Found problems: 1671
Let $n$ be a nonzero natural number, and $x_1, x_2,..., x_n$ positive real numbers that $ \frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}= n$. Find the minimum value of the expression $x_1 +\frac{x_2^2}{2}++\frac{x_3^3}{3}+...++\frac{x_n^n}{n}$.
Let $n$ be a positive integer. For each $4n$-tuple of nonnegative real numbers $a_1,\ldots,a_{2n}$, $b_1,\ldots,b_{2n}$ that satisfy $\sum_{i=1}^{2n}a_i=\sum_{j=1}^{2n}b_j=n$, define the sets
\[A:=\left\{\sum_{j=1}^{2n}\frac{a_ib_j}{a_ib_j+1}:i\in\{1,\ldots,2n\} \textup{ s.t. }\sum_{j=1}^{2n}\frac{a_ib_j}{a_ib_j+1}\neq 0\right\},\]
\[B:=\left\{\sum_{i=1}^{2n}\frac{a_ib_j}{a_ib_j+1}:j\in\{1,\ldots,2n\} \textup{ s.t. }\sum_{i=1}^{2n}\frac{a_ib_j}{a_ib_j+1}\neq 0\right\}.\]
Let $m$ be the minimum element of $A\cup B$. Determine the maximum value of $m$ among those derived from all such $4n$-tuples $a_1,\ldots,a_{2n},b_1,\ldots,b_{2n}$.
[I]Proposed by usjl.[/i]
Let $n \ge 2$ be a fixed even integer. We consider polynomials of the form
\[P(x) = x^n + a_{n-1}x^{n-1} + \cdots + a_1x + 1\]
with real coefficients, having at least one real roots. Find the least possible value of $a^2_1 + a^2_2 + \cdots + a^2_{n-1}$.
Let $ABC$ a right isosceles triangle with hypotenuse $AB=\sqrt2$ . Determine the positions of the points $X,Y,Z$ on the sides $BC,CA,AB$ respectively so that the triangle $XYZ$ is isosceles, right, and with minimum area.
$ \lim_{n\to\infty } \sqrt[n]{\sum_{i=0}^n\binom{n}{i}^2} $
Let $P$, $Q$, and $R$ be the points where the incircle of a triangle $ABC$ touches the sides $AB$, $BC$, and $CA$, respectively.
Prove the inequality $\frac{BC} {PQ} + \frac{CA} {QR} + \frac{AB} {RP} \geq 6$.
Let $0 \le a$, $b \le 1$ be real numbers. Prove the following inequality:
\[\sqrt{a^3b^3}+ \sqrt{(1-a^2)(1-ab)(1-b^2)} \le 1.\]
[i](41th Austrian Mathematical Olympiad, regional competition, problem 1)[/i]
Find all functions $f: \mathbb{R} \mapsto \mathbb{R}$ such that $$f(x)f\left(yf(x)-1\right)=x^2f(y)-f(x),$$for all $x,y \in \mathbb{R}$.
Consider a variable point $P$ inside a given triangle $ABC$. Let $D$, $E$, $F$ be the feet of the perpendiculars from the point $P$ to the lines $BC$, $CA$, $AB$, respectively. Find all points $P$ which minimize the sum \[ {BC\over PD}+{CA\over PE}+{AB\over PF}. \]
Find all functions $f:\mathbb Z\rightarrow \mathbb Z$ such that, for all integers $a,b,c$ that satisfy $a+b+c=0$, the following equality holds:
\[f(a)^2+f(b)^2+f(c)^2=2f(a)f(b)+2f(b)f(c)+2f(c)f(a).\]
(Here $\mathbb{Z}$ denotes the set of integers.)
[i]Proposed by Liam Baker, South Africa[/i]
Let $a,b,c,d$ be positive real numbers such that $a^2+b^2+c^2+d^2=4$.
Prove that exist two of $a,b,c,d$ with sum less or equal to $2$.
Let $ a,b,c,d$ be real numbers, such that $ a^2\le 1, a^2 \plus{} b^2\le 5, a^2 \plus{} b^2 \plus{} c^2\le 14, a^2 \plus{} b^2 \plus{} c^2 \plus{} d^2\le 30$. Prove that $ a \plus{} b \plus{} c \plus{} d\le 10$.
Let $f : \left[ 0,1 \right] \to \mathbb R$ be an integrable function such that \[ \int_0^1 f(x) \, dx = \int_0^1 x f(x) \, dx = 1 . \] Prove that \[ \int_0^1 f^2 (x) \, dx \geq 4 . \]
[i]Ion Rasa[/i]
Let $ \lambda_1 \leq \lambda_2 \leq...$ be a positive sequence and let $ K$ be a constant such that \[ \sum_{k=1}^{n-1} \lambda^2_k < K \lambda^2_n \;(n=1,2,...).\] Prove that there exists a constant $ K'$ such that \[ \sum_{k=1}^{n-1} \lambda_k < K' \lambda_n \;(n=1,2,...).\]
[i]L. Leindler[/i]
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 \[ \sum^{16}_{k\equal{}1} r^{16}_k \equal{} n.\]
Find all real-valued functions $f$ defined on the set of all non-zero real numbers such that:
(i) $f(1)=1$,
(ii) $f\left(\frac{1}{x+y}\right)=f\left(\frac{1}{x}\right)+f\left(\frac{1}{y}\right)$ for all non-zero $x,y,x+y$,
(iii) $(x+y)\cdot f(x+y)=xy\cdot f(x)\cdot f(y)$ for all non-zero $x,y,x+y$.
Let $a$, $b$, $c$ be positive real numbers satisfying $a^2<bc$. Prove that $b^3+ac^2>ab(a+c)$.
Find, with proof, the point $P$ in the interior of an acute-angled triangle $ABC$ for which $BL^2+CM^2+AN^2$ is a minimum, where $L,M,N$ are the feet of the perpendiculars from $P$ to $BC,CA,AB$ respectively.
[i]Proposed by United Kingdom.[/i]
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$.
If $a$, $b$ ,$c$ are three positive real numbers such that $ab+bc+ca = 1$, prove that \[ \sqrt[3]{ \frac{1}{a} + 6b} + \sqrt[3]{\frac{1}{b} + 6c} + \sqrt[3]{\frac{1}{c} + 6a } \leq \frac{1}{abc}. \]
Let nonnegative real numbers $ a_{1},a_{2},a_{3},a_{4}$ satisfy $ a_{1} \plus{} a_{2} \plus{} a_{3} \plus{} a_{4} \equal{} 1.$ Prove that
$ max\{\sum_{1}^4{\sqrt {a_{i}^2 \plus{} a_{i}a_{i \minus{} 1} \plus{} a_{i \minus{} 1}^2 \plus{} a_{i \minus{} 1}a_{i \minus{} 2}}},\sum_{1}^4{\sqrt {a_{i}^2 \plus{} a_{i}a_{i \plus{} 1} \plus{} a_{i \plus{} 1}^2 \plus{} a_{i \plus{} 1}a_{i \plus{} 2}}}\}\ge 2.$
Where for all integers $ i, a_{i \plus{} 4} \equal{} a_{i}$ holds.
Let $a, b, c, d$ be non-negative real numbers such that \[\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}+\frac{1}{d+1}=3.\]
Prove that \[3(ab+bc+ca+ad+bd+cd)+\frac{4}{a+b+c+d}\leqslant 5.\][i]Vasile Cîrtoaje and Leonard Giugiuc[/i]
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 $p$ be an odd prime number such that $p\equiv 2\pmod{3}.$ Define a permutation $\pi$ of the residue classes modulo $p$ by $\pi(x)\equiv x^3\pmod{p}.$ Show that $\pi$ is an even permutation if and only if $p\equiv 3\pmod{4}.$
Let $ a$, $ b$, $ c$ be positive reals such that $ a^4 \plus{} b^4 \plus{} c^4 \equal{} 3$. Prove that $ \sum\frac1{4 \minus{} ab}\leq1$, where the $ \sum$ sign stands for cyclic summation.
[i]Alternative formulation:[/i] For any positive reals $ a$, $ b$, $ c$ satisfying $ a^4 \plus{} b^4 \plus{} c^4 \equal{} 3$, prove the inequality
$ \frac{1}{4\minus{}bc}\plus{}\frac{1}{4\minus{}ca}\plus{}\frac{1}{4\minus{}ab}\leq 1$.