MHB Minimum of $a$ for $a^2+2b^2+c^2+ab-bc-ac=1$

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The problem involves finding the minimum value of the variable $a$ given the equation $a^2 + 2b^2 + c^2 + ab - bc - ac = 1$. Participants discuss various approaches to manipulate the equation and apply optimization techniques. The hint suggests considering specific values or constraints for $b$ and $c$ to simplify the problem. Several mathematical strategies, including completing the square and using Lagrange multipliers, are proposed to derive the minimum. Ultimately, the goal is to establish the conditions under which $a$ reaches its minimum while satisfying the given equation.
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Let $a,\,b,\,c$ be real numbers such that $a^2+2b^2+c^2+ab-bc-ac=1$. Find the minimum possible value of $a$.
 
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Hint:

Discriminant of quadratic equation helps...:)
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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