Problem Bank

Let

Determine, with proof, which of the following are true:

1. if $n = 2k$ ($k \in \mathbb{N}^*$), then $f(x)$ is even; 2. if $n = 2$ and $y = f(…

Let $H$ be the orthocenter of $\triangle ABC$, and

Find $\cos\angle AHB$.

A sequence $A \colon a_0, a_1, a_2, \dots, a_{20}$ satisfies $a_0 = 0$ and

For each of the values $0$, $2$, $10$, $12$, det…

Let $f(x) = \dfrac{a}{2}\ln(x+1) - \sqrt{x+2}$, where $a \in \mathbb{R}$.

1. When $a = \dfrac{8}{3}$, find the intervals on which $f(x)$ is monotonic. 2. If for all $x \geqslant 0…

Let $a, b$ be positive numbers and $\mathrm{e}$ the base of the natural logarithm. If

determine the relationship between $b$ and $\mathrm{e}^{…

Let $\{a_n\}$ be an arithmetic sequence with nonzero common difference and $a_5 = 6$, and let $\{b_n\}$ satisfy $b_1 = 3$ and

1. For $n \geqs…

Positive numbers $x, y$ satisfy

Find the maximum possible value of $y$.

A sample consists of the data $0, 1, 2, 3, 4$. A random value $X$ with

is appended to form a new sampl…

In square $ABCD$, $E$ is an arbitrary point on side $CD$. Draw $EF\perp AC$ at $F$, and let $BF$ extended meet the …

Let $M = \{t : t = m^2 - n^2,\ m, n \in \mathbb{Z}\}$.

1. Prove that the product of any two elements of $M$ is in $M$. 2. Are $32, 33, 34$ in $M$? Why? 3. Give (with proof) a nece…

A finite increasing sequence $a_1 < a_2 < \cdots < a_n$ ($n \ge 3$) has the property that for any indices $i < j < k$, at least one of the products $a_ia_j$, $a_ja_k$, $a_ia_k$ is …

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…