Practice (90/1000)
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Let real numbers $a_1$, $a_2$, $\cdots$, $a_{2016}$ satisfy $9a_i\ge 11a_{i+1}^2$ for $i=1, 2,\cdots, 2015$. Define $a_{2017}=a_1$, find the maximum value of $$P=\displaystyle\prod_{i=1}^{2016}(a_i-a_{i+1}^2)$$
As shown, two identical circles are internally tangent to each other and also tangent to a $5-12-13$ triangle. Find the circle's radius.
(British Flag Theorem) Let point $P$ lie inside rectangle $ABCD$. Draw four squares using each of $AP$, $BP$, $CP$, and $DP$ as one side. Show that $$S_{AA_1A_2P}+S_{CC_1C_2P}=S_{BB_1B_2P}+S_{DD_1D_2P}$$
(De Gua's Theorem) In a trirectangular tetrahedron $ABCD$ where $A$ is the shared right-angle corner. Show that $$S_{\triangle{BCD}}^2=S_{\triangle{ABC}}^2+S_{\triangle{ACD}}^2+S_{\triangle{ADB}}^2$$
Let $CD$ be the altitude in right $\triangle{ABC}$ from the right angle $C$. If inradii of $\triangle{ABC}$, $\triangle{ACD}$, and $\triangle{BCD}$ be $r_1$, $r_2$, and $r_3$, respectively, show that $$r_1 + r_2 + r_3 = CD$$
Three circles are tangent to each other and also a common line, as shown. Let the radii of circles $O_1$, $O_2$, and $O_3$ be $r_1$, $r_2$, and $r_3$, respectively. Show that $$\frac{1}{\sqrt{r_3}}=\frac{1}{\sqrt{r_1}} +\frac{1}{\sqrt{r_2}}$$
Given two segments $AB$ and $MN$, show that $$MN\perp AB \Leftrightarrow AM^2 - BM^2 = AN^2 - BN^2$$
Let point $P$ inside an equilateral $\triangle{ABC}$ such that $AP=3$, $BP=4$, and $CP=5$. Find the side length of $\triangle{ABC}$.
Let $M$ be a point inside $\triangle{ABC}$. Draw $MA'\perp BC$, $MB'\perp CA$, and $MC'\perp AB$ such that $BA'=BC'$ and $CA'=CB'$. Prove $AB'=AC'$.
Four sides of a concyclic quadrilateral have lengths of 25, 39, 52, and 60, in that order. Find the circumference of its circumcircle.
Let $a$ and $b$ be the two roots of $x^2 - 3x -1=0$. Try to solve the following problems without computing $a$ and $b$:
1) Find a quadratic equation whose roots are $a^2$ and $b^2$
2) Find the value of $\frac{1}{a+1}+\frac{1}{b+1}$
3) Find the recursion relationship of $x_n=a^n + b^n$
Find as many different solutions as possible.
Three of the roots of $x^4 + ax^2 + bx + c = 0$ are $2$, $−3$, and $5$. Find the value of $a + b + c$.
In $\triangle{ABC}$, let $a$, $b$, and $c$ be the lengths of sides opposite to $\angle{A}$, $\angle{B}$ and $\angle{C}$, respectively. $D$ is a point on side $AB$ satisfying $BC=DC$. If $AD=d$, show that
$$c+d=2\cdot b\cdot\cos{A}\quad\text{and}\quad c\cdot d = b^2-a^2$$
Suppose $a_1$, $b_1$, $c_1$, $a_2$, $b_2$, and $c_2$ are all positive real numbers. If both $a_1x^2 +b_1x+c_1=0$ and $a_2x^2+b_2x+c_2=$ are solvable in real numbers. Show that their roots must be all negative. Furthermore, prove equation $a_1a_2x^2+b_1b_2x+c_1c_2=0$ has two negative real roots too.
Let $x$, $y$, and $z$ be real numbers satisfying $x=6-y$ and $z^2=xy-9$. Show that $x=y$.
Let $\alpha_n$ and $\beta_n$ be two roots of equation $x^2+(2n+1)x+n^2=0$ where $n$ is a positive integer. Evaluate the following expression $$\frac{1}{(\alpha_3+1)(\beta_3+1)}+\frac{1}{(\alpha_4+1)(\beta_4+1)}+\cdots+\frac{1}{(\alpha_{20}+1)(\beta_{20}+1)}$$
Let real numbers $a$, $b$, and $c$ satisfy
$$
\left\{
\begin{array}{rcl}
a^2 - bc-8a +7&=&0\\
b^2 + c^2 +bc-6a+6&=&0
\end{array}
\right.
$$
Show that $1 \le a \le 9$.
Find one real solution $(a, b, c, d)$ to the following system:
$$
\left\{
\begin{array}{rcl}
a+b+c+d&=&-2\\
ab+ac+ad+bc+bd+cd&=&-3\\
abc+abd+acd+bcd&=&4\\
abcd&=&3
\end{array}
\right.
$$
If $m^2 = m+1, n^2-n=1$ and $m\ne n$, compute $m^7 +n^7$.
Find the range of real number $a$ if the two roots of $x^2+2ax+6-a=0$ satisfy one of the following condition:
- two roots are both greater than 1
- one root is greater than 1 and the other is less than 1
Solve equation $(6x+7)^2(3x+4)(x+1)=6$ in real numbers.
If $x^2 + 11x+16=0, y^2 + 11y+16=0$, and $x\ne y$, what is the value of $$\sqrt{\frac{x}{y}}-\sqrt{\frac{y}{x}}$$
Let $x_1$ and $x_2$ be two real roots of $x^2-x-1=0$. Find the value of $2x_1^5 + 5x_2^3$.
Find integer $m$ such that the equation $x^2+mx-m+1=0$ has two positive integer roots.
Let $\alpha$ and $\beta$ be two real roots of $x^4 +k=3x^2$ and also satisfy $\alpha + \beta = 2$. Find the value of $k$.