Found problems: 1671
Let a_1, a_2, a_3,... be a sequence of positive numbers. If there exists a positive number M such that for n = 1,2,3,...,
$a^{2}_{1}+a^{2}_{2}+...+a^{2}_{n}< Ma^{2}_{n+1}$
then prove that there exist a positive number M' such that for every n = 1,2,3,...,
$a_{1}+a_{2}+...+a_{n}< M'a_{n+1}$
The numbers $a, b$ and $c$ are such that $a^2 + b^2 + c^2 = 1$. Prove that $$a^4 + b^4 + c^4 + 2(ab^2 + bc^2 + ca^2)^2\le 1. $$ At what $a, b$ and $c$ does inequality turn into equality?
Find all positive real numbers $p$ such that $\sqrt{a^2 + pb^2} +\sqrt{b^2 + pa^2} \ge a + b + (p - 1) \sqrt{ab}$ holds for any pair of positive real numbers $a, b$.
Find all functions $ f: \mathbb R\longrightarrow \mathbb R$ such that for each $ x,y\in\mathbb R$:
\[ f(xf(y)) \plus{} y \plus{} f(x) \equal{} f(x \plus{} f(y)) \plus{} yf(x)\]
Find all the continuous bounded functions $f: \mathbb R \to \mathbb R$ such that
\[(f(x))^2 -(f(y))^2 = f(x + y)f(x - y) \text{ for all } x, y \in \mathbb R.\]
Prove that for all all non-negative real numbers $a,b,c$ with $a^2+b^2+c^2=1$
\[\sqrt{a+b}+\sqrt{a+c}+\sqrt{b+c} \geq 5abc+2.\]
If $a, b, c, d$ are reals with $a \ge b \ge c \ge d \ge 0$ and $b(b-a)+c(c-b)+d(d-c) \le 2 - \frac{a^2}{2}$, find the minimum value of the expression
\begin{align*}\frac{1}{b+2006c-2006d}+\frac{1}{a+2006b-2006c-d} + \frac{1}{2007a-2006b-c+d} + \frac{1}{a-b+c+2006d}.\end{align*}
Let $a_1,a_2,\dots,a_m$ be a finite sequence of positive integers. Prove that there exist nonnegative integers $b,c,$ and $N$ such that $$\left\lfloor \sum_{i=1}^m \sqrt{n+a_i} \right\rfloor =\left\lfloor \sqrt{bn+c} \right\rfloor$$ holds for all integers $n>N.$
[i]Proposed by Carl Schildkraut[/i]
Triangle $ABC$ has $AB = 40$, $AC = 31$, and $\sin A = \tfrac15$. This triangle is inscribed in rectangle $AQRS$ with $B$ on $\overline{QR}$ and $C$ on $\overline{RS}$. Find the maximum possible area of $AQRS$.
Find all functions $f : \mathbb{Z} \rightarrow \mathbb{Z}$ such that for all integers $m,n$,
\[f(m - n + f(n)) = f(m) + f(n).\]
Determine all triples of positive real numbers $(x, y, z)$ such that
\[x+y+z=6,\]\[\frac 1x + \frac 1y + \frac 1z = 2 - \frac{4}{xyz}.\]
Let $(a,b,c)$ be a Pythagorean triple, i.e. a triplet of positive integers with $ a^2\plus{}b^2\equal{}c^2$.
$a)$ Prove that $\left(\frac{c}{a}\plus{}\frac{c}{b}\right)^2>8$.
$b)$ Prove that there are no integer $n$ and Pythagorean triple $(a,b,c)$ satisfying $\left(\frac{c}{a}\plus{}\frac{c}{b}\right)^2\equal{}n$.
Is there a sequence $ a_1,a_2,\ldots$ of positive reals satisfying simoultaneously the following inequalities for all positive integers $ n$:
a) $ a_1\plus{}a_2\plus{}\ldots\plus{}a_n\le n^2$
b) $ \frac1{a_1}\plus{}\frac1{a_2}\plus{}\ldots\plus{}\frac1{a_n}\le2008$?
Let $a_1,a_2,\ldots, a_m$ be a set of $m$ distinct positive even numbers and $b_1,b_2,\ldots,b_n$ be a set of $n$ distinct positive odd numbers such that
\[a_1+a_2+\cdots+a_m+b_1+b_2+\cdots+b_n=2019\]
Prove that
\[5m+12n\le 581.\]
Let $S$ be a set of integers (not necessarily positive) such that
(a) there exist $a,b \in S$ with $\gcd(a,b)=\gcd(a-2,b-2)=1$;
(b) if $x$ and $y$ are elements of $S$ (possibly equal), then $x^2-y$ also belongs to $S$.
Prove that $S$ is the set of all integers.
Let $a, b, c $ be positive real numbers such that $abc = \frac {2} {3}. $ Prove that:
$$\frac {ab}{a + b} + \frac {bc} {b + c} + \frac {ca} {c + a} \geqslant \frac {a+b+c} {a^3+b ^ 3 + c ^ 3}.$$
If a, b, and c are non-negative real numbers satisfying $a + b + c = 400$, fi nd the maximum possible value of $\sqrt{2a+b}+\sqrt{2b+c}+\sqrt{2c+a}$.
Let $a,b\in\mathbb{R}_+$ such that $a+b=1$. Find the minimum value of the following expression:
\[E(a,b)=3\sqrt{1+2a^2}+2\sqrt{40+9b^2}.\]
Let $f: [0,1] \to (0, \infty)$ be an integrable function such that $f(x)f(1-x) = 1$ for all $x\in [0,1]$. Prove that $\int_0^1f(x)dx \geq 1$.
Let $n \geq 3$ be an integer. Let $t_1$, $t_2$, ..., $t_n$ be positive real numbers such that \[n^2 + 1 > \left( t_1 + t_2 + \cdots + t_n \right) \left( \frac{1}{t_1} + \frac{1}{t_2} + \cdots + \frac{1}{t_n} \right).\] Show that $t_i$, $t_j$, $t_k$ are side lengths of a triangle for all $i$, $j$, $k$ with $1 \leq i < j < k \leq n$.
Let $ a$, $ b$, $ c$, $ d$ be positive real numbers such that $ abcd \equal{} 1$ and $ a \plus{} b \plus{} c \plus{} d > \dfrac{a}{b} \plus{} \dfrac{b}{c} \plus{} \dfrac{c}{d} \plus{} \dfrac{d}{a}$. Prove that
\[ a \plus{} b \plus{} c \plus{} d < \dfrac{b}{a} \plus{} \dfrac{c}{b} \plus{} \dfrac{d}{c} \plus{} \dfrac{a}{d}\]
[i]Proposed by Pavel Novotný, Slovakia[/i]
Draw a $2004 \times 2004$ array of points. What is the largest integer $n$ for which it is possible to draw a convex $n$-gon whose vertices are chosen from the points in the array?
Suppose that $a_0=1$ and that $a_{n+1}=a_n+e^{-a_n}$ for $n=0,1,2,\dots.$ Does $a_n-\log n$ have a finite limit as $n\to\infty?$ (Here $\log n=\log_en=\ln n.$)
Find the roots $ r_i \in \mathbb{R}$ of the polynomial \[ p(x) \equal{} x^n \plus{} n \cdot x^{n\minus{}1} \plus{} a_2 \cdot x^{n\minus{}2} \plus{} \ldots \plus{} a_n\] satisfying \[ \sum^{16}_{k\equal{}1} r^{16}_k \equal{} n.\]
Let $a,b,c\geq 0$ and $a^2+b^2+c^2\leq 1.$ Prove that$$\frac{a}{a^2+bc+1}+\frac{b}{b^2+ca+1}+\frac{c}{c^2+ab+1}+3abc<\sqrt 3$$