Practice (Intermediate)
back to index | new
Let $\theta\in[0, 2\pi]$ satisfying $$\cos^5\theta -\sin^5\theta < 7(\sin^3\theta -\cos^3\theta)$$
Find the range of $\theta$.
In $\triangle{ABC}$, show that
\begin{align*}
&\sin^2A +\sin^2B+\sin^2C = 2 +2\cos A\cos B \cos C\\
&\cos^2A +\cos^2B + \cos^2C = 1-2\cos A\cos B\cos C
\end{align*}
Compute the values of
$$S=C_n^1\sin\theta + C_n^2\sin 2\theta + \cdots + C_n^n\sin n\theta$$
and
$$C=C_n^1\cos\theta + C_n^2\cos 2\theta + \cdots + C_n^n\cos n\theta$$
$$|\sin x + \cos x + \tan x + \cot x + \sec x + \csc x|$$
where $x$ is a real number.
Solve this inequality $$\frac{x}{\sqrt{x^2 +1}}+\frac{1-x^2}{1+x^2} > 0$$
Compute $\sin 15^\circ$ and $\cos 15^\circ$ using a geometry approach.
Find the range of real number $a$ if the following equation of $x$ is solvable in real number:
$$\sin^2 x + \cos x + a=0$$
Find the number of solutions to the equation $\sin x = \frac{x}{2018}$.
Prove that function $f(x)=\cos\sqrt{x}$ is not a periodical function.
Prove $\tan 20^\circ \tan 40^\circ \tan 60^\circ \tan 80^\circ=3$.
Show that $\cot 70^\circ + 4\cos 70^\circ = \sqrt{3}$.
Prove that $\cos 1^\circ$ is irrational.
In $\triangle{ABC}$, find the measurement of $C$ if
$$3\sin A + 4\cos B = 6\quad\text{and}\quad 4\sin B + 3\cos A = 1$$
Compute the value of $(\sqrt{3}\tan 18^\circ + \tan 18^\circ\tan 12^\circ +\sqrt{3}\tan 12^\circ)$.
Compute the value of $\cos 6^\circ\cos 42^\circ \cos 66^\circ\cos 78^\circ$.
Show that for any positive integer: $$\tan x \tan 2x +\tan 2x \tan 3x +\cdots + \tan(n-1)x\tan nx=\frac{\tan nx}{\tan x}-n$$
Show that $$\tan x + 2\tan 2x + 2^2\tan 2^2x +\cdots + 2^n\tan 2^nx = \cot x - 2^{n+1}\cot 2^{n+1}x$$
Show that $$\frac{1}{\sin 1^\circ\sin 2^\circ}+\frac{1}{\sin 2^\circ\sin 3^\circ}+\cdots+\frac{1}{\sin 89^\circ\sin 90^\circ}=\cos 1^\circ\csc^2 1^\circ$$
Find the remainder when $x^{2017}$ is divided by $(x+1)^2$.
Let $f(x)=2016x - 2015$. Solve this equation $$\underbrace{f(f(f(\cdots f(x))))}_{2017\text{ iterations}}=f(x)$$
In $\triangle{ABC}$, let $\angle{A}=120^\circ$. If $A'$, $B'$ and $C'$ are feet of the three interior angle bisectors as shown, prove $A'B'\perp A'C'$.
Take a list of positive integers $1$, $2$, $3$, $\cdots$, $2017$. At each step, pick up two of the numbers on the list, say $a$ and $b$, cross them out and replace them by the single number $(ab+a+b)$. Keep doing this until only a single number is left. What is (are) the possible value(s) of this last number?
As shown, both $ABCD$ and $OPRQ$ are squares. Additionally, $O$ is the center of $ABCD$, $OP=1$, $BP=\sqrt{2}$, and $CQ=\sqrt{5}$. Find the length of $DR$.
Does the expression $x+\sqrt{2x^2-2x+1}$ has either maximum or minimal value?
In trapezoid $ABCD$, $AD\parallel BC$ and $AD:BC=1:2$. Point $F$ lies on $AB$ and point $E$ is on $CF$. If $S_{\triangle{AOF}}:S_{\triangle{DOE}}=1:3$ and $S_{\triangle{BEF}}=24$, find the area of $\triangle{AOF}$.
