Found problems: 2529
Let $a,b,c,d$ be four real numbers such that $a+b+c+d=20$ and $ab+bc+cd+da=16$. Find the maximum possible value of $abc+bcd+cda+dab$.
[i]Proposed by Yannick Yao[/i]
Show that for any positive real numbers $a$, $b$, $c$ the following inequality takes place $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}+\frac{a+b+c}{\sqrt{a^2+b^2+c^2}} \geq 3+\sqrt{3}$
Prove that the largest pentagon (in terms of area) that can be inscribed in a circle of radius $1$ is regular (i.e., has equal sides).
Four positive integers $x,y,z$ and $t$ satisfy the relations
\[ xy - zt = x + y = z + t. \]
Is it possible that both $xy$ and $zt$ are perfect squares?
If $a,b,c>0$, what is the smallest possible value of $ \left\lfloor \dfrac {a+b}{c} \right\rfloor + \left\lfloor \dfrac {b+c}{a} \right\rfloor + \left\lfloor \dfrac {c+a}{b} \right\rfloor $? (Note that $ \lfloor x \rfloor $ denotes the greatest integer less than or equal to $x$.)
If $a, b, c$ are positive real numbers, prove that
$$\frac{a}{\sqrt{(a + 2b)^3}}+\frac{b}{\sqrt{(b + 2c)^3}} +\frac{c} {\sqrt{(c + 2a)^3}} \ge \frac{1}{\sqrt{a + b + c}}$$
Alexandru Mihalcu
If $57a + 88b + 125c \geq 1148$, where $a,b,c > 0$, what is the minimum value of
\[ a^3 + b^3 + c^3 + 5a^2 + 5b^2 + 5c^2? \]
Let $a,b,c$ be positive real numbers that satisfy $a+b+c=abc$. Prove that
$$\sqrt{(1+a^2)(1+b^2)}+\sqrt{(1+b^2)(1+c^2)}+\sqrt{(1+a^2)(1+c^2)}-\sqrt{(1+a^2)(1+b^2)(1+c^2)} \ge 4.$$
A triangle has side lengths $a$, $b$, $c$, and the altitudes have lengths $h_a$, $h_b$, $h_c$. Prove that
\[ \left( \frac{a}{h_a} \right)^2
+ \left( \frac{b}{h_b} \right)^2
+ \left( \frac{c}{h_c} \right)^2 \ge 4. \]
Solve the group of equations: \[\begin{cases} (x + y)^3 = z \\ (y + z)^3 = x \\ (z + x)^3 = y \end{cases}\]
Determine the greatest number, who is the product of some positive integers, and the sum of these numbers is $1976.$
$a_1,a_2,\cdots,a_n$ are positive real numbers, $a_1+a_2+\cdots,a_n=1$. Prove that
$$\sum_{m=1}^n\frac{a_m}{\prod\limits_{k=1}^m(1+a_k)}\leq1-\frac{1}{2^n}.$$
If $a_1,a_2,a_3\ldots a_n$ are positive numbers, find the maximum value of
\[\frac{a_1a_2\ldots a_{n-1}a_n}{(1+a_1)(a_1+a_2)\ldots (a_{n-1}+a_n)(a_n+2^{n+1})} \]
The real numbers $a_0,a_1,a_2,\ldots$ satisfy $1=a_0\le a_1\le a_2\le\ldots. b_1,b_2,b_3,\ldots$ are defined by $b_n=\sum_{k=1}^n{1-{a_{k-1}\over a_k}\over\sqrt a_k}$.
[b]a.)[/b] Prove that $0\le b_n<2$.
[b]b.)[/b] Given $c$ satisfying $0\le c<2$, prove that we can find $a_n$ so that $b_n>c$ for all sufficiently large $n$.
Let $a,b,c$ be positive real numbers such that $abc = 1$. Prove that:
$\displaystyle\prod(a^5+a^4+a^3+a^2+a+1)\geq 8(a^2+a+1)(b^2+b+1)(c^2+c+1)$
If positive reals $x,y$ are such that $2(x+y)=1+xy$, find the minimum value of expression $$A=x+\frac{1}{x}+y+\frac{1}{y}$$
Suppose that an interior point $O$ of a triangle $ABC$ is such that the angles $\angle BAO,\angle CBO, \angle ACO$ are all greater than or equal to $30^o$. Prove that the triangle $ABC$ is equilateral.
Define a sequence $\{a_n\}$ by\[S_1=1,\ S_{n+1}=\frac{(2+S_n)^2}{ 4+S_n} (n=1,\ 2,\ 3,\ \cdots).\]
Where $S_n$ the sum of first $n$ terms of sequence $\{a_n\}$.
For any positive integer $n$ ,prove that\[a_{n}\ge \frac{4}{\sqrt{9n+7}}.\]
At a tourist camp, each person has at least $50$ and at most $100$ friends among the other persons at the camp. Show that one can hand out a t-shirt to every person such that the t-shirts have (at most) $1331$ different colors, and any person has $20$ friends whose t-shirts all have pairwisely different colors.
Real numbers $x$, $y$, and $z$ satisfy the following equality: \[4(x+y+z)=x^2+y^2+z^2\] Let $M$ be the maximum of $xy+yz+zx$, and let $m$ be the minimum of $xy+yz+zx$. Find $M+10m$.
Let $n\ge 3$ be an integer, and suppose $x_1,x_2,\cdots ,x_n$ are positive real numbers such that $x_1+x_2+\cdots +x_n=1.$ Prove that $$x_1^{1-x_2}+x_2^{1-x_3}\cdots+x_{n-1}^{1-x_n}+x_n^{1-x_1}<2.$$
[i] ~Sutanay Bhattacharya[/i]
$ ABC$ is a triangle, the bisector of angle $ A$ meets the circumcircle of triangle $ ABC$ in $ A_1$, points $ B_1$ and $ C_1$ are defined similarly. Let $ AA_1$ meet the lines that bisect the two external angles at $ B$ and $ C$ in $ A_0$. Define $ B_0$ and $ C_0$ similarly. Prove that the area of triangle $ A_0B_0C_0 \equal{} 2 \cdot$ area of hexagon $ AC_1BA_1CB_1 \geq 4 \cdot$ area of triangle $ ABC$.
What is the largest possible number of subsets of the set $\{1, 2, \dots , 2n+1\}$ such that the intersection of any two subsets consists of one or several consecutive integers?
Four real numbers $ p$, $ q$, $ r$, $ s$ satisfy $ p+q+r+s = 9$ and $ p^{2}+q^{2}+r^{2}+s^{2}= 21$. Prove that there exists a permutation $ \left(a,b,c,d\right)$ of $ \left(p,q,r,s\right)$ such that $ ab-cd \geq 2$.
Let $o$, $r$, $g$, $t$, $n$, $i$, $z$, $e$, and $d$
be positive reals. Show that
\[
\sqrt{(d+o+t+t+e+d)(o+r+z+i+n+g)}
> \sqrt{ti} + \sqrt{go} + \sqrt[6]{orz}.
\]
when $d^2e \geq \tfrac{2}{1434}$.
Proposed by [i]David Fox[/i]