Problem Bank

If the constant term in the expansion of

equals $\dfrac{35}{2}$, find the positive integer $n$.

Four boys and three girls stand in a circle in random order. Find the probability that no two of the three girls are adjacent.

The ellipse $C \colon \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ (with $a > b > 0$) has left and right foci $F_1, F_2$ and eccentricity $e$. A point $P$ on the ellipse is such that t…

The ellipse $C: \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ ($a > b > 0$) has major axis of length $4$, and the vertical chord through its left focus has length $1$.

1. Find the equa…

Let $a\in\mathbb R$. If

for all $x>0$, find $a$.

The ellipse $C:\dfrac{y^2}{a^2}+\dfrac{x^2}{b^2}=1$ ($a>b>0$) has eccentricity $\dfrac{\sqrt 3}2$ and minor axis of length $2$; let $P$ be its right vertex. Consider the circle $Q:…

Real numbers $x,y$ satisfy

Find the range of $x$.

Suppose $|2x-a|\geqslant x-1$ for all real $x\in[0,2]$. Find the range of $a$.

Discuss the number of roots of the equation

in terms of the real parameter $k$.

Let $a > 0$, $A(2a, 0)$, $B(0, 2)$, and let $O$ be the origin.

1. Find the standard equation of the circle that has $OA$ as a chord and is tangent to line $AB$ at $A$. 2. This cir…

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

Find $x+y+z$.

The ellipse $M \colon \dfrac{x^2}{a^2} + \dfrac{y^2}{3} = 1$ (with $a > \sqrt{3}$) has right focus $F$, and the line $y = \dfrac{3}{\sqrt{7}}$ meets $M$ at $P$ and $Q$ with $PF \pe…