Toughest Question of World's Most Challenging GAOKAO Exam

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.