Problem Bank

Let $f(x)=a\sin x+b\cos x$, where $a,b\in\mathbb N$, $a>2$, and

Find the number of real solutions in $(0,30)$ of

Real numbers $x, y$ satisfy

Find the range of possible values of $x$.

Consider the parabola $C \colon y^2 = 2px$ (with $p > 0$). Through the point $P(2, 1)$, lines $l_1$ and $l_2$ with slopes $k_1$ and $k_2$ meet the parabola at $A, B$ and $M, N$ res…

Let $z_1, z_2, \dots, z_{673}$ be distinct complex numbers such that

where $g(x)$ is a p…

Let $A$ and $B$ be two points on the ellipse $\dfrac{x^2}4+\dfrac{y^2}2=1$, symmetric about the origin $O$, and let $P$ be a point on the ellipse $\dfrac{x^2}{12}+\dfrac{y^2}6=1$. …

In a pyramid $P\text{-}ABCD$, $AB=AD=\sqrt{10}$, $CB=CD=5$, $\angle BAD=90^\circ$, $PB=4$, $PC=3$. A point $Q$ inside $\triangle PBC$ is such that the pyramid $Q\text{-}ABCD$ and t…

Prove that the sequence

converges.

A die is rolled five times. Find the probability of the event: the five results are neither all equal nor all distinct, and from the second roll on, each result is at least the pre…

Prove that $2\mathrm e^x>x^3+3x$.

Let $f(x) = \left(1 + ax^2\right)\mathrm{e}^x - 1$.

1. For $a \geqslant 0$, discuss the monotonicity of $f(x)$. 2. Find the number of zeros of $f(x)$ on $[0, 1]$.

Let

The equation $f^2(x) - 3a|f(x)| + 2a^2 = 0$ has exactly …

In the coordinate plane, consider the ellipse $\Gamma \colon \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ (with $a > b > 0$). Let $A$ be an endpoint of the major axis, $B$ an endpoint …