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★ 8/10

An Angle Between a Line and a Plane After Projection

As shown in the figure, the equilateral triangle $A'BC$ has side length $1$ . Its vertex $A'$ lies in a plane $\alpha$ …

trigonometryExtremaParametrizationsolid geometry

★ 8/10

Triangles from 88 Lines

What is the maximum number of equilateral triangles that can be bounded by $88$ lines in the plane?

combinatoricsplane geometryCountingAM-GM

★ 8/10

A Logarithmic Telescope

Let $a > 0$ and

f(x) = \ln(1 + ax) - \frac{2x}{x + 2}.

1. When $a = \dfrac{1}{2}$ , find the extreme values of $f(x)$ . 2. Discuss the monotonicity of $f(x)$ on $(0, +\infty)$ …

ExtremacalculusMonotonicityTelescoping+1

★ 9/10

A Forced Large Gap

Find the largest positive integer $n$ with the property: however the numbers $1$ through $400$ are placed into a $20 \times 20$ grid, some row or column contains two numbers differ…

combinatoricsPigeonholeAM-GM

★ 8/10

A Logistic-Style Iteration

A sequence satisfies $2a_{n+1} = 1 - a_n^2$ with $0 < a_1 < 1$ .

1. Find the range of $a_3$ . 2. Prove by induction that for $n \ge 3$ ,

\left|a_n - \left(\sqrt2 - 1\right)\right|…

sequencesEstimationInduction

★ 7/10

A Cubic Radical Equation

Solve the equation

x^3-3x=\sqrt{x+2}.

trigonometrySubstitutionalgebraTrigonometric Identities

★ 6/10

Coordinates of a Point in a Shaded Region

As shown, $OM\parallel AB$ , and the point $P$ lies in the shaded region bounded by ray $OM$ , segment $OB$ , and the ex…

plane geometryLinear Programmingvectors

★ 8/10

An Order-Finding Problem

Let $n$ be the least positive integer with

3^3 \cdot 5^5 \cdot 7^7 \;\Big|\; 149^n - 2^n.

How many positive divisors does $n$ have?

number theoryModular Arithmetic

★ 7/10

A Rotating Recurrence Pair

Real sequences $\{a_n\}$ and $\{b_n\}$ (all terms nonzero) satisfy

\begin{cases}a_n=a_{n-1}\cos\theta-b_{n-1}\sin\theta,\\ b_n=a_{n-1}\sin\theta+b_{n-1}\cos\theta,\end{cases}

w…

complex numberstrigonometryGeometric Progressionsequences+1

★ 6/10

A Slope Through the Focus

A parabola $C$ has vertex at the origin $O$ and focus $F$ on the $x$ -axis. A line through the point $(2, 0)$ meets $C$ at two points $P$ and $Q$ with $OP \perp OQ$ , and $M$ is the …

Parametrizationconic sectionsAM-GManalytic geometry+1

★ 6/10

An Absolute-Value Domination

The inequality

ax^2 - |x + 1| + 3a \geqslant 0

holds for all real $x$ . Find the range of possible values of $a$ .

functionsSubstitutionAbsolute ValueAM-GM+1

★ 7/10

A Necessary Probe

The inequality

3\sin^2 x - \cos^2 x + 4a\cos x + a^2 \le 31

holds for all $x \in \mathbb{R}$ . Find the range of $a$ .

Quadratic EquationstrigonometryExtremaSubstitution+1