Practice (TheColoringMethod)

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Compute $$\cos\frac{\pi}{2n+1}\cdot\cos\frac{2\pi}{2n+1}\cdots\cos\frac{n\pi}{2n+1}$$

Compute $$\Big(1+\cos\frac{\pi}{5}\Big)\Big(1+\cos\frac{3\pi}{5}\Big)$$

Compute $$\sin^2 10^\circ + \cos^2 40^\circ + \sin 10^\circ \cos 40^\circ$$

Compute $$\sin^2 80^\circ -\sin^2 40^\circ +\sqrt{3}\sin 40^\circ \cos 80^\circ $$

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 $s(n) = 5$.


Show that for any right triangle whose sides' lengths are all integers, - one side's length must be a multiple of 3, and - one side's length must be a multiple of 4, and - one side's length must be a multiple of 5 Please note these sides may not be distinct. For example, in a 5-12-13 triangle, 12 is a multiple of both 3 and 4.

Let $a$, $b$, and $c$ form a geometric sequence. Can the last two digits of $N=a^3+b^3+c^3-3abc$ be 20?

Solve in integers the equation $x^2 + y^2 - 1 = 4xy$


Solve in $\textit{rational}$ numbers the equation $x^2 - dy^2 = 1$ where $d$ is an integer.

Let $x$ be a positive real number, and $\lfloor{x}\rfloor$ be the largest integer that not exceeding $x$. Prove that there exist infinity number of positive integers, $n$, such that $\lfloor{\sqrt{2}}\ n\rfloor$ is a perfect square.


Show that there are infinitely many integers $n$ such that $2n + 1$ and $3n + 1$ are perfect squares, and that such $n$ must be multiples of $40$.


Show that the equation $x^2 + y^3 = z^4$ has infinitely many integer solutions.


Prove that $5x^2\pm 4$ is a perfect square if and only if $x$ is a term in the Fibonacci sequence.

Find all $n\in\mathbb{N}$ such that $$\binom{n}{k-1} = 2 \binom{n}{k} + \binom{n}{k+1}$$

for some natural number $k < n$.


Prove that if $m=2+2\sqrt{28n^2 +1}$ is an integer for some $n\in\mathbb{N}$, then $m$ is a perfect square.


Prove that if the difference of two consecutive cubes is $n^2$, $n\in\mathbb{N}$, then $(2n-1)$ is a square.


If $n$ is an integer such that the values of $(3n+1)$ and $(4n+1)$ are both squares, prove that $n$ is a multiple of $56$.


Let $p$ be a prime. Prove that the equation $x^2-py^2 = -1$ has integral solution if and only if $p=2$ or $p\equiv 1\pmod{4}$.


Let integers $a$, $b$ and $c$ be the lengths of a right triangle's three sides, where $c > b > a$. Show that $\frac{(c-a)(c-b)}{2}$ must be a square number.

If $p$ is a prime of the form $4k+3$, show that exactly one of the equations $x^2-py^2=\pm 2$ has an integral solution.


Show that $3^n-2$ is a square only for $n=1$ and $n=3$.

Show that if $\frac{x^2+1}{y^2}+4$ is a perfect square, then this square equals $9$.