Practice (TheColoringMethod)
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Let $n$ be a positive integer. If the equation $x + 2y + 2z = n$ has exactly $28$ positive integer solutions, find the value of $n$.
Let $x$, $y$, and $z$ be three positive integers, If
- $7x^2 - 3y^2 + 4z^2 = 8$
- $16x^2 - 7y^2 + 9z^2=-3$
Find the value of $x^2 + y^2 + z^2$
Solve the equation in integers $(x+y)^x = y^x + 1413$
Solve the following system in integers:
$$
\left\{
\begin{array}{ll}
x_1 + x_2 + \cdots + x_n &= n \\
x_1^2 + x_2^2 + \cdots + x_n^2 &= n \\
\cdots\\
x_1^n + x_2^n + \cdots + x_n^n &= n
\end{array}
\right.
$$
Let $a_1, a_2, \cdots, a_{100}, b_1, b_2, \cdots, b_{100}$ be distinct real numbers. They are used to fill a $100 \times 100$ grids by putting the value of $(a_i + b_j)$ in the cell $(i, j)$ where $1 \le i, j \le 100$. Let $A_i$ be the product of all the numbers in column $i$, and $B_i$ be the product of all the numbers in row $i$. Show that if every $A_i$ equals to 1, then every $B_j$ equals to -1.
Suppose $x, y, z \in \mathbb{R}^+$, such that
\begin{align*}
x^2 + y^2 + xy &= 9\\
y^2 + z^2 + yz &= 16\\
z^2 + x^2 + zx &= 25
\end{align*}
Find the value of $xy+yz+zx$.
Suppose $P(x)$ is a monnic polynomial (meaning the leading coefficient is 1) with 20 roots, each distinct and of the form $\frac{1}{3^k}$ for $k=0, 1, \cdots, 19$. Find the coefficient of $x^{18}$ in $P(x)$.
Find the sum of the reciprocals of all perfect squares whose prime factorization contains only 3, 5, and 7, i.e. $$\frac{1}{9}+\frac{1}{25}+\frac{1}{49}+\frac{1}{9}+\frac{1}{81}+\frac{1}{225}+\frac{1}{441}+\frac{1}{625}+\cdots$$
Show that if $a$, $b$ and $c$ are odd integers, then the equation $ax^2 + bx + c=0$ has no integer solution.
Let $n$ be a positive integer, show that $11^{n+2} + 12^{2n+1}$ is a multiple of 133.
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.