Practice (90/1000)
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Show that for any positive integer $n$, the following relationship holds: $$2^n+2 > n^2$$
Show that the three medians of a triangle intersect at one point.
In $\triangle{ABC}$, let $AD$, $BE$, and $CF$ be the three altitudes as shown. If $AB=6$, $BC=5$, and $EF=3$, what is the length of $BE$?
Let $\triangle{ABC}$ be an acute triangle. If the distance between the vertex $A$ and the orthocenter $H$ is equal to the radius of its circumcircle, find the measurement of $\angle{A}$.
Let $AD$ be the altitude in $\triangle{ABC}$ from the vertex $A$. If $\angle{A}=45^\circ$, $BD=3$, $DC=2$, find the area of $\triangle{ABC}$.
Let $O$ be the centroid of $\triangle{ABC}$. Line $\mathcal{l}$ passes $O$ and intersects $AB$ and $AC$ at $P$ and $Q$, respectively. Point $D$, $E$, and $F$ are on the line $l$ such that $AD\perp l$, $BE \perp l$, and $CF\perp l$. Show that $AD = BE+CF$.
Let $O$ be the incenter of $\triangle{ABC}$. Connect $AO$, $BO$, and $CO$ and extends so that they intersect with $\triangle{ABC}$'s circumcircle at $D$, $E$, and $F$, respectively. Let $DE$ intersect $AC$ at $G$, and $DF$ intersects $AB$ at $H$. Show that $G$, $H$ and $O$ are collinear.
Let $ABC$ be an acute triangle. Circle $O$ passes its vertex $B$ and $C$, and intersects $AB$ and $AC$ at $D$ and $E$, respectively. If $O$'s radius equals the radius of $\triangle{ADE}$'s circumcircle, then the circle $O$ must passes $\triangle{ABC}$'s
(A) incenter (B) circumcenter (C) centroid (D) orthocenter.
Let $K$ is an arbitrary point inside $\triangle{ABC}$, and $D$, $E$, and $F$ be the centroids of $\triangle{ABK}$, $\triangle{BCK}$ and $\triangle{CAK}$, respectively. Find the value of $S_{\triangle{ABC}} : S_{\triangle{DEF}}$.
Let $I$ be the incenter of $\triangle{ABC}$. $AI$, $BI$, and $CI$ intersect $\triangle{ABC}$'s circumcircle at $D$, $E$, and $F$, respectively. Show that $EF \perp AD$
In $\triangle{ABC}$, $AB=AC$. Extending $CA$ to an arbitrary point $P$. Extending $AB$ to point $Q$ such that $AP=BQ$. Let $O$ be the circumcenter of $\triangle{ABC}$. Show that $A$, $P$, $Q$, and $O$ concyclic.
Find the range of the function $$y=x+\sqrt{x^2 -3x+2}$$
For any non-negative real numbers $x$ and $y$, the function $f(x+y^2)=f(x) + 2[f(y)]^2$ always holds, $f(x)\ge 0$, $f(1)\ne 0$. Find the value of $f(2+\sqrt{3})$.
Let $f(x)$ be a function defined on $\mathbb{R}$. If for every real number $x$, the relationships $$f(x+3)\le f(x)+3\quad\text{and}\quad f(x+2)\ge f(x)+2$$ always hold.
1) Show $g(x) = f(x)-x$ is a periodic function.
2) If $f(998)=1002$, compute $f(2000)$
Which pair contains same functions:
(A) $f(x)=\sqrt{(x-1)^2}$ and $g(x)=x-1$
(B) $f(x)=\sqrt{x^2 -1}$ and $g(x)=\sqrt{x+1}\cdot\sqrt{x-1}$
(C) $f(x)=(\sqrt{x -1})^2$ and $g(x)=\sqrt{(x-1)^2}$
(D) $f(x)\displaystyle\sqrt{\frac{x^2-1}{x+1}}$ and $g(x)=\displaystyle\frac{\sqrt{x^2-1}}{\sqrt{x+1}}$
Select all correct answers.
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
