Problem Bank

The equation

has exactly one real solution in $(0, +\infty)$. Find the range of $a$.

Unit vectors $\vec e_1, \vec e_2$ make an angle of $120^\circ$, and reals $x, y$ satisfy

Find the range of $\left|x\vec e_1 - y\v…

Consider the sequence $\{a_n\}$:

in which each positive odd number $k$ appears exactly $k$ times. Do there exist integers $r, s, t$ suc…

Three identical balls of radius $r$ cm rest inside a hemispherical bowl of radius $6$ cm: each ball is tangent to the bow…

In $\triangle ABC$, the centroid $G$ satisfies $AG \perp BG$, and $AB = 2$.

1. Find the minimum value of $\dfrac{1}{\tan A} + \dfrac{1}{\tan B}$. 2. Prove that $\dfrac{\tan C}{\ta…

Let $P(a, b)$ be any point on the ellipse $\dfrac{x^2}{3} + \dfrac{y^2}{2} = 1$. From the origin $O$, two tangent lines to the circle

The function $f(x) = ax^3 + bx^2 + cx + d$ has local extrema at the two points $O$ and $A$, where $O$ is the origin and $A$ lies on the curve

Let $abc \neq 0$ and $a + b + c = 0$. Find the value of

Positive reals $x, y$ satisfy

Find the maximum value of

In the triangular prism $ABC\text{-}A_1B_1C_1$, the lateral face $ABB_1A_1$ is perpendicular to the base $ABC$, wit…

A function $f$ satisfies, for all $x, y \in \mathbb{R}$,

Write down one such function $f(x)$.

Let $O$ be the center of an ellipse, and $A$ a point on the ellipse other than a vertex. Drop the perpendicular from $A$ to the major axis, with foot $M$. Extend $AO$ to meet the e…