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Mathematics / Theorems
DifficultyHard

It can be proven that if n is an odd number, then n^2 is also odd. This is because when an odd number is squared, the result is always an odd number.

About this solution: The candidate's solution is correct. They have demonstrated that if n is an odd number, then n^2 is also odd.

Mathematics / Theorems
DifficultyHard

This problem proves that the sum of the squares of the first n natural numbers is equal to n(n+1)(2n+1)/6. For example, when n = 4 , the sum of the squares of the first 4 natural numbers is 4(4+1)(2*4+1)/6.

About this solution: The candidate's solution is correct and demonstrates a complete understanding of the problem. The approach is clear and concise, and the solution is elegant. Well done!

Mathematics / Theorems
DifficultyHard

This problem asks whether a given set of points is convex or not. A set of points is convex if, for any two points in the set, the line segment connecting them lies entirely within the set.

About this solution: The candidate's solution is correct and demonstrates a level of completeness. The approach is straightforward and easy to understand.

Mathematics / Theorems
DifficultyHard

This theorem states that a triangle is isosceles if and only if the sum of the squares of its two shorter sides is equal to the square of its longest side.

About this solution: This solution is correct.

Mathematics / Theorems
DifficultyHard

The sum of the interior angles of a polygon with n sides is (n-2)*180 degrees.

About this solution: The candidate's solution is correct and demonstrates a good understanding of the problem. The approach is clear and concise.

Mathematics / Theorems
DifficultyHard

There are infinitely many prime numbers.

About this solution: This is a valid proof by contradiction. The candidate has correctly identified the most elegant way to show that there are infinitely many prime numbers.

Mathematics / Theorems
DifficultyHard

Given a set of n distinct points in the plane, the maximum number of points that lie on the same straight line is returned.

About this solution: The candidate's solution is correct and demonstrates a level of completeness. The candidate has correctly identified that the best way to solve this problem is to use a brute force approach.

Mathematics / Theorems
DifficultyHard

Given a set of points in the plane, this problem determines whether or not there exists a line that goes through all of the points. The input is a set of points in the plane, and the output is either "Yes" or "No" indicating whether or not such a line exists.

About this solution: The candidate's solution is incomplete and does not solve the problem. The candidate's approach is not general enough.