Practice (124)
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Let $f(x)=(x^2+3x+2)^{cos(\pi x)}$. Find the sum of all positive integers $n$ for which \[\left |\sum_{k=1}^nlog_{10}f(k)\right|=1.\]
Find the sum of all the positive solutions of
$2\cos2x \left(\cos2x - \cos{\left( \frac{2014\pi^2}{x} \right) } \right) = \cos4x - 1$
Rectangle $ABCD$ has $AB = 6$ and $BC = 3$. Point $M$ is chosen on side $AB$ so that $\angle AMD = \angle CMD$. What is the degree measure of $\angle AMD$?
Compute the value of $\sin{18^\circ}$ using regular geometry.
Let $F$ be a point inside $\triangle{ABC}$ such that $\angle{CAF} = \angle{FAB} = \angle{FBC} = \angle{FCA}$, show that the lengths of three sides form a geometric sequence.
Alice and Bob live $10$ miles apart. One day Alice looks due north from her house and sees an airplane. At the same time Bob looks due west from his house and sees the same airplane. The angle of elevation of the airplane is $30^\circ$ from Alice's position and $60^\circ$ from Bob's position. Which of the following is closest to the airplane's altitude, in miles?
Let sequence $\{a_n\}$ satisfy the condition: $a_1=\frac{\pi}{6}$ and $a_{n+1}=\arctan(\sec a_n)$, where $n\in Z^+$. There exists a positive integer $m$ such that $\sin{a_1}\cdot\sin{a_2}\cdots\sin{a_m}=\frac{1}{100}$. Find $m$.
Triangle $ABC$ has $AB=40,AC=31,$ and $\sin{A}=\frac{1}{5}$. This triangle is inscribed in rectangle $AQRS$ with $B$ on $\overline{QR}$ and $C$ on $\overline{RS}$. Find the maximum possible area of $AQRS$.
A regular pentagon is inscribed in a unit circle. Find the perimeter of this pentagon.
Two people located 500 yards apart have spotted a hot air balloon. The angle of elevation from one person to the balloon is $67^\circ$. From the second person to the balloon the angle of elevation is $46^\circ$. How high is the balloon when it is spotted? (You can use a calculator. Please keep the result to the 2 decimal places.)
In $\triangle{ABC}$, if $(a^2 +b^2)\sin(A-B)=(a^2-b^2)\sin(A+B)$, determine the shape of $\triangle{ABC}$.
Simplify $$\sin{x} + \sin{2x} + \cdots +\sin{nx}$$ and $$\cos{x} + \cos{2x} + \cdots + \cos{nx}$$
Solve the equation $\cos\theta + \cos 2\theta + \cos 3\theta = \sin \theta +\sin 2\theta + \sin 3\theta$.
Find the value of $\cos 20^\circ \cos 40^\circ \cos 80^\circ$ using at least two difference methods.
Simplofy $\sin\theta + \frac{1}{2}\cdot\sin 2\theta + \frac{1}{4}\cdot\sin 3\theta + \cdots$.
Solve the equation $\cos^2 x + \cos^2 2x +\cos^2 3x=1$ in $(0, 2\pi)$.
Prove for every positive integer $n$ and real number $x\ne \frac{k\pi}{2^t}$ where $t =0, 1, 2,\cdots$ and $k$ is an integer, the following relation always holds:
$$\frac{1}{\sin 2x}+\frac{1}{\sin 4x} + \cdots +\frac{1}{\sin 2^nx}=\frac{1}{\tan x}-\frac{1}{\tan 2^nx}$$
Without using a calculator, find the value of $\cos\frac{\pi}{13}+\cos\frac{3\pi}{13}+\cos\frac{9\pi}{13}$.
Compute $\cot 70^\circ + 4\cos 70^\circ$
Compute $4\cos\frac{2\pi}{7}\cos\frac{\pi}{7}-2\cos\frac{2\pi}{7}$