Problem Bank

Let $O$ be the origin and $F_1,F_2$ the two foci of the ellipse $C:\dfrac{x^2}{4}+y^2=1$. From $F_1$ and $F_2$, pe…

Fill a $3 \times 3$ table with $9$ natural numbers so that the three numbers in every row and in every column sum to $3$. How many different fillings are there?

The ellipse $C: \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ ($a > b > 0$) has eccentricity $\dfrac{\sqrt3}{2}$ and passes through $P(2, -1)$.

1. Find the equation of $C$. 2. Let $Q$ …

Consider the equation

1. Find the largest $n$ for which $E$ has a solution in positive integers $x_1, \dots, x_n$. 2. Call a po…

As shown in Figure 1, the line $y=kx-8k$ intersects the $x$-axis at a point $A$ and intersects the positive $y$-axis at a …

Let $a \neq 0$ be real and

1. If $a = 1$, find the minimum value of $f(x)$. 2. Discuss the monotonicity of $f(x)$. 3. If $f(x)$ has exactly thr…

Consider the ellipse $\dfrac{x^2}5+\dfrac{y^2}4=1$ and the circle $x^2+y^2=1$. Let $P$ be a moving point on the ellipse. A point $A$ on the ellipse is such that the line $AP$ passe…

Let $f(x)=5\cos x-\cos(5x)$.

1. Find the maximum value of $f(x)$ on $\left[0,\dfrac{\pi}4\right]$. 2. Given $\theta\in(0,\pi)$ and a real number $a$, prove that there exists $y\in…

A regular $2020$-gon $A_1A_2\cdots A_{2020}$ is inscribed in the unit circle $O$. Two distinct vertices $A_i, A_j$ are chosen at random.

1. Find the probability that $\overrightar…

Let

and let $g(x)=b-f(2-x)$ with $b\in\mathbb R$. If $y=f(x)-g(x)$ has exactly $4$ zeros, find the range of $b…

Find the area of the region defined by

Let $f(x) = |\sin x|$.

1. Prove that

2. Prove that for every positive integer $n$,