2025 Fall · 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, with this problem being written on August 10th, 2025 (also written

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

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Problem 2

Ishaan buys a variety pack of gum:

50

sticks in three flavors — strawberry, vanilla, and chocolate. There are

6

more strawberry sticks than vanilla, and three times as many chocolate sticks as vanilla. How many strawberry sticks are in the pack?

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

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Problem 3

At Ridgeville High, students may take any of Chinese, Spanish, and Russian (

0

,

1

,

2

, or

3

classes). There are

70

taking Chinese,

65

Spanish,

60

Russian,

25

Chinese and Spanish,

20

Chinese and Russian,

18

Spanish and Russian, and

10

taking all three. Given

154

students total, how many take no language classes?

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

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Problem 4

David leaves five perfectly spherical beach balls in a line, each with radius

16

. Overnight, four are punctured so that each ball has half the radius of the one before it. What is the total volume of air 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}

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Problem 5

One of Sonya's summer-reading books, The Leavers, is

338

pages. Assuming pages are numbered from page

2

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

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Problem 6

Kyle paints

4

schools in a day and Lyle paints

2

in a day; Miles is new and his rate is unknown. Kyle works an initial hour alone, two hours with Lyle, and one hour with both Lyle and Miles, and together they finish the entire 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

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Problem 7

Toyesh has the set

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

. If he picks a subset uniformly at random, what is the probability 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}

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Problem 8

In rhombus

ZAPY

,

m\angle ZAP = 120^\circ

and the perimeter is

24

units. What is the length of the longer diagonal?

\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}

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Problem 9

Joey's calculator spits out

6573

in base

9

,

13221

in base

4

,

3256

in base

7

, and

100111001

in base

2

. Find the sum of all these numbers in base

10

.

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

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Problem 10

There are

8

Ivy League schools, all within the T20. Assuming Jemail is accepted into a uniformly random set of

10

of the

20

T20 schools, what is the probability 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}

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Problem 11

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 of the result.

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Problem 12

From receipts:

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

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

After finding the price of each item, round each to the nearest integer and report the product of the rounded numbers.

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Problem 13

A practice track is made of a sixth of a circle with radius

6

, half of a regular hexagon with side length

4

, and a right triangle with legs

13

and

8

. The sum of the areas can be written as

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

. Find

\operatorname{lcm}(a,b,c,d)

.

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Problem 14

Bhuv's calculator has only two working functions:

\times 2

and

+1

. Starting from the number

1

, what is the fewest number of moves needed to reach

1500

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

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Problem 15

A FAIR archery game (expected value

0

) uses a target of concentric circles with radii

1,3,4,7,10

. A hit scores

+10

inside radius

1

;

+5

between radii

1

and

3

;

+2

between

3

and

4

; and

-1

between

4

and

7

. Assuming each player hits a uniformly random point on the target, how many points should the player lose in the outer ring (between radii

7

and

10

)? The answer, when simplified, is

\tfrac{a}{b}

; report

a+b

.

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