Problem Bank

Let $a_1, a_2, \ldots, a_9$ be any permutation of $1, 2, \ldots, 9$. Find the minimum value of

How many planes pass through at least $3$ of the midpoints of the edges of a cube?

Let $f(x) = \ln(x+1) + \sqrt{x+1} + ax + b$, where $a, b \in \mathbb{R}$ are constants, and suppose the curve $y = f(x)$ is tangent to the line $y = \dfrac{3}{2}x$ at the point $(0…

The ellipse $C:\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ ($a>b>0$) has left and right foci $F_1$, $F_2$; the point $M\left(1,\dfrac 32\right)$ lies on $C$, and in $\triangle F_1MF_2$, $…

The ellipse $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ ($a>b>0$) has right focus $F$ and upper vertex $B$, with eccentricity $\dfrac{2\sqrt 5}5$ and $|BF|=\sqrt 5$.

1. Find the equatio…

Let $A(2,0)$ be a fixed point in the plane, and let

be a moving poin…

Let $f(x) = kx^2 - kx$ and

Suppose the real number $k$ making $f(x) \geqslant g(…

The axial cross-section of an industrial groove is part of the hyperbola $y^2-x^2=1$ (with $y\in[1,10]$). A spherical cleaning ball is placed in the groove; it must be able to reac…

Solve over the reals:

In $\triangle ABC$, the sides opposite $A, B, C$ are $a, b, c$, and $a + c = 2b$. Prove that

Let $A$ and $B$ be the left and right vertices of the ellipse $C \colon \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ (with $a > b > 0$). The minor axis of $C$ has the same length as it…

Let $f(x) = x\ln x - ax^2 + a$ with $a > 0$.

1. If $f(x) < 0$ for all $x > 1$, find the range of possible values of $a$. 2. For integers $n \geqslant 2$, prove that