Practice (90/1000)
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Solve in positive integers $\big(1+\frac{1}{x}\big)\big(1+\frac{1}{y}\big)\big(1+\frac{1}{z}\big)=2$
Find all positive integers $n$ and $k_i$ $(1\le i \le n)$ such that $$k_1 + k_2 + \cdots + k_n = 5n-4$$ and $$\frac{1}{k_1} + \frac{1}{k_2} + \cdots + \frac{1}{k_n}=1$$
Solve in positive integers the equation $$3(xy+yz+zx)=4xyz$$
There are $2015$ people standing in a circle, counting $1$ and $2$ in turn continuously. Those who count $2$ will be out. For example, people who stand at initial positions of $2, 4, \dots, 2014, 1, 3, \dots$ etc will be out. The game goes on until there is only one person remaining in the circle. What is his initial position?
Let complex number $z_1=2-i\cos\theta$, $z_2=2-i\sin\theta$. Find the maximum value of $|z_1z_2|$.
Let $z$ be a complex number, $w=z+\frac{1}{z}$ be a real number, and $-1 < w < 2$. Find $|z|$ and the range $Re(z)$.
In the diagram $ABCDEFG$ is a regular heptagon (a 7 sided polygon). Shown is the star $AEBFCGD$. The degree measure of the obtuse angle formed by $AE$ and $CG$ is $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
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Ten unfair coins with probability of $1, \frac{1}{2}, \frac{1}{3}, \dots, \frac{1}{10}$ of showing heads are flipped. What is the probability that odd number of heads are shown?
Let $a$, $b$, and $c$ be three odd integers. Prove the equation $ax^2 + bx + c=0$ does not have rational roots.
Show that the difference of two squares of odd numbers must be a multiple of $8$.
Find the least positive integer $n$ such that for every prime number $p$, $p^2 + n$ is never prime.
Ten unfair coins with probability of $1, \frac{1}{3}, \frac{1}{4}, \dots, \frac{1}{11}$ of showing heads are flipped. What is the probability that odd number of heads are shown?
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 \(k\).
Let $x, y \in [-\frac{\pi}{4}, \frac{\pi}{4}], a \in \mathbb{Z}^+$, and
$$
\left\{
\begin{array}{rl}
x^3 + \sin x - 2a &= 0 \\
4y^3 +\frac{1}{2}\sin 2y +a &=0
\end{array}
\right.
$$
Compute the value of $\cos(x+2y)$
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}
Show that
$$\sin^2\alpha - \sin^2\beta = \sin(\alpha + \beta)\sin(\alpha-\beta)$$
$$\cos^2\alpha - \cos^2\beta = - \sin(\alpha + \beta)\sin(\alpha-\beta)$$
Compute $$\sin^410^{\circ} +\sin^450^{\circ}+\sin^470^\circ$$
Simplify $$\sin^2\alpha + \sin^2\Big(\alpha + \frac{\pi}{3}\Big)+\sin^2\Big(\alpha - \frac{\pi}{3}\Big)$$
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 $11$?
Let $F(1)=1, F(2)=1, F(n+2)= F(n+1)+F(n)$ be the Fibonacci sequence. Prove if $i | j$, then $F(i) | F(j)$. In another word, every $k^{th}$ element is a multiple of $F(k)$.
The ratio $\frac{10^{2000}+10^{2002}}{10^{2001}+10^{2001}}$ is closest to which of the following numbers?
According to the standard convention for exponentiation,
$$2^{2^{2^2}} = 2^{\left(2^{\left(2^2\right)}\right)} = 2^{16} = 65,536$$
If the order in which the exponentiations are performed is changed, how many other values are possible?
For how many positive integers $m$ is there at least 1 positive integer $n$ such that $mn \le m + n$?
Each of the small circles in the figure has radius one. The innermost circle is tangent to the six circles that surround it, and each of those circles is tangent to the large circle and to its small-circle neighbors. Find the area of the shaded region.
