2025 Spring · Introductory Algebra

Problem 1

Derek's math final is coming up, and he really needs an A in his class to avoid summer school! His class is weighted Homework:

20\%

, Quizzes:

20\%

, Tests:

50\%

, Final:

10\%

, and he has averages of

100

,

70

, and

90

in homework, quizzes, and tests respectively. What does Derek need on his final exam to get an A (

90

or higher)?

\text{(A) } 70\% \quad \text{(B) } 80\% \quad \text{(C) } 90\% \quad \text{(D) } 100\% \quad \text{(E) } 110\%

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Problem 2

In right triangle

ZAP

(with

\angle A = 90^\circ

),

AY

is the altitude to hypotenuse

ZP

where

Y

lies on

ZP

. Given

|ZA|=n

,

|ZP|=3n-6

, and

|ZY|=\dfrac{3n}{n-2}

, the length

AP=m\sqrt{n}

in simplest radical form. Find

m+n

.

\text{(A) } 16 \quad \text{(B) } 18 \quad \text{(C) } 20 \quad \text{(D) } 22 \quad \text{(E) } 24

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Problem 3

Enan is being haunted by the GOTG (Ghost of the Geometry), and the only way to fend it off is to answer correctly. How many distinct ways are there to rearrange the letters in the word

\text{GEOMETRY}

, where repeated letters are not considered distinct?

\text{(A) } 10800 \quad \text{(B) } 20160 \quad \text{(C) } 40320 \quad \text{(D) } 45480 \quad \text{(E) } 50400

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Problem 4

David's calculator broke and keeps repeating the digits "378" over and over. How many positive factors does the number

378{,}378

have?

\text{(A) } 72 \quad \text{(B) } 84 \quad \text{(C) } 96 \quad \text{(D) } 108 \quad \text{(E) } 120

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Problem 5

Cinco de Mayo's date

5/5/25

(M/D/Y) has the special property that the month times the date equals the last two digits of the year (

5\cdot 5 = 25

). Given that May 5, 2025 is a Monday and 2025 is not a leap year, what is the day of the week of the last time this property was satisfied? (The property: a date

a/b/c

with

a\cdot b=c

, where

c

is the last two digits of the year.)

\text{(A) } \text{Tuesday} \quad \text{(B) } \text{Wednesday} \quad \text{(C) } \text{Thursday} \quad \text{(D) } \text{Friday} \quad \text{(E) } \text{Saturday}

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Problem 6

April's Excel Math Competition had

10

multiple-choice questions, each with five answer choices (ignore the free-response). Bhuv, Zoohas, and Rodium each guessed randomly on every question. What is the expected total number of correct answers among the three of them?

\text{(A) } 2 \quad \text{(B) } 3 \quad \text{(C) } 4 \quad \text{(D) } 6 \quad \text{(E) } 9

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Problem 7

Consider a regular polygon with

n

sides. How many values of

n

exist such that the measure of an interior angle is a positive integer (in degrees)?

\text{(A) } 21 \quad \text{(B) } 22 \quad \text{(C) } 23 \quad \text{(D) } 24 \quad \text{(E) } 25

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Problem 8

In COD, Rishan's character has three attacks: a punch (

1

HP), a fireball (

3

HP), and a sword slash (

6

HP). To eliminate an enemy with

50

HP he must deal EXACTLY

50

HP. How many different combinations of attacks can he use? (Order does not matter—three fireballs and a punch is the same as one punch and three fireballs.)

\text{(A) } 79 \quad \text{(B) } 80 \quad \text{(C) } 81 \quad \text{(D) } 82 \quad \text{(E) } 83

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Problem 9

Kha builds an

8\times 8\times 8

cube out of unit cubes and paints the outside. The structure is knocked over and he picks up a random unit cube. What is the probability that this cube has either

1

or

2

painted sides?

\text{(A) } \tfrac{65}{128} \quad \text{(B) } \tfrac{33}{64} \quad \text{(C) } \tfrac{17}{32} \quad \text{(D) } \tfrac{9}{16} \quad \text{(E) } \tfrac{1}{2}

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Problem 10

Which answer choice is logically equivalent to: "If David is eating and watching TV, then he's not writing math problems."?

\begin{aligned}&\text{(A) If David is writing math problems, then he is not eating or not watching TV.}\\&\text{(B) If David is not eating or not watching TV, then he is writing math problems.}\\&\text{(C) If David is not writing math problems, then he is eating and watching TV.}\\&\text{(D) If David is eating or watching TV, then he is not writing math problems.}\\&\text{(E) If David is eating and watching TV, then he is writing math problems.}\end{aligned}

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Problem 11

What are the tens digit and units digit of

7^{2025}

? If the integer value is

\ldots abc

, your answer should be of the form

0bc

.

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Problem 12

Express the repeating decimal

1.0\overline{375}=1.037537537\ldots

as a fraction in simplest form; report the numerator.

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Problem 13

Santa Clop gives gifts to all

270{,}000

students at RHS.

40\%

want a basketball,

30\%

shoes,

30\%

a puppy. Of basketballs: half NBA (each

\tfrac13

green/blue/red), half NCAA (each

\tfrac12

gold/silver);

\tfrac14

of basketballs are women's. Of shoes: twice as many Nike as Adidas; of Nikes

\tfrac14

cleats,

\tfrac14

basketball shoes,

\tfrac12

sandals; of Adidas

\tfrac15

loafers,

\tfrac{3}{10}

sneakers,

\tfrac12

dress shoes; across all shoes the colors are

15\%

green,

20\%

red,

30\%

brown,

35\%

gold. Find (green women's NCAA basketballs)

+

(brown Nike cleats)

+

(gold Adidas sneakers). Take the resulting five-digit integer

\overline{abcde}

and report

\overline{abc}-\overline{de}

.

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Problem 14

There are several integers

n

with

800<n<1000

whose digit-reversal (so

\overline{abc}\to\overline{cba}

) equals

p\%

of

n

for some integer

p

with

1\le p\le 100

. Find the ONLY such

n

in this interval for which

p\neq 1,10,100

.

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Problem 15

Right triangle

ABC

has right angle at

B

,

AB=8

,

BC=15

. On hypotenuse

AC

lie the foot

D

of the angle bisector from

B

, the foot

E

of the median from

B

, and the foot

F

of the altitude from

B

. If

|DE|\cdot|AF|\cdot|FC|=\dfrac{m}{n}

in simplest form, find the sum of the largest prime factors of

m

and

n

.

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