The roots of the polynomial
x³ - 2x² - 11x + k
are r₁, r₂, and r₃.

Given that
r₁ + 2r₂ + 3r₃ = 0,
what is the product of all possible values of k?

In triangle ABC, the lengths |AB|, |BC|, and |CA| are proportional to 4, 5, and 6, respectively. Points D and E lie on segment [BC] such that the angles ∠BAD, ∠DAE, and ∠EAC are all equal. What is the measure of angle ∠AEB in degrees?

fair coins are flipped. If of them land heads, find the expected value of .

Let ABC be a triangle with orthocentre H. The tangent lines from A to the circle with diameter BC touch this circle at P and Q. Prove that H, P and Q are collinear.

On the sides AB and AD of the rectangle ABCD, points M and N are taken such that MB = ND. Let P be the intersection of BN and CD, and Q be the intersection of DM and CB. How can we prove that PQ || MN?
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Let and .Prove that
Let and .Prove that

"[(x - 2)^2 + 4] * (x + (1/x)) = 10. What is the sum of the elements in the solution set of this equation?

Let a be a root of the equation x^3-x-1=0 , with a>1
What is the value of the expression:
∛(3a^2 - 4a) + ∛(3a^2 + 4a + 2)?

Let Prove that
https://artofproblemsolving.com/community/c6h107534p608181:

Let a, b, c be positive real numbers. Prove the inequality:

sqrt(a² - ab + b²) + sqrt(b² - bc + c²) ≥ sqrt(a² + ac + c²)