mathrm d. ~4/5 \qquad \mathrm e. ~3/5
a
.
6/7
b
.
5/7
c
.
1/2
d
.
4/5
e
.
3/5
#24
1
Hide problems
Bob is practicing
Bob is practicing addition in base
2.
2.
2.
Each time he adds two numbers in base
2
,
2,
2
,
he counts the number of carries. For example, when summing the numbers
1001
1001
1001
and
1011
1011
1011
in base
2
,
2,
2
,
1
1
1
0
1
0
0
1
0
1
0
1
1
1
0
1
0
0
\begin{array}{ccccc} \overset{1}{}&& \overset {1}{}&\overset {1}{} \\ 0&1&0&0&1\\0&1&0&1&1 \\ \hline 1&0&1&0&0 \end{array}
1
0
0
1
1
1
0
1
0
0
1
1
0
1
0
1
1
0
there are three carries (shown on the top row). Suppose that Bob starts with the number
0
,
0,
0
,
and adds
111
(
111~(
111
(
i.e.
7
7
7
in base
2
)
2)
2
)
to it one hundred times to obtain the number
1010111100
(
1010111100~(
1010111100
(
i.e.
700
700
700
in base
2
)
.
2).
2
)
.
How many carries occur (in total) in these one hundred calculations?
a
.
280
b
.
289
c
.
291
d
.
294
e
.
297
\mathrm a. ~ 280\qquad \mathrm b.~289\qquad \mathrm c. ~291 \qquad \mathrm d. ~294 \qquad \mathrm e. ~297
a
.
280
b
.
289
c
.
291
d
.
294
e
.
297