Practice (EndingDigits,TheDivideByNineMethod,MODBasic)

2238

Find all non-negative integers $n$ such that $2^{200}+2^{192}\cdot 15+2^n$ is a perfect square

2239

Prove that $\frac{5^{125}-1}{5^{25}-1}$ is composite.

2240

Find all integers $a$, $b$, $c$ with $1 < a < b < c$ such that the number $(a-1)(b-1)(c-1)$ is a divisor of $(abc-1)$.

2241

Solve in positive integers $\big(1+\frac{1}{x}\big)\big(1+\frac{1}{y}\big)\big(1+\frac{1}{z}\big)=2$

2242

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 $$\frac{1}{k_1} + \frac{1}{k_2} + \cdots + \frac{1}{k_n}=1$$

2243

Solve in positive integers the equation $$3(xy+yz+zx)=4xyz$$

2244

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?

2245

Let complex number $z_1=2-i\cos\theta$, $z_2=2-i\sin\theta$. Find the maximum value of $|z_1z_2|$.

2246

Let $z$ be a complex number, $w=z+\frac{1}{z}$ be a real number, and $-1 < w < 2$. Find $|z|$ and the range $Re(z)$.

2247

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 $m + n$. %

2248

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?

2249

Let $a$, $b$, and $c$ be three odd integers. Prove the equation $ax^2 + bx + c=0$ does not have rational roots.

2250

Show that the difference of two squares of odd numbers must be a multiple of $8$.

2251

Find the least positive integer $n$ such that for every prime number $p$, $p^2 + n$ is never prime.

2252

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?

2253

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 $k$.

2254

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 $\cos(x+2y)$

2255

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}

2256

Show that $$\sin^2\alpha - \sin^2\beta = \sin(\alpha + \beta)\sin(\alpha-\beta)$$ $$\cos^2\alpha - \cos^2\beta = - \sin(\alpha + \beta)\sin(\alpha-\beta)$$

2257

Compute $$\sin^410^{\circ} +\sin^450^{\circ}+\sin^470^\circ$$

2258

Simplify $$\sin^2\alpha + \sin^2\Big(\alpha + \frac{\pi}{3}\Big)+\sin^2\Big(\alpha - \frac{\pi}{3}\Big)$$

2259

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 $$x_1^2 +x_2^2+x_3^2=y_1^2+y_2^2+y_3^2=\frac{3}{2}$$

2260

How many among the first $1000$ Fibonacci numbers are multiples of $11$?

2261

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 $F(k)$.

2263

The ratio $\frac{10^{2000}+10^{2002}}{10^{2001}+10^{2001}}$ is closest to which of the following numbers?

Practice (EndingDigits,TheDivideByNineMethod,MODBasic)

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