Avenues Institute for Advanced Mathematics
The mathematician cannot create things at will, any more than the geographer can; one can only discover what is there and give it a name.
— Gottlob Frege
The Foundations of Arithmetic, 1884
The Avenues Institute for Advanced Mathematics (AIAM) offers a path of advanced study in mathematics for secondaryaged students at Avenues campuses around the world. More than just an advanced math class, the Institute is an exceptionally selective global program designed for students who see mathematics as a source of beauty and inspiration in the world, and who feel driven to produce groundbreaking work in pure and applied mathematics. If you perceive nature as a collection of patterns, hear number ratios in music, and find yourself explaining the phenomena around you in terms of mathematics, the Institute could be the right next step for you.
AIAM provides a rigorous and highly demanding mathematics curriculum beyond the level of any advanced math class currently taught at Avenues. As such, the Institute replaces a student’s regular math classes. Students will learn from a combination of independent research, peer group discussion, handson projects, and the constant construction of mathematical models. Students will examine major mathematical breakthroughs from 17,000 BCE to the present day, incorporating cultural and historical perspectives into their analysis.
The Experience
Instruction is individualized and the learning environment is flexible. Each student completes the material in a manner and at a pace that maximizes their learning. The scope and sequence of the work are strictly determined by the student’s pace, progress, and passion. There are no “classrooms” where students sit together grouped by age or grade level; rather, students benefit from a rhythm of intensive independent study followed by regular group or oneonone meetings with their teacher — either in person or online.
The program is based on two kinds of work: modules and projects. Modules consist of curricular materials, which students work on independently, discussing their solutions with their teacher and peer group on a regular basis. Projects are handson and often collaborative, challenging students to explore applications of mathematics in engineering, science, or art.
Who
Current 6th, 7th, 8th, and 9th graders working at high levels in math at any Avenues campus may apply. Interested students should consult with their deans. Students are required to complete the challenge task below and attend an online interview as part of their application. Once enrolled, students may continue in the program through graduation.
Applying
We will be accepting applications on Friday, March 4, 2022.
Step 1 
Solve the Challenge
Follow the instructions in the video, then answer the three questions below.
Download a Printable Golden Rectangle (pdf)
Question 1
Start with a golden rectangle of arbitrary length (size and units of measure are your choice). Following the labeling OABC where O is the origin at the left bottom corner, OA is the length of the rectangle on the xaxis, and OC is the width on the yaxis.
Find the coordinates of the limit point where the cut rectangles are shrinking to a single point. The coordinates will depend on your initial choice of rectangle length, so make sure you clearly indicate the units you use.
Question 2
Generalize the case for any rectangle whose length is a and width is b, and find the coordinates of this point in terms of a.
Question 3
It turns out that the same point is also the intersection point of two lines: the diagonal CA and the perpendicular line that goes from B to CA. By finding the equations of these lines,
 find the coordinates of this point in terms of a,
 show by a method of your choice that these coordinates yield the same values as the ones you found in Question 1.
Be prepared to discuss: Is this problem related to the Fibonacci sequence and the Golden Spiral in any ways? Explain in detail.
Prepare your solution:
 Create a video of yourself explaining your solution in detail. Your video should be around 3 minutes long. You can use notes, but explain your solution without reading from a script. Be creative with your approach, presentation and video editing.
 Create a PDF file of your mathematical solution, showing your work that leads to an answer in an organized way. Variables should be defined, all diagrams should be labeled, and each mathematical step should be justified.

Step 2 
Let’s Discuss Your Solution
 Your submission will be reviewed in a few days and you will be invited to an online interview.
 In the interview, we will discuss your solution and talk about extensions of the challenge, and you will receive feedback on your work.

Step 3 
Get Your Results
 After your online interview, you will hear from the Institute within two weeks about your admission status.

Application Deadline
Applications are due on March 25, 2022. Please submit all your work before or on this date.
Apply Now
Thank you for your interest in the Avenues Institute for Advanced Mathematics.
The application period has ended and we are no longer accepting new submissions.
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