Category Archives: Algebra

Let $P(x)$ is a polynomial with integer coefficients so that $P(d)=\dfrac{2025}{d}$, where $d$ is a positive divisor of $2025$. Find $P(x)$.🔑 Claim: There is no $P(x)$ to satisfy $P(d)=\dfrac{2025}{d}$, where $d$ is a positive divisor of $2025$. Lemma: If $P(x)$ … Continue reading →

Let $n$ be the real number. Define $[n]$ be the integer part of $n$, and $\{n\}$ be the decimal part of $n$. Solve the following equations: $$\begin{array}{ccccccc} \{x\} & + & [y] & + & \{z\} & = 2.9\\ \{y\} … Continue reading →

Find the real solutions for the following equations: $$a^2+b^2\ \ \ \ \ \ \ \ \ \ \ \ \ =1\tag{1}$$ $$b^2+c^2+\sqrt{3}bc=1\tag{2}$$ $$c^2+a^2+\ \ \ \ \ ca=1\tag{3}$$ Click here for the solution. Solution: If $c=0$, based on equation … Continue reading →

For integer $n>0$, find the values of $$\sum\limits_{i=1}^{n}(-1)^{i+1}\cdot i\cdot\binom{n-1}{i-1}$$ Click here for the solution. Solution: If $n=1$, then \begin{flalign*} \sum\limits_{i=1}^{n}(-1)^{i+1}\cdot i\cdot\binom{n-1}{i-1} &= (-1)^{1+1}\cdot 1\cdot\binom{1-1}{1-1}& \\ &=1& \end{flalign*} If $n=2$, then \begin{flalign*} \sum\limits_{i=1}^{n}(-1)^{i+1}\cdot i\cdot\binom{n-1}{i-1} &=(-1)^{1+1}\cdot 1\cdot\binom{2-1}{1-1}+(-1)^{2+1}\cdot 2\cdot\binom{2-1}{2-1}& \\ &=1-2& \\ &=-1& … Continue reading →