Practice (TheColoringMethod)

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Consider a non-empty set of segments of length 1 in the plane which do not intersect except at their endpoints. (In other words, if point $P$ lies on distinct segments $a$ and $b$, then $P$ is an endpoint of both $a$ and $b$.) This set is called $3-amazing$ if each endpoint of a segment is the endpoint of exactly three segments in the set. Find the smallest possible size of a 3-amazing set of segments.

Let $z_1$, $z_2 \in \mathbb{C}$. Prove that the number $E= z_1\cdot z_2 + \overline{z_1}\cdot \overline{z_2}$ is a real number.

Prove that if $|z_1| = |z_2| = 1$ and $z_1z_2 \ne 1$, then $\displaystyle\frac{z_1 + z_2}{1 + z_1z_2}$ is a real number.

Consider the complex numbers $z_1, z_2, \cdots , z_n$ with $|z_1| = |z_2| = \cdots = |z_n| = r > 0$. Prove that the number $$E =\frac{(z_1 + z_2)(z_2 + z_3)\cdots(z_{n-1} + z_n)(z_n + z_1)}{z_1z_2 \cdots z_n}$$ is real.

Prove $E = (2 + i\sqrt{5})^7 + (2-i\sqrt{5})^7 \in \mathbb{R}$.

Prove $E=(\frac{19+7i}{9-i})^n + (\frac{20+5i}{7+6i})^n \in \mathbb{R}$

Let complex numbers $z_1$, $z_2$, and $z_3$ satisfy $|z_1|=|z_2|=|z_3| = r > 0$. If $z_1+z_2+z_3 \ne 0$, prove $$\frac{z_1z_2+z_2z_3+z_3z_1}{z_1+z_2+z_3}=r$$

Let $z$ be a complex number, and $|z|=1$. Find the maximal value of $u=|z^3-3z+2|$.

Let points $A$, $B$, and $C$ represent $z$, $\overline{z}$, and $\frac{1}{z}$ on the complex plane. If $\triangle ABC$ is a right triangle, find the trajectory of $A$.

On the complex plane, $z_1$ moves along the segment defined by $1+i$ and $1-i$. $z_2$ moves along the unit circle whose center is at the origin. 1) Find the trajectory of $z_1^2$ 2) Find the area that is defined $z_1z_2$ 3) Find the area that is defined by $z_1+z_2$

Let positive real number $x$, $y$, and $z$ satisfy $x+y+z=1$. Find the minimal value of $u=\sqrt{x^2 + y^2 + xy} + \sqrt{y^2 +z^2 +yz} +\sqrt{z^2 +x^2 + xz}$

Let integer $n\ge 2$, prove $$\sin{\frac{\pi}{n}}\cdot\sin{\frac{2\pi}{n}}\cdots\sin{\frac{(n-1)\pi}{n}}=\frac{n}{2^{n-1}}$$

Let $a_n=\binom{2020}{3n-1}$. Find the vale of $\displaystyle\sum_{n=1}^{673}a_n$.


Let $A=x\cos^2{\theta} + y\sin^2{\theta}$, $B=x\sin^2{\theta}+y\sin^2{\theta}$, where $x$, $y$, $A$, and $B$ are all real numbers. Prove $x^2 + y^2 \ge A^2 + B^2$

Let $S_n$ be the minimal value of $\displaystyle\sum_{k=1}^n\sqrt{(2k-1)^2+a_k^2}$, where $n\in\mathbb{N}$, $a_1, a_2, \cdots, a_n\in\mathbb{R}^+$, and $a_1+a_2+\cdots a_n = 17$. If there exists a unique $n$ such that $S_n$ is also an integer, find $n$.

Let sequences {$a_n$} and {$b_n$} satisfy: $a_n=a_{n-1}\cos{\theta} - b_{n-1}\sin{\theta}$ and $b_n=a_{n-1}\sin{\theta}+b_{n-1}\cos{\theta}$. If $a_1=1$ and $b_1=\tan{\theta}$, where $\theta$ is a known real number, find the general formula for {$a_n$} and {$b_n$}.

Let polynomials $P(x)$, $Q(x)$, $R(x)$, and $S(x)$ satisfy: $$P(x^5) + xQ(x^5)+x^2R(x^5)=(x^4+x^3+x^2+x+1)S(x)$$ Prove: $(x-1) | P(x)$

Let $f(z) = z^2 + az + b$, where both $a$ and $b$ are complex numbers. If for all $|z|=1$, find the values of $a$ and $b$.

Can a five-digit number consisting of 5 distinct even digits a perfect square?

Let $F(z)=\dfrac{z+i}{z-i}$ for all complex numbers $z \neq i$, and let $z_n=F(z_{n-1})$ for all positive integers $n$. Given that $z_0=\dfrac{1}{137}+i$ and $z_{2002}=a+bi$, where $a$ and $b$ are real numbers, find $a+b$.

Month $A$ has three Wednesdays, but neither its first day nor its last day is Wednesday. What day of the week is the first day of month $A$?

Joe puts $63$ cards, number from $1$ to $63$, on a regular chessboard in sequence. The last space on the chessboard is left empty. A card can be moved to a neighboring space if that space is empty. Joe wants to just switch the card $1$ and card $2$, but leave all other cards at their original spaces, after a series of moves. Is it possible?

Find the smallest $n$ such that $\frac{1}{n}(1^2 + 2^2 + \cdots + n^2)$ is a square of an integer.

Solve in integers $y^2=x^4 + x^3 + x^2 +x +1$.

Solve in integers $x^3 + (x+1)^3 + \cdots + (x+7)^3 = y ^3$