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Three Values Bound a Quadratic

The quadratic $f(x) = ax^2 + bx + c$ ( $a \ne 0$ ) satisfies

|f(0)| \le 2, \qquad |f(2)| \le 2, \qquad |f(-2)| \le 2.

Find the maximum possible value of $|f(x)|$ over $x \in [-…

functionsQuadratic EquationsAbsolute ValueCompleting the Square+1

Escaping Past $m$

A recursive sequence $\{x_n\}$ satisfies

x_{n+1} = x_n^2 - 4x_n, \quad n = 1, 2, \dots

Suppose that for every first term $x_1 \in \mathbb{R}$ with $x_1 \neq 0$ , the sequence …

Quadratic EquationsRecursionalgebrasequences+1

Multiplicative on Squares

A sequence $\{a_n\}$ of positive integers satisfies, for all positive integers $m, k$ ,

a_{m^2} = a_m^2, \qquad a_{m^2 + k^2} = a_m a_k.

Find a closed formula for $a_n$ .

functionsnumber theorysequencesInduction+2

Four Roots in Geometric Progression

Let $a, b \in \mathbb{R}$ . The four real roots of

\left(x^2 - ax + 1\right)\left(x^2 - bx + 1\right) = 0

form a geometric progression with common ratio $q \in \left[\tfrac13,…

Geometric ProgressionSubstitutionalgebrasequences+1

Nested Region Inclusions

Consider the three point sets in the plane

M=\{(x,y)\mid |x|+|y|<1\},

Set TheoryAbsolute Valueconic sectionsanalytic geometry

An Arctangent Mean Bound

For $0 < a < b$ , prove that

\frac{\arctan b - \arctan a}{b - a} > \frac{1}{\sqrt{1+a^2}\cdot\sqrt{1+b^2}}.

trigonometrycalculusSubstitutionMonotonicity+1

A Strict Symmetric Bound

Reals $a, b, c$ satisfy $a + b + c = 1$ and $abc > 0$ . Prove that

ab + bc + ca < \frac{\sqrt{abc}}{2} + \frac14.

algebraAM-GMinequalitySymmetry

A Law-of-Cosines System

Let $x,y,z$ be positive numbers satisfying

\begin{cases}x^2+y^2+xy=3,\\ y^2+z^2+yz=4,\\ z^2+x^2+zx=7.\end{cases}

Find $x+y+z$ .

SubstitutionalgebraSymmetry

Iterated Quadratic Roots

Let $P(x)$ be a monic quadratic with complex coefficients. The equation $P(P(x)) = 0$ has four distinct solutions $x = 3, 4, a, b$ . Find the sum of all possible values of $(a+b)^2$ …

complex numbersalgebraPolynomialsCasework+1

A Positive Gap

Let $f(x) = \mathrm{e}^x\left(x\ln x + \dfrac{2}{\mathrm{e}}\right)$ .

1. Find the intervals on which $h(x) = f(x) - \mathrm{e}^x\left(1 + \dfrac{2}{\mathrm{e}}\right)$ is monotoni…

ExtremacalculusMonotonicityinequality

Forcing a Right Angle

The angles of a non-obtuse triangle $ABC$ satisfy

\cos^2 A + \cos^2 B = \sin C.

Prove that $C$ is a right angle.

trigonometryTrigonometric IdentitiesTriangle Geometry

A Cubic and Its Chord

Let $f(x) = \dfrac{1}{3}x\left(x^2 + 3x + a\right)$ .

1. Find the intervals on which $f(x)$ is monotonic, discussing all cases. 2. Suppose $f(x)$ has two extreme points $x_1, x_2$ ,…

functionscalculusMonotonicityPolynomials+2

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