2025 Summer · Introductory Algebra

Problem 1

Welcome back to the August Excel Introductory Algebra Test; we hope you all had a great summer! The test writers procrastinated a bit on writing this test, with this problem being written on August 10th, 2025 (also represented as 8/10/25). What is the sum of the factors of

8 \cdot 10 \cdot 25

?

\text{(A) } 4836 \quad \text{(B) } 2001 \quad \text{(C) } 8370 \quad \text{(D) } 4826 \quad \text{(E) } 2340

Community solutions

No community solutions yet. Be the first.

Checking sign-in…

Problem 2

Ishaan has decided to prepare for the new school year by buying a variety pack of gum. The package comes with

50

sticks of gum in three flavors: strawberry, vanilla, and chocolate. Given that there are

6

more strawberry sticks of gum than vanilla, and three times as many chocolate sticks of gum as vanilla, how many strawberry sticks of gum does Ishaan have in the pack?

\text{(A) } 9 \quad \text{(B) } 15 \quad \text{(C) } 21 \quad \text{(D) } 27 \quad \text{(E) } 36

Community solutions

No community solutions yet. Be the first.

Checking sign-in…

Problem 3

It's time to schedule language classes at Ridgeville High. Students may take Chinese, Spanish, and Russian, in any number (

0

,

1

,

2

, or

3

). There are

70

taking Chinese,

65

taking Spanish,

60

taking Russian,

25

taking Chinese and Spanish,

20

taking Chinese and Russian,

18

taking Spanish and Russian, and

10

taking all three. If there are

154

students total, how many are taking no language classes?

\text{(A) } 8 \quad \text{(B) } 11 \quad \text{(C) } 12 \quad \text{(D) } 15 \quad \text{(E) } 22

Community solutions

No community solutions yet. Be the first.

Checking sign-in…

Problem 4

During the summer, David loves going to the beach, and especially loves playing with beach balls! He leaves five perfectly spherical beach balls in a line, each with radius

16

. Overnight, four are punctured so that in the line of five, each ball has half the radius of the one before it. What is the total volume of air present in all five beach balls?

\text{(A) } \tfrac{16384\pi}{3} \quad \text{(B) } \tfrac{18734\pi}{3} \quad \text{(C) } \tfrac{18744\pi}{3} \quad \text{(D) } \tfrac{18724\pi}{3} \quad \text{(E) } \tfrac{18688\pi}{3}

Community solutions

No community solutions yet. Be the first.

Checking sign-in…

Problem 5

Sonya's taking AP Lit, and she REALLY doesn't want to do her summer reading. One of her books, The Leavers, is

338

pages. Assuming the pages are numbered starting at page

2

and going up to page

338

, how many total

3

's appear in the page numbers?

\text{(A) } 108 \quad \text{(B) } 110 \quad \text{(C) } 112 \quad \text{(D) } 115 \quad \text{(E) } 119

Community solutions

No community solutions yet. Be the first.

Checking sign-in…

Problem 6

Kyle and his two brothers, Lyle and Miles, help RHS paint the entire school. Kyle can paint

4

schools in a day and Lyle can paint

2

in a day; Miles is new, and no one knows his rate. Kyle works an initial hour by himself, two hours with Lyle, and one hour with both Lyle and Miles, and together they finish exactly one school. How many schools can Miles paint in a day?

\text{(A) } 1 \quad \text{(B) } 2 \quad \text{(C) } \tfrac{3}{2} \quad \text{(D) } 3 \quad \text{(E) } 4

Community solutions

No community solutions yet. Be the first.

Checking sign-in…

Problem 7

Toyesh has the set of the first six positive integers

\{1,2,3,4,5,6\}

. He picks a subset uniformly at random from all subsets. What is the probability that the subset contains at least one odd number?

\text{(A) } \tfrac{1}{2} \quad \text{(B) } \tfrac{3}{4} \quad \text{(C) } \tfrac{13}{16} \quad \text{(D) } \tfrac{29}{32} \quad \text{(E) } \tfrac{7}{8}

Community solutions

No community solutions yet. Be the first.

Checking sign-in…

Problem 8

In rhombus

ZAPY

,

m\angle ZAP = 120^{\circ}

and the perimeter is

24

units. What is the length of the longer diagonal of the rhombus?

\text{(A) } 4\sqrt{3} \quad \text{(B) } 6 \quad \text{(C) } 6\sqrt{3} \quad \text{(D) } 8 \quad \text{(E) } 8\sqrt{3}

Community solutions

No community solutions yet. Be the first.

Checking sign-in…

Problem 9

Joey's calculator keeps spitting out numbers in different bases:

6573

in base

9

,

13221

in base

4

,

3256

in base

7

, and

100111001

in base

2

. Find the sum of all these numbers, expressed in base

10

.

\text{(A) } 6795 \quad \text{(B) } 6785 \quad \text{(C) } 6825 \quad \text{(D) } 6805 \quad \text{(E) } 6815

Community solutions

No community solutions yet. Be the first.

Checking sign-in…

Problem 10

It's college application season for all the rising seniors! There are

8

Ivy League schools, all within the T20. Assuming Jemail is accepted into

10

schools chosen uniformly at random from the T20, what is the probability that he gets into all

8

Ivy League schools?

\text{(A) } \tfrac{15}{22} \quad \text{(B) } \tfrac{66}{125970} \quad \text{(C) } \tfrac{132}{184756} \quad \text{(D) } \tfrac{132}{125970} \quad \text{(E) } \tfrac{66}{184756}

Community solutions

No community solutions yet. Be the first.

Checking sign-in…

Problem 11

The test writers saw Fantastic Four: First Steps a week before writing this test; in that spirit, here's an easy question! Find how many different ways there are to arrange the letters in FANTASTIC (identical letters are interchangeable), multiply that count by four, and report the sum of the digits in the result.

Community solutions

No community solutions yet. Be the first.

Checking sign-in…

Problem 12

Ken and Tony go school shopping at the Pencil, Notebook, Eraser, and Puppy Store. From receipts they find

P+N+E+U = 16,\quad 2P+N+E+U = 19.50,

P+2N+3E+U = 28,\quad 2P+N+E+2U = 25,

where

P,N,E,U

are the prices of a pencil, notebook, eraser, and puppy. After finding each price, round it to the nearest integer and report the product of the rounded numbers.

Community solutions

No community solutions yet. Be the first.

Checking sign-in…

Problem 13

Rishan sets up a practice track made of a sixth of a circle with radius

6

, half of a regular hexagon with side length

4

, and a right triangle with leg lengths

13

and

8

. The sum of the three areas can be written as

a + b\pi + c\sqrt{d}

. Find the least common multiple of

a

,

b

,

c

, and

d

.

Community solutions

No community solutions yet. Be the first.

Checking sign-in…

Problem 14

Bhuv's calculator has only two working functions:

\times 2

(multiply by two) and

+1

(add one). Starting from the number

1

, what is the fewest number of moves needed to reach

1500

? (A move is one use of either function.)

Community solutions

No community solutions yet. Be the first.

Checking sign-in…

Problem 15

The Summer Carnival's archery game is completely FAIR (its expected value is

0

). The target is concentric circles of radii

1, 3, 4, 7, 10

. A hit in the smallest circle scores

10

; the next ring scores

5

; the next scores

2

; and the ring between radii

4

and

7

loses

1

point. A player hits a uniformly random point on the target. How many points should the player lose for hitting the outer ring (between radii

7

and

10

)? The answer, in lowest terms, is

\tfrac{a}{b}

; report

a+b

.

Community solutions

No community solutions yet. Be the first.

Checking sign-in…