Problem Bank

Let $A = 5555^{5555}$, let $B$ be the digit sum of $A$, $C$ the digit sum of $B$, and $D$ the digit sum of $C$. Find $D$.

For $0 \le x, y \le 1$, prove that

List all positive roots of the equation $\tan x = x$ in increasing order, and let $r_n$ denote the $n$-th one. Prove that for every positive integer $n$,

In the right triangular prism $ABC$-$A_1B_1C_1$ shown in the figure, $AB=AC=5$, and the quadrilateral $B_1BCC_1$ is a s…

Let $\vec a$ be a given nonzero plane vector. Consider the following claims about decomposing $\vec a$ (in each, the vectors $\vec b,\vec c,\vec a$ lie in one plane and are pairwis…

The hyperbola $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ ($a>0$, $b>0$) has left and right foci $F_1$, $F_2$. The parabola $y^2=2px$ ($p>0$) has focus $F_2$, and meets the hyperbola at …

An ellipse $E$ centered at the origin $O$ with foci on the $x$-axis passes through $A\left(\sqrt{2}, -\dfrac{\sqrt{2}}{2}\right)$ and $B(0, 1)$.

1. Find the standard equation of $…

Let $\theta$ be a constant, and define a sequence of functions by

Acute angles $\alpha, \beta$ satisfy

Find the maximum of $\tan\alpha$.

Let $f(x) = |x - m|$ and $g(x) = x|x - m| + m^2 - 7m$.

1. If the equation $f(x) = |m|$ has two distinct real roots in $[-4, +\infty)$, find the range of possible values of $m$. 2.…

Let $f_1(x)=f(x)=x(x-1)$ and $f_n(x)=f(f_{n-1}(x))$ for $n\geqslant 2$. Prove:

1. For $x>2$, $f_n(x)$ has no zeros. 2. For $1\leqslant x\leqslant 2$, $f_n(x)$ has at least $n$ zer…

Let $\triangle ABC$ be obtuse, with sides $a,b,c$ opposite the angles $A,B,C$. Given $a=2$, $b\sin A=\sqrt 3$, and $c=3$, find $b$.