Problem Bank

In triangle $ABE$, points $C$ and $D$ lie on segment $BE$ with $BE = 3$, $\angle BAC = \angle EAD$, and

When $\angle AEB$ attai…

A regular tetrahedron can contain $10$ balls of radius $1$. Find the minimum possible edge length of the tetrahedron.

The ellipse $C:\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ ($a>b>0$) has eccentricity $\dfrac{\sqrt 2}2$ and major axis of length $4$.

1. Find the equation of $C$. 2. A line $l$ through …

Find the number of positive integers $n\leqslant 2015$ such that the units digit of $1^n+2^n+3^n+4^n$ is $0$.

A Cassini oval is the locus of points whose distances to two fixed points have constant product. In the coordinate plane, let $M(-2, 0)$ and $N(2, 0)$, and let $P$ satisfy

In a cube $ABCD\text{-}A_1B_1C_1D_1$, let $P$ be a moving point on the four sides of the square $A_1B_1C_1D_1$, $O$ the center of the square $ABCD$, and $M,N$ the midpoints of $AB,…

In an acute triangle $ABC$ with $A = \dfrac{\pi}{6}$, let $H$ be the orthocenter, and suppose $AH = \sqrt{3}$. Find the range of possible values of

A sequence $a_1a_2\cdots a_n\cdots$ with $a_i \in \{0, 1\}$ is a $0$-$1$ periodic sequence if there is a positive integer $m$ with $a_{i+m} = a_i$ for all $i$; the least such $m$…

Let

1. Give one value of $a$ for which $f(x)$ has no maximum value. 2. Find the range of value…

The hyperbola $C:\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ ($a,b>0$) has eccentricity $2$ and left and right foci $F_1$, $F_2$. A line $l$ through $F_2$ meets the right branch of $C$ at…

Triangle $ABC$ has side lengths $AB=120$, $BC=220$, $AC=180$. Lines $\ell_A$, $\ell_B$, $\ell_C$ are parallel to $BC$, $AC$, $AB$ respectively, and the segments cut by $\triangle A…

In $\triangle ABC$, $AC = 5$ and

Find $BC + AB$.