Source: Grade 10 Selection Test for Gifted Math Students, Academic Year 2016 - 2017, High School for the Talented, HCMC, Vietnam.
Here I post the entire test. Among of them are some problems which I couldn't solve when doing the test (so hard!). Try to solve if you are interested.
1. Type-2 Symmetric System of Equations:
Given this system ( are variables, is a parameter).
Solve the system when .
Find so that the system has at least 1 solution satisfy and .
2. Ugly Vieta's in Quadratic Equation:
Find so that the equation has 2 different solutions satisfy .
3. Vieta Jumping, Where Are You?
Let be positive integers satisfy .
Prove that are coprime odds.
Let . Prove that and .
Bonus (not in the test): How many different values of are there?
Let be reals satisfy and .
Find the maximum value of .
5. Geometry - Welcome to Russia:
Acute triangle with . Draw squares outside the triangle . intersects at . intersects at .
Prove that triangle and triangle are similar. Prove that is concylic.
Let be midpoint of segment . Prove that is the circumcenter of triangle .
intersects at . and intersect at . Tangents of at and intersect at . Prove that are colinear.
Clarification: denotes the circumcircle of the figure.
6. Integral Sequence About Factorization:
For each positive integer greater than , let be the second greatest factor of .
For example, for any prime .
Consider the following sequence .
Prove that for any given value of , there exists an integer such that .
(Is this too large?) Find if .