Found problems: 592
Let $n\geqslant 1$ be an integer, and let $x_0,x_1,\ldots,x_{n+1}$ be $n+2$ non-negative real numbers that satisfy $x_ix_{i+1}-x_{i-1}^2\geqslant 1$ for all $i=1,2,\ldots,n.$ Show that \[x_0+x_1+\cdots+x_n+x_{n+1}>\bigg(\frac{2n}{3}\bigg)^{3/2}.\][i]Pakawut Jiradilok and Wijit Yangjit, Thailand[/i]
Prove the following inequality:
\[\prod^k_{i=1} x_i \cdot \sum^k_{i=1} x^{n-1}_i \leq \sum^k_{i=1}
x^{n+k-1}_i,\] where $x_i > 0,$ $k \in \mathbb{N}, n \in
\mathbb{N}.$
Suppose that $a$, $b$, $c$, and $d$ are real numbers such that $a+b+c+d=8$. Compute the minimum possible value of \[20(a^2+b^2+c^2+d^2)-\sum_{\text{sym}}a^3b,\] where the sum is over all $12$ symmetric terms.
[i]Derek Liu[/i]
Let $a,b,c,d$ real numbers such that:
$$
a+b+c+d=0 \text{ and } a^2+b^2+c^2+d^2 = 12
$$
Find the minimum and maximum possible values for $abcd$, and determine for which values of $a,b,c,d$ the minimum and maximum are attained.
Show that
\[ \frac{1 - s^a}{1 - s} \leq (1 + s)^{a-1}\]
holds for every $1 \neq s > 0$ real and $0 < a \leq 1$ rational.
Find the smallest positive real number $\lambda$ such that for every numbers $a_1,a_2,a_3 \in \left[0, \frac{1}{2} \right]$ and $b_1,b_2,b_3 \in (0, \infty)$ with $\sum\limits_{i=1}^3a_i=\sum\limits_{i=1}^3b_i=1,$ we have $$b_1b_2b_3 \le \lambda (a_1b_1+a_2b_2+a_3b_3).$$
Let $a_{ij}, i = 1, 2, \dots, m$ and $j = 1, 2, \dots, n$ be positive real numbers. Prove that
\[ \sum_{i = 1}^m \left( \sum_{j = 1}^n \frac{1}{a_{ij}} \right)^{-1} \le \left( \sum_{j = 1}^n \left( \sum_{i = 1}^m a_{ij} \right)^{-1} \right)^{-1} \]
The positive integers $x_1, \cdots , x_n$, $n \geq 3$, satisfy $x_1 < x_2 <\cdots< x_n < 2x_1$. Set $P = x_1x_2 \cdots x_n.$ Prove that if $p$ is a prime number, $k$ a positive integer, and $P$ is divisible by $pk$, then $\frac{P}{p^k} \geq n!.$
Problem 4. Let a,b be positive real numbers and let x,y be positive real numbers less than 1, such that:
a/(1-x)+b/(1-y)=1
Prove that:
∛ay+∛bx≤1.
For positive real numbers $x$, $y$ and $z$ holds $x^2+y^2+z^2=1$. Prove that $$\frac{x}{1+x^2}+\frac{y}{1+y^2}+\frac{z}{1+z^2} \leq \frac{3\sqrt{3}}{4}$$
For any polynomial $P(x)=a_0+a_1x+\ldots+a_kx^k$ with integer coefficients, the number of odd coefficients is denoted by $o(P)$. For $i-0,1,2,\ldots$ let $Q_i(x)=(1+x)^i$. Prove that if $i_1,i_2,\ldots,i_n$ are integers satisfying $0\le i_1<i_2<\ldots<i_n$, then: \[ o(Q_{i_1}+Q_{i_2}+\ldots+Q_{i_n})\ge o(Q_{i_1}). \]
Let $n$ and $k$ be positive integers such that $1 \leq n \leq N+1$, $1 \leq k \leq N+1$. Show that: \[ \min_{n \neq k} |\sin n - \sin k| < \frac{2}{N}. \]
If $a$ and $b$ are arbitrary positive real numbers and $m$ an integer, prove that
\[\Bigr( 1+\frac ab \Bigl)^m +\Bigr( 1+\frac ba \Bigl)^m \geq 2^{m+1}.\]
Find the minimum value of
\[\max(a + b + c, b + c + d, c + d + e, d + e + f, e + f + g)\]
subject to the constraints
(i) $a, b, c, d, e, f, g \geq 0,$
(ii)$ a + b + c + d + e + f + g = 1.$
For $a,b,c$ positive real numbers such that $ab+bc+ca=3$, prove:
$ \frac{a}{\sqrt{a^3+5}}+\frac{b}{\sqrt{b^3+5}}+\frac{c}{\sqrt{c^3+5}} \leq \frac{\sqrt{6}}{2}$
[i]Proposed by Dorlir Ahmeti, Albania[/i]
[b]Q.[/b] Prove that:
$$\sum_{\text{cyc}}\tan (\tan A) - 2 \sum_{\text{cyc}} \tan \left(\cot \frac{A}{2}\right) \geqslant -3 \tan (\sqrt 3)$$where $A, B$ and $C$ are the angles of an acute-angled $\triangle ABC$.
[i]Proposed by SA2018[/i]
Let $a_1,a_2,...,a_{2020}$ be positive real numbers. Prove that:
$$\max{(a^2_1-a_2,a^2_2-a_3,...,a^2_{2020}-a_1)}\ge\max{(a^2_1-a_1,a^2_2-a_2,...,a^2_{2020}-a_{2020})}$$
a) Prove that for all real numbers $k,l,m$ holds :
$$(k+l+m)^2 \ge 3 (kl+lm+mk)$$
When does equality holds?
b) If $x,y,z$ are positive real numbers and $a,b$ real numbers such that $$a(x+y+z)=b(xy+yz+zx)=xyz,$$
prove that $a \ge 3b^2$. When does equality holds?
Let $a, b, c$ be real numbers with the property as $ab + bc + ca = 1$. Show that:
$$\frac {(a + b) ^ 2 + 1} {c ^ 2 + 2} + \frac {(b + c) ^ 2 + 1} {a ^ 2 + 2} + \frac {(c + a) ^ 2 + 1} {b ^ 2 + 2} \ge 3 $$.
Let $A_1A_2 \ldots A_n$ be a convex polygon. Point $P$ inside this polygon is chosen so that its projections $P_1, \ldots , P_n$ onto lines $A_1A_2, \ldots , A_nA_1$ respectively lie on the sides of the polygon. Prove that for arbitrary points $X_1, \ldots , X_n$ on sides $A_1A_2, \ldots , A_nA_1$ respectively,
\[\max \left\{ \frac{X_1X_2}{P_1P_2}, \ldots, \frac{X_nX_1}{P_nP_1} \right\} \geq 1.\]
[i]Proposed by Nairi Sedrakyan, Armenia[/i]
Let $a,b,c$ be positive real numbers such that $a+b+c=1$. Prove that $$\frac{a^2}{(b+c)^3}+\frac{b^2}{(c+a)^3}+\frac{c^2}{(a+b)^3}\geq \frac98$$
Let $x$ and $y$ are real numbers such that $x^2y^2+2yx^2+1=0.$ If $S=\frac{2}{x^2}+1+\frac{1}{x}+y(y+2+\frac{1}{x})$, find
(a)max$S$ and
(b) min$S$.
Let $M$ be the set of all positive integers that do not contain the digit $9$ (base $10$). If $x_1, \ldots , x_n$ are arbitrary but distinct elements in $M$, prove that
\[\sum_{j=1}^n \frac{1}{x_j} < 80 .\]
Given the inequalities:
$a)$ $\left(\frac{2a}{b+c}\right)^2+\left(\frac{2b}{a+c}\right)^2+\left(\frac{2c}{a+b}\right)^2\geq \frac{a}{c}+\frac{b}{a}+\frac{c}{b}$
$b)$ $\left(\frac{a+b}{c}\right)^2+\left(\frac{b+c}{a}\right)^2+\left(\frac{c+a}{b}\right)^2\geq \frac{a}{b}+\frac{b}{c}+\frac{c}{a}+9$
For each of them either prove that it holds for all positive real numbers $a$, $b$, $c$ or present a counterexample $(a,b,c)$ which doesn't satisfy the inequality.
Let $q$ be a positive integer which is not a perfect cube. Prove that there exists a positive constant $C$ such that for all natural numbers $n$, one has
$$\{ nq^{\frac{1}{3}} \} + \{ nq^{\frac{2}{3}} \} \geq Cn^{-\frac{1}{2}}$$
where $\{ x \}$ denotes the fractional part of $x$.