2025 Spring · Intermediate Algebra

Problem 1

Welcome (or welcome back) to the ExcelAcademe Math Competition! As the last full month of spring, everyone's hopefully looking forward to a nice summer break! During summer, Earth is at the farthest point from the Sun in its elliptical orbit, at about

152

million kilometers away. The time when Earth is closest to the Sun (which is at the focus of its orbit) is in winter, at about

147

million kilometers away. Which of the following is closest to the eccentricity of the Earth's orbit around the sun?

\text{(A) } \tfrac{1}{120} \quad \text{(B) } \tfrac{1}{60} \quad \text{(C) } \tfrac{1}{45} \quad \text{(D) } \tfrac{1}{30} \quad \text{(E) } \tfrac{1}{15}

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Problem 2

Consider equilateral

\triangle RAL

with side length

1

and equilateral

\triangle DMP

with parallel sides to

\triangle RAL

,

D

at the center of

\triangle RAL

, and side length

2

. Find the area of convex pentagon

RAMPL

.

\text{(A) } \tfrac{\sqrt{3}}{2} \quad \text{(B) } \sqrt{3} \quad \text{(C) } \tfrac{3\sqrt{3}}{2} \quad \text{(D) } 2\sqrt{3} \quad \text{(E) } \tfrac{5\sqrt{3}}{2}

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Problem 3

David, Michael, and Brendan are scrambling to finish their engineering projects! Each of them has to choose a distinct project idea out of a pool of

5

ideas. However, they all have a

50\%

chance of completely failing to complete any given project and having to pick another un-chosen idea from the pool of remaining ideas. What is the probability that at least one of the three engineers runs out of possible project ideas before they all complete a project?

\text{(A) } \tfrac{1}{6} \quad \text{(B) } \tfrac{1}{5} \quad \text{(C) } \tfrac{1}{4} \quad \text{(D) } \tfrac{1}{3} \quad \text{(E) } \tfrac{1}{2}

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Problem 4

Zohar can bench

7

lbs. Alex can bench

6

lbs. Siyuan can bench

3

lbs. What is the maximum number of lbs they cannot bench in total, if they all bench a positive number of times without tiring?

\text{(A) } 20 \quad \text{(B) } 21 \quad \text{(C) } 24 \quad \text{(D) } 27 \quad \text{(E) } 31

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Problem 5

This MAO national convention, a certain school has

40

attendees. Grister Migelis knows that

25\%

of these attendees ordered a pepperoni pizza, and the rest ordered a cheese pizza. However, some of the students are very greedy and say they ordered pepperoni while having ordered only cheese! He doesn't notice this until he has already given out

n

pepperoni pizzas and

2n

cheese pizzas. Trying to salvage the situation, Grister Migelis compromises with the remaining students by mixing and matching slices and giving each of them a pizza which is

\tfrac{1}{n}

'th pepperoni and the rest cheese. He manages to do this with all of the remaining students successfully. What is the sum of the possible values of

n

?

\text{(A) } 5 \quad \text{(B) } 8 \quad \text{(C) } 11 \quad \text{(D) } 13 \quad \text{(E) } 15

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Problem 6

A

\textit{steradian}

is a unit (denoted

\Omega

) used to measure the solid angle in 3-D space, and is defined as the ratio between the surface area covered on a sphere and

R^2

, the squared radius of that sphere. The earth's surface is about

71\%

water. The steradian measure of the earth which is covered by

\textbf{land}

can be written as

k\pi

. Find

\lfloor 10k \rfloor

.

\text{(A) } 11 \quad \text{(B) } 14 \quad \text{(C) } 17 \quad \text{(D) } 20 \quad \text{(E) } 23

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Problem 7

Consider

f(x)=x^3-2025x^2-4x+1

with roots

r,s,t

. Find the sum of the digits of

\frac{r+s}{t}+\frac{s+t}{r}+\frac{r+t}{s}.

\text{(A) } 20 \quad \text{(B) } 24 \quad \text{(C) } 27 \quad \text{(D) } 30 \quad \text{(E) } 36

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Problem 8

Three chords are independently drawn at random in circle

\Omega

. What is the probability that they form a non-degenerate triangle entirely contained within

\Omega

?

\text{(A) } \tfrac{1}{18} \quad \text{(B) } \tfrac{1}{15} \quad \text{(C) } \tfrac{1}{7} \quad \text{(D) } \tfrac{1}{5} \quad \text{(E) } \tfrac{1}{3}

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Problem 9

There exist real numbers

a_n

with

n\ge 0

such that

f(x)=\frac{2x^2+3}{x-4}=\sum_{n=0}^{\infty} a_n x^n

for

|x|<4

. For the purposes of this question you may assume this is simply an equivalent way to represent the function

f

on this interval. Determine

\sum_{n=0}^{\infty} a_{2n+1}.

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

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Problem 10

Find the area between the

x

-axis and the graph of

f(x)=\lfloor x\rfloor+\lceil x\rceil-x

from

x=0

to

x=6

.

\text{(A) } 6 \quad \text{(B) } 9.5 \quad \text{(C) } 12 \quad \text{(D) } 15.5 \quad \text{(E) } 18

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Problem 11

Let

f(k)

be the concatenation of every

1

in the base

2

representation of

k

. For example,

f(5)=11

as

5=101_2

. Find the number of positive integers

n<256

satisfying

37 \mid f(n)

.

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Problem 12

n

circles of radius

1

are each pairwise externally tangent to

2

of the others and arranged such that their centers lie on a larger common circle. As an example, the

n=6

case is provided.

A_n

is defined for

n\ge 3

as the area bounded by the inner-facing portions of the circles, as shown above in the shaded region for

A_6

. Given that

A_{2025}

can be written as

a\tan(b\pi)-c\pi

for real numbers

a,b,c

, determine

\dfrac{c}{ab}

.

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Problem 13

Satvik has been gifted a certain sequence

S_n=\{s_1,s_2,\dots\}

for his birthday.

S_n

is special in that the differences between the

(n+1)

'th and

n

'th terms for

n\ge 1

form a geometric sequence with common ratio

r

. Satvik knows the following about his sequence:

r=3

,

s_3=15

, and

S_n

consists of strictly increasing, positive, integer terms. Help Satvik find the sum of possible values of

s_1

.

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Problem 14

We define the Möbius function

\mu(n)

on a positive integer

n

as

\mu(n)=\begin{cases}1 & \text{if } n=1\\ (-1)^{k} & \text{if } n \text{ is the product of } k \text{ distinct primes}\\ 0 & \text{if } n \text{ is divisible by a perfect square} > 1\end{cases}

If

S

is the set of all positive integer divisors of

15!

, determine

\frac{1}{10}\sum_{x\in S} x\cdot\mu(x).

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Problem 15

Call

\Omega

the set of distinct circles centered at the origin which are tangent to the graph of

\tan(xy)=1

at some point. Note: these circles may intersect the graph at other points. Call

A(\omega)

for a circle

\omega\in\Omega

the area of that circle. The following sum can be written as

\tfrac{1}{a}\pi^{b}

for integers

a,b

. Find

a-b

.

\sum_{\omega\in\Omega}\frac{1}{\big(A(\omega)\big)^2}

Hint!

\dfrac{1}{1^2}+\dfrac{1}{3^2}+\dfrac{1}{5^2}+\dfrac{1}{7^2}+\cdots=\dfrac{\pi^2}{8}

.

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