Sure! Here's the same translation without LaTeX:


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The graph of the quadratic polynomial with real coefficients y = px^2 + qx + r, called G1, intersects the graph of the polynomial y = x^2, called G2, at points A and B. The lines tangent to G2 at points A and B intersect at point C. It is known that point C lies on G1. What is the value of p?

Let Find the value of

I have noticed thousands upon thousands of inequalities that you have posted to HSO and was wondering where you get the inspiration, imagination, and even the validation that such inequalities are true? Also, what do you find particularly appealing and important about specifically inequalities rather than other branches of mathematics? Thank you :)

So a while ago, I compiled some tricks on inequalities. You are welcome to post solutions below!

The point is the midpoint of the segment connecting the orthocentre of the scalene triangle and the point . The incircle of triangle incircle is tangent to and at points and respectively. Prove that point , the point symmetric to point with respect to line , lies on the line that passes through both the circumcentre and the incentre of triangle .

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Note that for an odd natural number Prove that divides for all .

Sheldon was really annoying Leonard. So to keep him quiet, Leonard decided to do something. He gave Sheldon the following grid



and asked him to transform it to the new grid below



by only applying the following algorithm:

At each step, Sheldon must choose either two rows or two columns.

For two columns , if are entries in respectively, then we say that and are corresponding if they belong to the same row. Similarly we define corresponding entries of two rows. So for Sheldon's choice, if two corresponding entries have the same parity, he should do nothing to them, but if they have different parities, he should add 1 to both of them.

Leonard hoped this would keep Sheldon occupied for some time, but Sheldon immediately said, "But this is impossible!". Was Sheldon right? Justify.

Let such that Prove thatWhere
Let such that Prove that

Find all functions such that for all

Let are positive real numbers. Prove that