AwesomeMath Test C

March 16 – March 28, 2024

PLEASE READ CAREFULLY and THOROUGHLY

DEADLINE: Thursday, March 28, 2024, 5:00 PM CT

INSTRUCTIONS

• Students who wish to take Level 1 & Level 2 classes must submit solutions to Part I only.

• Students who wish to take Level 3 & Level 4 classes must submit solutions to Part II only.

• Only one test and one part (Part I or Part II) will be graded per student. Do NOT submit solutions to Part I

and then additional solutions to Part II as a separate submission.

• Students accepted for Level 1 and Level 2 will not be able to take Level 3 and 4 classes, however students

accepted for Level 3 and 4 are allowed to switch to lower-level classes if it turns out that their initial

classes are too difficult.

• This test is not timed, therefore, do not rush. Show us your best work. Your solutions will be evaluated

against mathematical experience and any math competition test scores provided on your application.

• We want you to have a positive and successful experience at our camps, therefore your work should be

yours and yours alone. The purpose of the test is to ensure that you are placed in classes that fit your level

and skills. If you consult outside resources (other people or online), then you are compromising your ability

to succeed in our challenging program.

• Do not be discouraged if you cannot solve all the questions. We want to see the solutions you come up

with no matter how many problems you solve. Some of the problems involve complex mathematical ideas,

but all of them can be solved using only elementary techniques, admittedly combined in clever ways.

• Include all significant steps in your reasoning and computation. We are interested in your ability to

present your work, so unsupported answers will receive less credit than well-reasoned progress towards a

solution without a correct answer.

• In this document, you will find an answer sheet. Print out or make several copies of the blank answer

sheet. Fill out the top of each answer sheet as you go. Start each problem on a new answer sheet.

• You may handwrite or type your solutions. If you type your solutions, your answer sheets should still

include the same information as shown on the test packet answer sheets (9-digit UIN, First and Last Name,

Problem #, etc.)

SUBMISSION REQUIREMENTS

• Your solutions must be submitted as a single pdf document. Do NOT submit test solutions as individual

pages.

• UPLOAD your solutions file in accordance with the directions provided on your student dashboard.

• DO NOT email us your solutions as it will significantly delay receipt and grading of your solutions.

AwesomeMath Admission Test

Cover Sheet

PLEASE PRINT LEGIBLY and DO NOT LEAVE ANY FIELDS BLANK

First Name:

A B C

Part I Part II

· Make sure this COVER SHEET is the first page of your submission, and that it is completely filled out.

· UPLOAD your solutions file in accordance with the directions provided on your student dashboard.

· Your solutions must be submitted as a single pdf document. Do NOT submit test solutions as individual pages.

· DO NOT email us your solutions as it will significantly delay receipt and grading of your solutions.

· Students who wish to take Level 1 & Level 2 classes must submit solutions to Part I only

· Students who wish to take Level 3 & Level 4 classes must submit solutions to Part II only

· Only one test and one part (Part I or Part II) will be graded per student. Do NOT submit solutions to Part I and

then submit additional solutions to Part II.

D E

9‐Digit UIN:

Last Name:

Admission Test (check one):

Part Completed (check one):

Email:

Important Reminders:

AwesomeMath Admission Test Answer Sheet

Problem Number:

First Name:

9-Digit UIN:

Page Number: out of total pages being submitted

(including cover sheet)

Write neatly! All work should be inside the box. Do NOT write on the back of the page!

Last Name:

© AwesomeMath, 2024

All materials contained within this document are protected by U.S. copyright law and may not

be reproduced, distributed, transmitted, displayed, published, or broadcast without prior, express written

permission of AwesomeMath LLC, except for non-commercial educational purposes only.

Test C

March 16 – 28, 2024

Part I Levels 1 & 2

1. Find all 4-tuples (p,q,r,s) of primes such that

p2 + q2 + 4r2 + 6s2 = 2024.

2. Find the least positive integer n for which n4

4−2024n2 + 1 is a product of four primes, not

necessarily distinct.

3. Two lotteries each draw balls marked 1 through n. The first draws 6 out of the n balls and the

second 5 out of n. Playing the same number of tickets at one drawing, the odds of hitting the

jackpot at the second lottery is 10 times bigger than at the first. Find n.

4. Write 2·20243 as

ab·bc·cd·de·ef·fg·gh

and 3·20243 as

pq·qr·rs·st·tu·uv·vw,

where xy is the two-digit number whose digits are x and y.

5. For some positive integer n, in the equality

10n

k=2

k−2 +

k2

−k+ 1

=

(10n + 1)!

100...01 ,

the denominator of the fraction in the right-hand side has 2024 zeros. Find n.

6. Let nbe a positive integer such that 6n2 has precisely 2024 positive divisors and 4n2 has precisely

2025 positive divisors. Find n.

7. Let ABC be a triangle with AB= AC and area 2024. Point P lies on base BC and Q and R

are the orthogonal projections of P onto AB and AC, respectively. Given that PQ+ PR = 22,

evaluate BC.

8. Solve in integers the equation

x2

−15 min(x,y) + y2 = 2024.

9. Let ABC be a triangle with AB= AC = 2024 for which there is a point D on side BC such that

AD= 1981 and |BD−CD|= 1846. Evaluate ∠BAC.

10. Solve the equation

(16x−{x})⌊x⌋= 2024,

where ⌊x⌋and {x}are the greatest integer less than or equal to x and the fractional part of x,

respectively.

© AwesomeMath, 2024

All materials contained within this document are protected by U.S. copyright law and may not

be reproduced, distributed, transmitted, displayed, published, or broadcast without prior, express written

permission of AwesomeMath LLC, except for non-commercial educational purposes only.

Part II Levels 3 & 4

1. Solve in prime numbers the equation

p3 + 2q+ r3 = 2024.

2. Let ak = k2

−78k+ 2028, k= 1,2,3,.... Evaluate

a1

+

a2

+···+

a77

3. Solve in integers the equation

x2 + 2yz y2 + 2zx z2 + 2xy min(x,y,z) = 2024.

4. Find the least positive integer n such that S(n−1)S(n+ 1) = 2024, where S(N) is the sum of

the digits of N.

5. Let a and b be real numbers for which there is an integer n such that a3 + nab+ b3 = 107 and

9 a2

−ab+ b2 + n(n+ 3(a+ b)) = 2024

n−3(a+ b).

Find n.

6. Let a∈ 0,

π

such that

sin2 acos2 a+ csc2 2a= 2024.

Given that cos 4a = m√n−p, where m, n, p are positive integers and n is square-free, find

m+ n+ p.

7. Solve in real numbers the system of equations

xy+ z+ w= 50, yz+ w+ x= 90, zw+ x+ y= 184, wx+ y+ z = 2024.

8. Quadrilateral ABCD is inscribed in a circle with center O. Given that AB = 13, BC = 23,

CD= 35, and 39AC = 25BD, evaluate ∠AOD.

9. In the rectangular box ABCDA′B′C′D′

, AB = 2024, AA′ + BD = 2074, and A′D = 102. Find

the volume of the rectangular box.

10. Solve the equation

45 2x−{x}2

= ⌊x⌋2 + 2024,

where ⌊x⌋and {x}denote the integer part and the fractional part of x, respectively.