Practice (Intermediate)
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Solve in positive integers the equation $$3(xy+yz+zx)=4xyz$$
If the circle \(x^2 + y^2 = k^2\) covers at least one maximum and one minimal of the curve \(f(x)=\sqrt{3}\sin\frac{\pi x}{k}\), find the range of \(k\).
Prove the following identities
\begin{align}
\sin (3\alpha) &= 4\cdot \sin(60-\alpha)\cdot \sin\alpha\cdot \sin(60+\alpha)\\
\cos (3\alpha) &= 4 \cdot\cos(60-\alpha)\cdot \cos\alpha\cdot \cos(60+\alpha)\\
\tan (3\alpha) &= \tan(60-\alpha) \cdot\tan\alpha \cdot\tan(60+\alpha)
\end{align}
Compute $$\sin^410^{\circ} +\sin^450^{\circ}+\sin^470^\circ$$
Let $A (x_1, y_1)$, $B (x_2, y_2)$, and $C (x_3, y_3)$ be three points on the unit circle, and $$x_1 + x_2 + x_3 = y_1+y_2+y_3=0$$ Prove $$x_1^2 +x_2^2+x_3^2=y_1^2+y_2^2+y_3^2=\frac{3}{2}$$
How many among the first $1000$ Fibonacci numbers are multiples of $11$?
Compute $(1+\tan 1^\circ)(1+\tan 2^\circ)\cdots(1+\tan 44^\circ)(1+\tan 45^\circ)$
Compute $$\cos\frac{\pi}{2n+1}\cdot\cos\frac{2\pi}{2n+1}\cdots\cos\frac{n\pi}{2n+1}$$
Compute $$\sin^2 20^\circ -\sin 5^\circ (\sin 5^\circ +\frac{\sqrt{6}-\sqrt{2}}{2}\cos 20^\circ)$$
Let $\alpha, \beta \in (0, \frac{\pi}{2})$. Show that $\alpha + \beta = \frac{\pi}{2}$ if and only if $$\frac{\sin^4 \alpha}{\cos^2 \beta} + \frac{\cos^4\alpha}{\sin^2\beta} = 1$$
If $\sin\alpha + \sin\beta = \frac{3}{5}$ and $\cos\alpha+\cos\beta=\frac{4}{5}$, compute $\cos(\alpha -\beta)$ and $\sin(\alpha+\beta)$.
For each positive integer $n$, let $s(n)$ denote the number of ordered positive integer pair $(x, y)$ for which $\frac{1}{x} + \frac{1}{y} = \frac{1}{n}$ holds. Find all positive integers $n$ for which $s(n) = 5$.
Solve in integers the equation $x^2 + y^2 - 1 = 4xy$
Solve in non-negative integers the equation $$x^3 + 2y^3 = 4z^3$$
For any given positive integer $n > 2$, show that there exists a right triangle with all sides' lengths are integers and one side's length equals $n$.
Show that there exists an infinite sequence of positive integers $a_1, a_2, \cdots$ such that $$S_n=a_1^2 + a_2^2 + \cdots + a_n^2$$ is square for any positive integer $n$.
Solve in positive integers the equation $$m^2 - n^2 - 3n = 5$$
Let $ ABC$ be acute triangle. The circle with diameter $ AB$ intersects $ CA,\, CB$ at $ M,\, N,$ respectively. Draw $ CT\perp AB$ and intersects above circle at $ T$, where $ C$ and $ T$ lie on the same side of $ AB$. $ S$ is a point on $ AN$ such that $ BT = BS$. Prove that $ BS\perp SC$.
Let $ a,\, b,\, c$ be side lengths of a triangle and $ a+b+c = 3$. Find the minimum of \[ a^{2}+b^{2}+c^{2}+\frac{4abc}{3}\]
Find all the Pythagorean triangles whose two sides are consecutive integers.
Suppose the point $F$ is inside a square $ABCD$ such that $BF=1$, $FA=2$, and $FD=3$, as shown. Find the measurement of $\angle{BFA}$.
Solve in integers the equation $$(x+y)^2 = x^3 + y^3$$
Find all positive integer $n$ such that $n$ is a square and its last four digits are the same.
Find a four-digit square number whose first two digits are the same and the last two digits are the same too.
Let $A$ and $B$ be two positive integers and $A=B^2$. If $A$ satisfies the following conditions, find the value of $B$:
- $A$'s thousands digit is $4$
- $A$'s tens digit is $9$
- The sum of all $A$'s digits is $19$