Found problems: 2529
Given positive real numbers $a,b,c,d$ such that
$ \frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}=6 \quad \text{and} \quad \frac{b}{a}+\frac{c}{b}+\frac{d}{c}+\frac{a}{d}=36.$
Prove the inequality
${{a}^{2}}+{{b}^{2}}+{{c}^{2}}+{{d}^{2}}>ab+ac+ad+bc+bd+cd.$
In $ABC$ with $\angle C = 60^{\circ}$, prove that
\[\frac{c}{a} + \frac{c}{b} \ge2.\]
Prove that the polynomial equation $x^{8}-x^{7}+x^{2}-x+15=0$ has no real solution.
Let $a$ and $b$ be positive real numbers such that $ab(a-b)=1$. Which of the followings can $a^2+b^2$ take?
$
\textbf{(A)}\ 1
\qquad\textbf{(B)}\ 2
\qquad\textbf{(C)}\ 2\sqrt 2
\qquad\textbf{(D)}\ \sqrt {11}
\qquad\textbf{(E)}\ \text{None of the preceding}
$
Let $x, y, z$ be positive real numbers. Prove that:
$\frac{x + 2y}{z + 2x + 3y}+\frac{y + 2z}{x + 2y + 3z}+\frac{z + 2x}{y + 2z + 3x} \le \frac{3}{2}$
Let $x, y,$ and $z$ be positive real numbers such that $xy + yz + zx = 27$. Prove that
$x + y + z \ge \sqrt{3xyz}$.
When does equality hold?
Let $a, b$ and $c$ be positive real numbers such that $abc = 1$. Prove the inequality
$\Big(ab + bc +\frac{1}{ca}\Big)\Big(bc + ca +\frac{1}{ab}\Big)\Big(ca + ab +\frac{1}{bc}\Big)\ge (1 + 2a)(1 + 2b)(1 + 2c)$.
We have two positive integers both less than $1000$. The arithmetic mean and the geometric mean of these numbers are consecutive odd integers. Find the maximum possible value of the difference of the two integers.
Let $ AB$ be the diameter of a given circle with radius $ 1$ unit, and let $ P$ be a given point on $ AB$. A line through $ P$ meets the circle at points $ C$ and $ D$, so a convex quadrilateral $ ABCD$ is formed. Find the maximum possible area of the quadrilateral.
Let $a, b$ and $k$ be positive integers with $k> 1$ such that
$lcm (a, b) + gcd (a, b) = k (a + b)$.
Prove that $a + b \geq 4k$
If $a,b,c$ are sides of a triangle, prove that \[ \frac{a}{c+a-b} + \frac{b}{a+b-c} + \frac{c}{b+c-a} \geq 3. \]
Given an integer $ m\geq 2$, and two real numbers $ a,b$ with $ a > 0$ and $ b\neq 0$. The sequence $ \{x_n\}$ is such that $ x_1 \equal{} b$ and $ x_{n \plus{} 1} \equal{} ax^{m}_{n} \plus{} b$, $ n \equal{} 1,2,...$. Prove that
(1)when $ b < 0$ and m is even, the sequence is bounded if and only if $ ab^{m \minus{} 1}\geq \minus{} 2$;
(2)when $ b < 0$ and m is odd, or when $ b > 0$ the sequence is bounded if and only if $ ab^{m \minus{} 1}\geq\frac {(m \minus{} 1)^{m \minus{} 1}}{m^m}$.
If $ a,b,c$ are three positive real numbers, prove that $ \frac {a^{2}\plus{}1}{b\plus{}c}\plus{}\frac {b^{2}\plus{}1}{c\plus{}a}\plus{}\frac {c^{2}\plus{}1}{a\plus{}b}\ge 3$
Show that
\[\left(a+2b+\dfrac{2}{a+1}\right)\left(b+2a+\dfrac{2}{b+1}\right)\geq 16\]
for all positive real numbers $a$ and $b$ such that $ab\geq 1$.
Let $ a,b,c$ be positive real numbers. Prove the inequality:
$ \frac {a^3}{b^2} \plus{} \frac {b^3}{c^2} \plus{} \frac {c^3}{a^2}\ge \frac {a^2}{b} \plus{} \frac {b^2}{c} \plus{} \frac {c^2}{a}$
Determine the smallest positive real number $\alpha$ such that there exists a sequence of positive real numbers $(a_n)$, $n \in \mathbb{N}$, with the property that for every $n \in \mathbb{N}$ it holds that:
\[
a_1 + \cdots + a_{n+1} < \alpha \cdot a_n.
\]
[i]Proposed by Pavle Martinović[/i]
A polyomino is a connected figure constructed by joining one or more unit squares edge-to-edge. Determine, with proof, the number of non-congruent polyominoes with no holes, perimeter $180,$ and area $2024.$
Let $a,b,c$ be positive real numbers such that $2a^2 +b^2=9c^2$.Prove that $\displaystyle \frac{2c}{a}+\frac cb \ge\sqrt 3$.
Let $ n$ be a positive integer and let $ a_1,a_2,\ldots,a_n$ be positive real numbers such that:
\[ \sum^n_{i \equal{} 1} a_i \equal{} \sum^n_{i \equal{} 1} \frac {1}{a_i^2}.
\]
Prove that for every $ i \equal{} 1,2,\ldots,n$ we can find $ i$ numbers with sum at least $ i$.
Prove that the following inequality holds for all positive real numbers $x,y,z$:
\[\dfrac{x^3}{y^2+z^2}+\dfrac{y^3}{z^2+x^2}+\dfrac{z^3}{x^2+y^2}\ge \dfrac{x+y+z}{2}.\]
For any two positive integers $n>m$ prove the following inequality:
$$[m,n]+[m+1,n+1]\geq \dfrac{2nm}{\sqrt{m-n}}$$
As always, $[x,y]$ means the least common multiply of $x,y$.
[I]Proposed by A. Golovanov[/i]
Solve in positive integers the equation $$m^{\frac{1}{n}}+n^{\frac{1}{m}}=2+\frac{2}{mn(m+n)^{\frac{1}{m}+\frac{1}{n}}}.$$
An infinite rectangular stripe of width $3$ cm is folded along a line. What is the minimum possible area of the region of overlapping?
The positive numbers $a$, $b$, and $c$ satisfy $abc = 1$. Show that:
$$
\frac{1}{a^2+a}+\frac{1}{b^2+b}+\frac{1}{c^2+c} \ge \frac{3}{2}.
$$
Let $a,b,c$ be positive reals satisfying $a+b+c = \sqrt[7]{a} + \sqrt[7]{b} + \sqrt[7]{c}$. Prove that $a^a b^b c^c \ge 1$.
[i]Proposed by Evan Chen[/i]