Problem Bank

Solve the equation

Consider the cubic equation $x^3+ax+b=0$, where $a,b$ are real. Determine, with proof, which of the following conditions guarantee that the equation has exactly one real root.

1. …

Let $O$ be the origin and let $F$ be the focus of the parabola $C_1:\ x^2=2py$ (with $p>0$). The tangent line to $C…

The figure shows the planar unfolding (net) of a tetrahedron $P$-$ABC$, in which $D$, $E$, and $F$ mark the positions tak…

An infinite sequence $P \colon a_1, a_2, \ldots$ satisfies $a_i \in \mathbb{N}^*$ and $a_i \leqslant a_{i+1}$. For $k \in \mathbb{N}^*$, let $T_k(P)$ be the smallest element of ${…

Discrete random variables $X, Y$ satisfy $2 \leqslant X \leqslant 3$ and $XY = 1$. Find the range of possible values of $E(X)E(Y)$.

In a trapezoid $ABCD$, $\overrightarrow{BC}=2\overrightarrow{AD}$ and $AB=AD=CD$. A point $P$ in the plane satisfies $\overrightarrow{PB}\cdot\overrightarrow{PC}=0$ and

Let $f(x) = (x-2)\mathrm{e}^x + a(x-1)^2$.

1. Discuss the monotonicity of $f(x)$. 2. If $f(x)$ has two zeros, find the range of possible values of the real number $a$.

A quadrilateral inscribed in the unit circle has perpendicular diagonals. Find the sum of the squares of its four side lengths.

An equilateral triangle is divided by line segments into $9$ congruent small equilateral triangles, two of which are labeled $P$ …

In triangle $ABC$, the sides opposite angles $A, B, C$ are $a, b, c$ respectively. Given that

and the area of triangle $ABC$ is $\d…

In $\triangle ABC$, a point $D$ on $BC$ is such that $AD$ bisects $\angle BAC$, and the incenter of $\triangle ADC$ coincides with the circumcenter of $\triangle ABC$. Find $\angle…