Problem Bank

Let $P$ be a point of the ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ (with $a > b > 0$), and let $A, B$ move along the two asymptotes of the hyperbola $\dfrac{x^2}{a^2} - \d…

A sequence $\{x_n\}$ has $x_1 = 1$ and

Find $\displaystyle\sum_{n=1}^{2008} x_n$.

In the coordinate plane, the L-distance between points $P_1(x_1,y_1)$ and $P_2(x_2,y_2)$ is defined as

Let $F_1$, $F_2$ be two distinct fixed …

Let $f(x) = x^2 + 2x + 1$. Suppose there is a real $t$ such that

Find the maximum possible value of the real number …

Eleven students volunteer as museum guides. From 9 am to 5 pm there is one talk each hour ($8$ talks in total), and each talk needs exactly one student speaker. To avoid fatigue, n…

In $\triangle ABC$, $\angle A=60^\circ$, and $\angle BAP=\angle CAP$ with $P$ inside the triangle. The extension of $BP$ meets $AC$ at $Q$, and

From an arbitrary point $P$ of the line $l \colon x + y = 2$, two tangent lines are drawn to the circle $C \colon x^2 + y^2 = 1$, touching it at $A$ and $B$. Let $Q$ be the midpoin…

Let

Arrange $a, b, c$ in increasing order, with proof.

A line $l$ through $P(0, 2)$ meets the ellipse $\dfrac{x^2}{5} + y^2 = 1$ at points $M$ and $N$, and $\overrightarrow{PM} = \lambda\overrightarrow{PN}$. Find the range of possible …

Let $f(x) = \dfrac{e^x - 1}{x}$.

1. Find the intervals of monotonicity of $f(x)$. 2. Suppose

and let $\lambd…

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

Let $f(x)=x^3+ax^2+b$ ($a,b\in\mathbb R$).

1. Discuss the monotonicity of $f(x)$. 2. Suppose $b=c-a$, where $c$ is a constant independent of $a$. If the set of values of $a$ for w…