Problem Bank

Let $A, B, C$ be the interior angles of a triangle. Find the minimum value of

Let $a_1, a_2, \ldots, a_8$ be $8$ positive integers, and let

be…

Observe:

…

The ellipse $C:\dfrac{x^2}9+\dfrac{y^2}3=1$ has left vertex $A$. A line $l$ through the point $B(1,0)$ meets $C$ at two points $P$ and $Q$. Let $N$ be the circumscribed circle of $…

A sequence $\{a_n\}$ is periodic with period $T$ if there is a positive integer $T$ with $a_{n+T}=a_n$ for all $n$. Let $\{a_n\}$ satisfy $a_1=m$ ($m>0$) and

Consider the circle $C \colon x^2 + y^2 = 2$ and the line $l \colon x + 2y - 4 = 0$, and let $P(x_0, y_0)$ be a point on $l$. If there exists a point $Q$ on the circle $C$ such tha…

Let $f(x)=|x+1|+|ax-2|$ ($a>0$) with domain $\mathbb R$, and let $M(a)$ be its minimum value. Determine, with proof, which of the following claims are true.

1. The minimum value o…

In a cube $ABCD\text{-}A_1B_1C_1D_1$ with edge length $3$, let $E$ and $F$ be the centroids of $\triangle A_1DB$ and $\triangle A_1DC_1$ respectively. Find the length of the chord …

Let $n \in \mathbb{N}^*$. Prove that

Let $a, b, c > 0$ with $a + b + c = 3$. Prove that

Let $P(x_0, y_0)$ be a point inside the ellipse $E \colon \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ (with $a > b > 0$). A line $l$ through $P$ meets the ellipse at $A$ and $B$, and …

Let

Find the units digit of $\left\lfloor a_{2017} \right\rfloor$.