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Completing The Square

Learn how to eliminate the linear term and see through messy quadratics. You’ll know just how high to shoot your three-pointer to avoid tall defenders.

Level 4

         

The numbers \( a\) and \(b \) are positive integers both greater than 5. The function \[ f(x) = \cos ^2 x - a \sin x + b \] has a maximum of \(34\) and a minimum of \(12\). What is the value of the product \( a \times b\)?

The Queen plans to build a number of new cities in the wilderness, connected by a network of roads. She expects the annual tax revenues from each city to be numerically equal to the square of the population. Road maintenance will be expensive, though. The annual cost for each road is expected to be numerically equal to the product of the populations of the two cities that road connects. To keep the costs for road maintenance down, the Queen decrees that there should be only one way to get from any city to any other, and not more than three roads should connect to any given city (the attached design would be acceptable, for example). What is the largest number of cities the Queen can build without risking to lose money, regardless of the populations of the new cities and the design of the roads (subject to the two constraints mentioned above)?

Inspiration

\[P(x)=(x^2+5000x-1)^2+(2x+5000)^2\]

Let \(P(x)\) be a polynomial, then find the sum of all the real solutions to the equation \[P(x)=\text{min}(P(x)).\]

Image Credit: Wikimedia Five Thousand

If \(x, y, z \in\mathbb{R}\)

\[x^2+11z=-40.25\] \[y^2-9x=-28.25\] \[z^2+7y=5.75\]

Then the value of \[|x|+|y|+|z|\] is


Note

You can find more such problems here

How many values of \(\lambda\) lie between \([-1000\pi,1000\pi]\) for which the polynomial \[x^4-2^{\tan\lambda}.x^2+(\cos\lambda+\cos2\lambda).x+2^{\tan\lambda-2}\] is the square of the quadratic trinomial with respect to \(x\) ?

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