Solving an incredibly difficult question from Turkey

Thanks to Alp for the suggestion! This question is adapted and translated from a Turkish University Entrance Exam. The first step is the Basic Proficiency Test (TYT) which about 3 million students aged around 18 years take. Then there are Field Proficiency Tests (AYT) in Turkish language and literature, social sciences, mathematics, and science. About 2 million students take this. The AYT can be quite challenging.

Turkey Student Selection and Placement System
2021 AYT Zor Polinom Sorusu Çözümü
Duvar English average score and number students
AACRAO minutes
TÜM ZAMANLARIN ÇIKMIŞ EN ZOR POLİNOM SORUSU 🔥ÖSYM SEVER 🤠

Send me suggestions by email (address at end of many videos). I may not reply but I do consider all ideas!

If you purchase through these links, I may be compensated for purchases made on Amazon. As an Amazon Associate I earn from qualifying purchases. This does not affect the price you pay.

If you purchase through these links, I may be compensated for purchases made on Amazon. As an Amazon Associate I earn from qualifying purchases. This does not affect the price you pay.

Book ratings are from January 2023.

My Books (worldwide links)

My Books (US links)
Mind Your Decisions: Five Book Compilation
A collection of 5 books:
"The Joy of Game Theory" rated 4.3/5 stars on 290 reviews
"The Irrationality Illusion: How To Make Smart Decisions And Overcome Bias" rated 4.1/5 stars on 33 reviews
"40 Paradoxes in Logic, Probability, and Game Theory" rated 4.2/5 stars on 54 reviews
"The Best Mental Math Tricks" rated 4.3/5 stars on 116 reviews
"Multiply Numbers By Drawing Lines" rated 4.4/5 stars on 37 reviews

Mind Your Puzzles: Collection Of Volumes 1 To 3
A collection of 3 books:
"Math Puzzles Volume 1" rated 4.4/5 stars on 112 reviews
"Math Puzzles Volume 2" rated 4.2/5 stars on 33 reviews
"Math Puzzles Volume 3" rated 4.2/5 stars on 29 reviews

2017 Shorty Awards Nominee. Mind Your Decisions was nominated in the STEM category (Science, Technology, Engineering, and Math) along with eventual winner Bill Nye; finalists Adam Savage, Dr. Sandra Lee, Simone Giertz, Tim Peake, Unbox Therapy; and other nominees Elon Musk, Gizmoslip, Hope Jahren, Life Noggin, and Nerdwriter.

My Blog

Twitter

Instagram

Merch

Patreon

Press

[artemysreborn]

One of the best problems I've seen on this channel and anywhere, really. Didnt expect that you'd be able to solve for the actual polynomial, and the solution is super elegant.

[sinavcekelimhadi]

Çok şükür kendi ülkemin sorusunun kaliteli bir kanal tarafından çözüldüğünü görebildim😊

[Miliradian]

As the saying goes:
"You don't enter the exam, the exam enters you"

[enesa]

As a Turk, I can say asking harder questions and try to teach many subjects in a year doesn't make us learn Math better. Because you forget them all together. That is why we forget the simplest things.

[Aphelia.]

I followed you for years and I never expected a question from my homeland, let alone an exam question I had to solve in 2021 💜 thank you

[adandap]

I like the approach to the first problem, and it's more efficient than mine. I stuck with p(x), realising that it had to be tangent to y=x at x=1 and x=3. So you have 5 equations for the 5 unknowns in a x^4 + b x^3 + c x^2 + d x + e, knowing p(x) at 1, 2 and 3 and that p'(x) = 1 at x = 1 and 3. I wrote them in matrix form and inverted the matrix to get p(x) = 2 x^4 - 16 x^3 + 44 x^2 - 47 x + 18, which has value 22 at x=4.

[MathFromAlphaToOmega]

A similar trick can be used to find the unique line tangent to a quartic at two points (if the quartic has 3 local extreme values). If you start with y=x^4+ax^3+bx^2+cx+d, you can find u and v such that (x^4+ax^3+bx^2+cx+d)-(ux+v) is a perfect square. Alternatively, you can first shift x to get rid of the x^3 term, then subtract the equation of the tangent. That gives a perfect square with no x^3 term, so it has to be (x^2+k)^2 for some constant k. Then it's easy to compute k from comparing coefficients.

[kicorse]

I like your method. To me the obvious thing to do was to use the fact that the derivative of P(x) had to be 1 at x=1 and x=3 to complete the system of equations. But simultaneously solving 5 equations by hand is laborious, and your approach avoided this.

[AltuğBeyhan]

In my opinion, this question is actually very good and mind-blowing, even though it is very hard and not suitable for the university entrance exam. In the exam, I was able to solve the question thanks to background on mathematical olympiad.

[exosproudmamabear]

In the last years questions started to become like iq tests rather than questioning your knowledge. 10 years ago it wasnt this bad but since youth unemployment rose and need for high education in good unis rose too.This caused a surging competition that only grew bigger as the economic crisis got worse. As everyone knows once the competition climbs it is really difficult to get less difficult questions in exams. Korean and Japanese exams are a nice example to that but Turkey's education system is not good as them, in fact it has been steadily declining for at least 7-8 years.

[yanggang]

Shout-out to all Turkish people keeping it real 🇹🇷

[homosacrilegus]

I was one of the rare students who answered all 40 Math questions correctly in the exam. The hardness of the exam doesn't come from the questions themselves, it comes from the time pressure and stress you're experiencing. You're whole future is being determined by a 3 hour exam. I remember it felt like as if there was a gun on my head and people were yelling at me to solve the questions. Thank you so much for remarking the hardness of the exam.

[MichaelRothwell]

Thanks for this nice problem.
To my surprise, I found it quite straight forward.

Let f(x)=P(x)-x, a fourth degree polynomial that is always non-negative.
Then
f(1)=P(1)-1=0
f(2)=P(2)-2=2
f(3)=P(3)-3=0
So f has zeros at 1 & 3. These must be roots of even order as f is always non-negative (if a polynomial f has a root α of odd order n, then f(x)=(x-α)ⁿg(x) where (x-α)ⁿ changes sign at x=α but g doesn't, so f changes sign at x=α).
Hence each root is of order 2.
So f(x)=a(x-1)²(x-3)² for some real a≥0.
From f(2)=2 we get
a(2-1)²(2-3)²=2
So a=2
f(x)=2(x-1)²(x-3)²
P(x)=2(x-1)²(x-3)²+x

[trnfncb]

Regardless of the whole min/max analysis, it is obvious that Q'(1)=Q'(3)=0 because Q(x) cannot cross the zero. So setting Q(x)=(x-1)(x-3)(ax^2+b.x+c) and solving the system Q'(1)=Q'(3)=0 and Q(2)=2 gives a, b and c right away.

[MathsMadeSimple]

Here’s the ‘Before 3 million subscribers button’

[delta]

Whenever I see an international channel mention this exam I feel so weird because I was one of the many students who participated this exam and solved this questions.I was lucky enough to solve most of the questions. (regarding to the fact mathematic was my favorite lesson I can even say I was lucky that they did math extra difficult)
I can appreciate hard Harvard-Oxford or Olimpiad questions but oddly enough seeing my exam's questions in this channels gives me a strange out of reality feeling... Idk why...
Nonetheless thank you for bringing attention to this exam and solution... I enjoyed watching it :)

[mike.]

I rather liked this problem! I feel like your justification of the multiplicity of the two roots could have been slimmed down to simply that polynomial roots either cross or touch and turn. Since crossing would make the polynomial go negative, they are touch and turn roots. Since the degree is 4, that makes them both multiplicity 2. No derivative or talking about turning points needed!

[gabrielbarrantes]

There is a way to non use calculus to complete the exercise and do it using pure algebraic properties...
Just note that Q(x)= P(x)-x has roots at x=1and x=3, this means that Q(x)= (x-1)(x-3)H(x), with H a 2 degree polynomial, then given that P(x)>=x, we see that Q(x)>=0 meaning that H(x) Is a quadratic polynomial that just happens to match the signs of (x-1)(x-3) to be always positive, given that (x-1)(x-3) is negative in ]1, 3[ and positive outside that interval, H is forced to match this while being a polynomial of degree 2, negative in ]1, 3[ and positive outside that interval, meaning that is actually h*(x-1)(x-3) with h>0, now evaluate Q(x) in x=2 to get the value of h and you get the result.
Is less clean but can be technically solved without using calculus tools.

[ue]

Old Turkish student here. This is not the method of solving this type of questions. Because a student is expected solve the hard questions under 2-3 minutes, you cannot apply all the logic to solve the questions step by step. The magic lies in your preparation. High calibre students often solve thousands of (not kidding) questions just for some topics in the exam such as calculus part. So a Turkish student should recognise the pattern of the question and skip few steps. That’s how you ace turkish uni entrance exam :)

[subaykrmas]

I made 40/40 at maths this year and video title make me proud but since i chose medicine faculty i wont need maths for my all life ;(