Toughest Question of World's Most Challenging GAOKAO Exam
This transcript explains a solution to one of the toughest questions from China's GAOKAO exam, described as 10 times harder than India's IIT JEE and UPSC. The speaker substitutes a variable k to simplify logarithmic expressions for x, y, and z, then uses graphical analysis to determine which ordering relationship among x, y, z is impossible. The conclusion is that x > z > y is the only relationship that can never occur.
Summary
The speaker introduces the GAOKAO, China's notoriously difficult college entrance exam, contextualizing its difficulty by noting that 15 million Chinese students appear for it, but only 10,000 seats exist at top universities — making it roughly 10 times harder than India's IIT JEE and UPSC combined. The video focuses on the most difficult level of question from this exam.
The core mathematical problem involves three variables x, y, and z defined through logarithmic equations. To simplify, the speaker sets the expression '2 + log base 2 of x' equal to a common variable k. From this substitution, x becomes 2^(k-2), y becomes 3^(k-3), and z becomes 5^(k-5).
The speaker then analyzes how the relative sizes of x, y, and z change depending on the value of k. For small values of k (such as k=1), the ordering is x > y > z. As k grows very large (e.g., k=1000), z dominates because of its larger base, followed by y, then x.
To make this analysis rigorous, the speaker uses a graphical approach, plotting x, y, and z as functions of k. Each variable remains below 1 until k reaches its respective base value (2 for x, 3 for y, 5 for z), after which it grows exponentially. By dividing the k-axis into segments — before k=5, between k=3 and k=5, between k=2 and k=3, and beyond k=5 — the speaker identifies all possible orderings: x>y>z, y>x>z, y>z>x, and z>y>x. The ordering x>z>y (option D) never appears in any segment, making it the impossible relationship.
Key Insights
- The speaker claims GAOKAO is approximately 10 times harder than India's IIT JEE and UPSC, with 15 million students competing for only 10,000 seats at top universities.
- The speaker substitutes '2 + log base 2 of x = k' to unify all three variables into a single parameter, yielding x = 2^(k-2), y = 3^(k-3), and z = 5^(k-5).
- The speaker argues that the relative ordering of x, y, and z is entirely dependent on the value of k — for small k, x dominates, while for very large k (like 1000), z dominates due to its larger base.
- Through graphical analysis, the speaker identifies four distinct segments of k that produce four different valid orderings: x>y>z, y>x>z, y>z>x, and z>y>x.
- The speaker concludes that the ordering x > z > y (option D) is the only relationship that is impossible and never occurs across any segment of k values.
Topics
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