Practice (TheColoringMethod)
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Let function $f(x)$ is defined as the following: $$ f(x)= \left\{ \begin{array}{ll} x+2 &, \text{if } x \le -1\\ 2x &, \text{if } -1 < x < 2\\ \displaystyle\frac{x^2}{2} &, \text{if } x \ge 2 \end{array} \right. $$ (A) Compute $f(f(f(-\frac{7}{4})))$ (B) If $f(a)=3$, find the value of $a$
Let $a$ and $k$ be two positive integers, and function $f(x)=3x+1$. If $f(x)$'s domain is $\{1, 2, 3, k\}$ and range is $\{4, 7, a^4, a^2 + 3a\}$, find the value of $a$ and $k$.
Which of the following function is the same as $y=\sqrt{-2x^3}$?
(A) $y=x\sqrt{-2x}\qquad$ (B) $y=-x\sqrt{-2x}\qquad$ (C) $y=-\sqrt{2x^3}\qquad$ (D) $y=x^2\sqrt{-2/x}$
Let $f(x)$ be an odd function and $g(x)$ be an even function. If $f(x)+g(x)=\frac{1}{x-1}$, find $f(x)$ and $g(x)$.
If $f\Big(\displaystyle\frac{x+1}{x}\Big)=\displaystyle\frac{x^2+x+1}{x^2}$, find $f(x)$.
If $f(x)$ is an odd function defined on $\mathbb{R}$, compute $f(0)$.
Let function $f(x)$ satisfy $f(a)+f(b)=f(ab)$, and $f(2)=2$ and $f(3)=3$. Compute $f(72)$.
In triangle ABC, M is median of BC. O is incenter. AH is altitude. MO and AH intersect at E. Prove that AE equal to the radius of incircle
Let $G$ be the centroid of $\triangle{ABC}$, $L$ be a straight line. Prove that $$GG'=\frac{AA'+BB'+CC'}{3}$$ where $A'$, $B'$, $C'$ and $G'$ are the feet of perpendicular lines from $A$, $B$, $C$, and $G$ to $L$.
Let $A$, $B$, $C$ and $D$ be four distinct points on a line, in that order. The circles with diameters $AC$ and $BD$ intersect at the points $X$ and $Y$. The line $XY$ meets $BC$ at the point $Z$. Let $P$ be a point on the line $XY$ different from $Z$. The line $CP$ intersects the circle with diameter $AC$ at the points $C$ and $M$, and the line $BP$ intersects the circle with diameter $BD$ at the points $B$ and $N$. Prove that the lines $AM$, $DN$ and $XY$ are concurrent
(Euler Line) For any triangle $ABC$, show that the cicumcenter $O$, centroid $G$, and the orthocenter $H$ are collinear. Moreover, we have $OG:GH=1:2$.
(Euler Theorem) Let $ABC$ be a triangle, $O$, $I$ be respectively the circumcetner and incenter. Then $$OI^2 = R^2 - 2Rr$$, where $R$ denotes the circumradius and $r$ denotes the inradius.
(Nine-point circle) For any triangle $AB$C, the feet of three altitudes, the midpoints of three sides, and the midpoints of the segments from the vertices to the orthocenter, all lie on the same circle. This circle is of radius half as the circumradius of triangle $ABC$.
Show that the nine-point center $N$ lies on the Euler line, and is the middle point of $O$, the circumcenter, and $H$, the orthocenter.
Show that the triangles $ABC$, $ABH$, $BCH$, and $CAH$ all have the same nine-point circle, providing that the orthocenter $H$ does not coincide with each of the vertices $A$, $B$, $C$.
Let $a, b, c$ be respectively the lengths of three sides of a triangle, and $r$ be the triangle's inradius. Show that $$r = \frac{1}{2}\sqrt{\frac{(b+c-a)(c+a-b)(b+a-c)}{a+b+c}}$$
Let $a, b, c$ be respectively the lengths of three sides of a triangle, and $R$ be the triangle's circumradius. Show that $$R = \frac{abc}{\sqrt{(a+b+c)(b+c-a)(c+a-b)(b+a-c)}}$$
Prove the angle bisector theorem
Prove the median's length formula: $$m_a = \frac{1}{2}\sqrt{2b^2 + 2c^2 - a^2}$$
Prove the triangle's altitude formula: $$h_a = \frac{2}{a}\sqrt{p(p-a)(p-b)(p-c)}$$
Prove the triangle's angle bisector length formula: $$t_a=\frac{2}{b+c}\sqrt{bcp(p-a)}=\sqrt{bc\Big(1-(\frac{a}{b+c})\Big)^2}$$
Cozy the Cat is going up a staircase of $10$ steps. She can either walk up $1$ step a time or jump $2$ steps a time. How many different ways can she reach the top of this staircase?
Find the general formula of the sequence defined as $a_1=6$ and $a_n=\frac{1}{2}a_{n-1}+4$.
Colour all the points on a plane either white or black randomly. Show that it is always possible to find a triangle whose vertices have the same colour and its side length is either $1$ or $\sqrt{3}$.
Randomly colour all the points one a plane either black or white. Show that if any two points with a distance of $2$ units have the same colour, then all the points on this plane have the same colour.