MathDB

1

Swapping First and Last Digits

N_{15}

be the answer to problem 15.
For a positive integer

x

expressed in base ten, let

x'

be the result of swapping its first and last digits (for example, if

x = 2024

, then

x' = 4022

). Let

C

be the number of

N_{15}

-digit positive integers

x

with a nonzero leading digit that satisfy the property that both

x

and

x'

are divisible by

11

(note:

x'

is allowed to have a leading digit of zero). Compute the sum of the digits of

C

when

C

is expressed in base ten.

15

2

Sum of Binomial Coefficient Products is Divisible by 89

Let

\ell = 1

,

M = 23

,

N = 45

, and

u = 67

. Compute the number of ordered pairs of nonnegative integers

(X, Y)

with

X \leq M - \ell

and

Y \leq N + u

such that the sum

\sum_{k=\ell}^{u} \binom{X + k}{M}\cdot\binom{Y - k}{N}

is divisible by

89

(for integers

a

and

b

, define the binomial coefficient

\tbinom{a}{b}

to be the number of

b

-element subsets of any given

a

-element set, which is

0

when

a < 0

,

b < 0

, or

b > a

).

Rectangle Inscribed in Rectangle with Tangent Condition

Let

N_{14}

be the answer to problem 14.
Rectangle

ABCD

has area

\sqrt{2N_{14}}

. Points

E

,

F

,

G

, and

H

lie on the rays

\overrightarrow{AB}

,

\overrightarrow{BC}

,

\overrightarrow{CD}

, and

\overrightarrow{DA}

, respectively, such that

EFGH

is a rectangle with area

2\sqrt{2N_{14}}

that contains all of

ABCD

in its interior. If

\tan\angle AEH = \tan\angle BFE = \tan\angle CGF = \tan\angle DHG = \sqrt{\frac{1}{48}},

then

EG=\tfrac{m\sqrt{n}}{p}

, where

m

,

n

, and

p

are positive integers,

m

and

p

are relatively prime, and

n

is not divisible by the square of any prime. Compute

m + n + p

.

14

2

Trigonometric Expression with Strange Angles

Compute

\frac{1}{2}\sin\frac{3\pi}{7}+\sin\frac{2\pi}{7}\cos\frac{3\pi}{7}.

Infinite Product of Trinomials

Let

N_{13}

be the answer to problem 13, and let

k = \tfrac{1}{N_{13} + 6}

.
Compute the infinite product

(1 - k + k^2)(1 - k^3 + k^6)(1 - k^9 + k^{18})(1 - k^{27} + k^{54})\cdots,

where the factors take the form

(1 - k^{3^a} + k^{2\cdot 3^a})

for all nonnegative integers

a

.

13

2

Prism with Kite Bases

In prism

JHOPKINS

, quadrilaterals

JHOP

and

KINS

are parallel and congruent bases that are kites, where

JH = JP = KI = KS

and

OH = OP = NI = NS

; the longer two sides of each kite have length

\tfrac{4 + \sqrt{5}}{2}

, and the shorter two sides of each kite have length

\tfrac{5 + \sqrt{5}}{4}

. Assume that

\overline{JK}

,

\overline{HI}

,

\overline{ON}

, and

\overline{PS}

are congruent edges of

JHOPKINS

perpendicular to the planes containing

JHOP

and

KINS

. Vertex

J

is part of a regular pentagon

JAZZ'Y

that can be inscribed in prism

JHOPKINS

such that

A \in \overline{HI}

,

Z \in \overline{NI}

,

Z' \in \overline{NS}

,

Y \in \overline{PS}

,

AI = YS

, and

ZI = Z'S

. Compute the height of

JHOPKINS

(that is, the distance between the bases).

Remainders of Squares Modulo 31

Compute the largest nonnegative integer

T \leq 30

that is the remainder when

T^2 + 4

is divided by

31

.

12

2

Three Sequences

Let

\{ a_n \}_{n=0}^{\infty}

,

\{ b_n \}_{n=0}^{\infty}

, and

\{ c_n \}_{n=0}^{\infty}

be sequences of real numbers such that for all

k\geq 1

, \begin{align*} a_k&=\left\lfloor \sqrt{2}+\frac{k-1}{2024} \right\rfloor+a_{k-1} \\ b_k+c_k&=1 \\ a_{k-1}b_k&=a_kc_k. \end{align*} Suppose that

a_0=1

,

b_0=2

, and

c_0=3

. Given that

\sqrt2\approx1.4142

, compute

\sum_{k=1}^{2024}(a_kb_k-a_{k-1}c_k).

Concave Heptagon with Square Components

Let

N_{11}

be the answer to problem 11.
Concave heptagon

HOPKINS

, where

180^\circ<\angle HOP<270^\circ

, has area

N_{11}

, and

HP=NI\sqrt{24}

. Suppose that

HONS

and

OPKI

are congruent squares. Compute the common area of each of these squares.

11

2

Convenient Combinatorics

Call a positive integer convenient if its digits can be partitioned into two collections of contiguous digits whose element sums are

7

and

11

. For example,

3456

is convenient, but

4247

is not. Compute the number of convenient positive integers less than or equal to

10^5

.

Sum of Squares Greater than Product Diophantine

Let

N_{10}

be the answer to problem 10.
Compute the number of ordered pairs of integers

(m,n)

that satisfy the equation

m^2+n^2=mn+N_{10}.

10

2

Inscribed Sphere of Tetrahedron

One triangular face

F

of a tetrahedron

\mathcal{T}

has side lengths

\sqrt{5}

,

\sqrt{65}

, and

2\sqrt{17}

. The other three faces of

\mathcal{T}

are right triangles whose hypotenuses coincide with the sides of

F

. There exists a sphere inside

\mathcal{T}

tangent to all four of its faces. Compute the radius of this sphere.

Expectation of Gecko Jumps

Let

N_9

be the answer to problem 9.
In a rainforest, there is a row of nine rocks labeled from

1

to

N_9

. A gecko is standing on rock

1

. The gecko jumps according to following rules:
[*] if it is on rock

1

, then it will jump with equal probability to any of the other rocks. [*] if it is on rock

R

and

R

is prime, then it will jump to rock

N_9

. [*] if it is on rock

4

, then it will jump with equal probability to rock

1

or rock

6

. [*] if it is on rock

R

and

R

is composite with

4<R<N_9

, then it will jump with equal probability to rock

Q

or

S

, where

Q

is the greatest composite number less than

R

, and

S

is the smallest composite number greater than

R

. [*] if it is on rock

N_9

, then it stops jumping.
The expected number of jumps the gecko will take to reach rock

N_9

is

\tfrac{p}{q}

, where

p

are

q

relatively prime positive integers. Compute

p+q

.

9

2

Last Two Digits and 21st Power

Let

N \in \{10, 11, \ldots, 99\}

be a two-digit positive integer. Compute the number of values of

N

for which the last two digits in the decimal expansion of

N^{21}

are the digits of

N

in the same order.

Area Involving Minimum and Sum of Squares

Compute the smallest positive integer

k

such that the area of the region bounded by

k\min(x,y)+x^2+y^2=0

exceeds

100

.

8

2

Concyclic Points and Angle Bisectors

Points

A

,

B

,

C

, and

D

lie on a circle

\Gamma

, in that order, with

AB=5

and

AD=3

. The angle bisector of

\angle ABC

intersects

\Gamma

at point

E

on the opposite side of

\overleftrightarrow{CD}

as

A

and

B

. Assume that

\overline{BE}

is a diameter of

\Gamma

and

AC=AE

. Compute

DE

.

Coloring Sides of a Regular Polygon

Let

N_7

be the answer to problem 7.
Each side of a regular

N_7

-gon is colored with a single color from a set of two given colors. Two colorings that can be obtained from one another by a rotation or a reflection of the entire figure are considered the same. Compute the number of possible different colorings.

7

2

Sines and Floors

Compute the sum of all real solutions

\alpha

(in radians) to the equation

|\sin\alpha|=\left\lfloor \frac{\alpha}{20} \right\rfloor.

Units Digit of Tetration

Let

N_6

be the answer to problem 6.
Given positive integers

n

and

a

, the

n

th tetration of

a

is defined as

^{n}a=\underbrace{a^{a^{\cdot^{\cdot^{\cdot^a}}}}}_{n \text{ times}}.

For example,

^{4}2=2^{2^{2^2}}=2^{2^4}=2^{16}=65536

. Compute the units digit of

^{2024}N_6

.

6

2

Median is an Element

Compute the number of nonempty subsets

S

of

\{ 1,2,3,4,5,6,7,8,9,10 \}

such that the median of

S

is an element of

S

.

Altitudes, Midpoints, and Similar Triangles

Let

N_5

be the answer to problem 5.
Triangle

JHU

satisfies

JH=N_5

and

JU=6

. Point

X

lies on

\overline{HU}

such that

\overline{JX}

is an altitude of

\triangle{JHU}

, point

Y

is the midpoint of

\overline{JU}

, and

\overline{JX}

and

\overline{HY}

intersect at

Z

. Assume that

\triangle{HZX}

is similar to

\triangle{JZY}

(in this vertex order). Compute the area of

\triangle{JHU}

.

5

2

Cevian and Equal Perimeters

Triangle

JHU

has side lengths

JH=13

,

HU=14

, and

JU=15

. Point

X

lies on

\overline{HU}

such that

\triangle{JHX}

and

\triangle{JUX}

have equal perimeters. Compute

JX^2

.

Difference Between Quadratic Roots

Compute the positive difference between the two solutions to the equation

2x^2-28x+9=0

.

4

2

Cube Root Condition

Let

x

be a real number satisfying

\sqrt[3]{125-x^3}-\sqrt[3]{27-x^3}=7.

Compute

|\sqrt[3]{125-x^3}+\sqrt[3]{27-x^3}|

.

Exponential Equation

Let

N_3

be the answer to problem 3.
Compute the sum of all real solutions

x

to the equation

50^x+72^x+(N_3)^x=800^x.

3

2

Painted Cube Probability

Amelia has

27

unit cubes. She selects one and paints one of its faces. She then randomly glues all

27

cubes together to form a

3 \times 3 \times 3

cube (with all possible arrangements of the unit cubes being equally likely). Compute the probability that the resulting cube appears unpainted.

Circling Positive Integers

Let

N_2

be the answer to problem 2.
On a number line, Tanya circles the first

\ell

positive integers. Then, starting with the greatest number in the most recent circle, she circles the next

\ell

positive integers, so that the two circles have exactly one number in common; she repeats this until

N_2

is in a circle. Compute the sum of all possible values of

\ell

for which

N_2

is the greatest number in a circle.

2

Quadratic with Unique Zero

Let

Q

be a quadratic polynomial with a unique zero. Suppose

Q(12)=Q(16)

and

Q(20)=24

. Compute

Q(28)

.

Triangles Formed by Intersections

Let

N_1

be the answer to problem 1.
On square

JHMT

, point

X

lies on

\overline{HM}

, and point

Y

is the intersection point of lines

JM

and

TX

. Assume that

\tfrac{TY}{XY}=\sqrt5

and the area of

\triangle{XYM}

is

N_1

. Compute the area of

\triangle{JYT}

.

1

2

Powers of 2 and Multiples of 2024

Compute the smallest positive integer

N

for which

N \cdot 2^{2024}

is a multiple of

2024

.

Squares in a Grid

Compute the number of squares of positive area whose vertices all are points on the grid shown below. [asy] unitsize(1cm); dot((0,0)); dot((1,0)); dot((2,0)); dot((3,0)); dot((0,1)); dot((1,1)); dot((2,1)); dot((3,1)); dot((0,2)); dot((1,2)); dot((2,2)); dot((3,2)); dot((0,3)); dot((1,3)); dot((2,3)); dot((3,3)); [/asy]

2024 JHMT HS

Subcontests

Swapping First and Last Digits

Sum of Binomial Coefficient Products is Divisible by 89

Rectangle Inscribed in Rectangle with Tangent Condition

Trigonometric Expression with Strange Angles

Infinite Product of Trinomials

Prism with Kite Bases

Remainders of Squares Modulo 31

Three Sequences

Concave Heptagon with Square Components

Convenient Combinatorics

Sum of Squares Greater than Product Diophantine

Inscribed Sphere of Tetrahedron

Expectation of Gecko Jumps

Last Two Digits and 21st Power

Area Involving Minimum and Sum of Squares

Concyclic Points and Angle Bisectors

Coloring Sides of a Regular Polygon

Sines and Floors

Units Digit of Tetration

Median is an Element

Altitudes, Midpoints, and Similar Triangles

Cevian and Equal Perimeters

Difference Between Quadratic Roots

Cube Root Condition

Exponential Equation

Painted Cube Probability

Circling Positive Integers

Quadratic with Unique Zero

Triangles Formed by Intersections

Powers of 2 and Multiples of 2024

Squares in a Grid