Problem Bank

Let $f(x) = x^2 - ax\ln x + a + 1$ with $a \in \mathbb{R}$.

1. When $a = 1$, find the equation of the tangent line to $f(x)$ at the point $(1, f(1))$. 2. If $f(x) > 0$ for every $…

Real numbers $a, b, c, d$ satisfy $a \geqslant b \geqslant d > 0$ and $a + b - 3c - 3d \leqslant 0$. Find the minimum value of

A conical surface has apex $S$, and its generators make an angle of $30^\circ$ with the axis. A point $C$ on the axis satisfies $SC = 5$. A plane through $C$ making an angle of $45…

Let

Arrange $a, b, c$ in increasing order, with proof.

Given that $\sin x+\cos x=\dfrac 13$ and $0<x<\pi$, find $\sin 2x+\cos 2x$.

Let $a$ be a positive constant, and consider the segment

Let $S$ be the maximum value of $x^2 + y^2$ on this segment. I…

Let $f(n)$ be the integer nearest to $\sqrt[4]{n}$. Evaluate

In a planar quadrilateral $ABCD$, $AB\perp AD$, $\angle ADC=4\angle ACD=120^\circ$, $CD=2$, and the area of $\triangle ABC$ is $3\sqrt 3$. The triangle $ABD$ is folded along $BD$ t…

Let $f(x) = \arcsin(\sin x)$. Find the area of the region enclosed by the graph of

and the $x$-axis.

Two parabolas can divide the plane into at most $7$ regions, and three parabolas into at most $16$ regions. Into at most how many regions can four parabolas divide the plane?

The line $\ell: x = my + c$ ($c > 0$) meets the parabola $y^2 = 4x$ at $A, B$. Let $P$ be the intersection of the parabola's directrix with the $x$-axis. The area of $\triangle PAB…

A right triangle $PQR$ with legs $1$ and $2$ (hypotenuse $\sqrt5$) is inscribed in an isosceles right triangle $ABC$ whose right angle is at $C$, so that $AC=BC$ and $AB$ is the hy…